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ProblemsinQuantum Computing

byWilli-Hans SteebInternational School for Scientific ComputingatUniversity of Johannesburg, South Africa

Yorick HardyDepartment of Mathematical SciencesatUniversity of South Africa

updated: August 4, 2017

Preface

The purpose of this book is to supply a collection of problems in quantumcomputing.

Prescribed books for problems.

1) Problems and Solutions in Quantum Computing and Quantum Informa-tion (third edition)

by Willi-Hans Steeb and Yorick HardyWorld Scientific, Singapore, 2011ISBN-13 978-981-4366-32-8http://www.worldscibooks.com/physics/8249.html

2) Classical and Quantum Computing with C++ and Java Simulations

by Yorick Hardy and Willi-Hans SteebBirkhauser Verlag, Boston, 2002ISBN 376-436-610-0

3) Matrix Calculus and Kronecker Product

by Willi-Hans SteebWorld Scientific Publishing, Singapore 2010ISBN 978-981-4335-31-7http://www.worldscibooks.com/mathematics/8030.html

4) Problems and Solutions in Introductory and Advanced Matrix Calculus

by Willi-Hans SteebWorld Scientific Publishing, Singapore 2006ISBN 981 256 916 2http://www.worldscibooks.com/mathematics/6202.html

5) Continous Symmetries, Lie Algebras, Differential Equations and Com-puter Algebra, second edition

by Willi-Hans SteebWorld Scientific Publishing, Singapore 2007ISBN 981-256-916-2http://www.worldscibooks.com/physics/6515.html

v

http://www.worldscibooks.com/physics/8249.html

http://www.worldscibooks.com/mathematics/8030.html

http://www.worldscibooks.com/mathematics/6202.html

http://www.worldscibooks.com/physics/6515.html

The International School for Scientific Computing (ISSC) provides certifi-cate courses for this subject. Please contact the authors if you want to dothis course or other courses of the ISSC.

e-mail addresses of the author:

[email protected][email protected]

Home page of the author:

http://issc.uj.ac.za

vi

Contents

Preface v

Notation x

1 Qubits 1

2 Kronecker and Tensor Product 12

3 Matrix Properties 25

4 Density Operators 67

5 Partial Trace 87

6 Reversible Logic Gates 89

7 Unitary Transformations and Quantum Gates 95

8 Entropy 106

9 Measurement 110

10 Entanglement 113

11 Bell Inequality 130

12 Quantum Channels 132

13 Miscellaneous 138

Bibliography 150

Index 155

viii

x

Notation

:= is defined as∈ belongs to (a set)/∈ does not belong to (a set)∩ intersection of sets∪ union of sets∅ empty setN set of natural numbersZ set of integersQ set of rational numbersR set of real numbersR+ set of nonnegative real numbersC set of complex numbersRn n-dimensional Euclidean space

space of column vectors with n real componentsCn n-dimensional complex linear space

space of column vectors with n complex componentsH Hilbert spacei

√−1

<z real part of the complex number z=z imaginary part of the complex number z|z| modulus of complex number z

|x+ iy| = (x2 + y2)1/2, x, y ∈ RT ⊂ S subset T of set SS ∩ T the intersection of the sets S and TS ∪ T the union of the sets S and Tf(S) image of set S under mapping ff g composition of two mappings (f g)(x) = f(g(x))x column vector in CnxT transpose of x (row vector)0 zero (column) vector‖ . ‖ normx · y ≡ x∗y scalar product (inner product) in Cnx× y vector product in R3

A,B,C m× n matricesdet(A) determinant of a square matrix Atr(A) trace of a square matrix Arank(A) rank of matrix AAT transpose of matrix A

xi

A conjugate of matrix AA∗ conjugate transpose of matrix AA† conjugate transpose of matrix A

(notation used in physics)A−1 inverse of square matrix A (if it exists)In n× n unit matrixI unit operator0n n× n zero matrixAB matrix product of m× n matrix A

and n× p matrix BA •B Hadamard product (entry-wise product)

of m× n matrices A and B[A,B] := AB −BA commutator for square matrices A and B[A,B]+ := AB +BA anticommutator for square matrices A and BA⊗B Kronecker product of matrices A and BA⊕B Direct sum of matrices A and Bδjk Kronecker delta with δjk = 1 for j = k

and δjk = 0 for j 6= kλ eigenvalueε real parametert time variableH Hamilton operator

The Pauli spin matrices are used extensively in the book. They are givenby

σx :=(

0 11 0

), σy :=

(0 −ii 0

), σz :=

(1 00 −1

).

In some cases we will also use σ1, σ2 and σ3 to denote σx, σy and σz .

Chapter 1

Qubits

Problem 1. Let |0〉, |1〉 be the standard basis in the Hilbert space C2,i.e.

|0〉 =(

10

), |1〉 =

(01

).

Let (0 ≤ θ < π4 )

|Ψ+(θ)〉 := cos(θ)|0〉+ sin(θ)|1〉, |Ψ−(θ)〉 := cos(θ)|0〉 − sin(θ)|1〉.

(i) Find the scalar product 〈Ψ−(θ)|Ψ+(θ)〉. Discuss.(ii) Consider the states

|+〉 :=1√2

(|0〉+ |1〉), |−〉 :=1√2

(|0〉 − |1〉)

and the projection operators (projection matrices)

Π+ := |+〉〈+|, Π− := |−〉〈−|.

Find

〈Ψ+(θ)|Π+|Ψ+(θ)〉, 〈Ψ+(θ)|Π−|Ψ+(θ)〉, 〈Ψ−(θ)|Π+|Ψ−(θ)〉, 〈Ψ−(θ)|Π−|Ψ−(θ)〉

and the 2× 2 matrices Π+ + Π− and Π+Π−. Discuss.

Problem 2. (i) Consider the normalized vector in the Hilbert space C3

n =

sin(θ) cos(φ)sin(θ) sin(φ)

cos(θ)

.

1

2 Problems and Solutions

Show that the vector is normalized.(ii) Calculate the 2× 2 matrix

U(θ, φ) = n · σ ≡ n1σ1 + n2σ2 + n3σ3

where σ1, σ2, σ3 are the Pauli spin matrices.(iii) Is the matrix U(θ, φ) unitary? Find the trace and the determinant. Isthe matrix U(θ, φ) hermitian?(iv) Find the eigenvalues and normalized eigenvectors of U(θ, φ).

Problem 3. Consider the states

ψ1(φ) =(

cos(φ/2)sin(φ/2)

), ψ2(φ) =

(− sin(φ/2)cos(φ/2)

).

in the Hilbert space C2.(i) Show that these states can be generated from the standard basis usingthe exponential function and the Pauli matrix σ2, i.e. calculate

exp(−iφ

2σ2

)(10

), exp

(−iφ

2σ2

)(01

).

(ii) Find the states after the transformation φ→ φ+ 2π.

Problem 4. Let σ1, σ2, σ3 be the Pauli spin matrices and I2 the 2 × 2identity matrix. Find the eigenvalues and normalized eigenvectors of theHamilton operator

H = ε0I2 + ~ωσ3 + ∆1σ1 + ∆2σ2

where ε0 > 0. Are the normalized eigenvectors orthonormal to each other?

Problem 5. Let H be a 2×2 hermitian matrix. Consider the normalizedstate

|ψ〉 =(eiφ cos(θ)

sin(θ)

)in the Hilbert space C2. Assume that

〈ψ|H|ψ〉 = ~ω cos(φ) sin(2θ), 〈ψ|H2|ψ〉 = ~2ω2.

Reconstruct the hermitian matrix H from these three assumptions. Notethat

cos(θ) sin(θ) ≡ 12

sin(2θ), eiφ = cos(φ)+i sin(φ), e−iφ = cos(φ)−i sin(φ).

Qubits 3

Problem 6. (i) Consider the symmetric matrix over R

H =(h11 h12

h12 h22

)and the state

|ψ〉 =(

cos(θ)sin(θ)

).

Calculate the variance

VH(|ψ〉) = 〈ψ|H2|ψ〉 − (〈ψ|H|ψ〉)2.

(ii) Consider the Hadamard matrix

H =1√2

(1 11 −1

)and the state

|ψ〉 =(

cos(θ)sin(θ)

).

Calculate the variance

VH(ψ) := 〈ψ|H2|ψ〉 − (〈ψ|H|ψ〉)2

and discuss the dependence on θ.

Problem 7. Let A and B be n × n hermitian matrices. Let |ψ〉 be anormalized state in the Hilbert space Cn. Then we have the inequality

(∆A)(∆B) ≥ 12|〈[A, B]〉|

where∆A :=

√〈A2〉 − 〈A〉2, ∆B :=

√〈B2〉 − 〈B〉2

and〈A〉 := 〈ψ|A|ψ〉, 〈B〉 := 〈ψ|B|ψ〉.

Consider the hermitian spin- 12 matrices

s1 =12

(0 11 0

), s2 =

12

(0 −ii 0

), s3 =

12

(1 00 −1

).

Let A = s1 and B = s2. Find states |ψ〉 such that

(∆A)(∆B) =12|〈[A, B]〉|


i.e. the inequality given above should be an equality.

Problem 8. Given the two normalized states

|ψ〉 =1√2

(11

), |φ〉 =

1√2

(1−1

).

Find a unitary matrix U such that |ψ〉 = U |φ〉. Give the eigenvalues of U .

Problem 9. Let

A =3∑k=0

akσk, B =3∑`=0

b`σ`

where σ0 = I2 and ak, b` ∈ R with a3 6= 0 and b1 = a1b3/a3, b2 = a2b3/a3.Calculate the commutator [A,B].

Problem 10. Let σ1, σ2, σ3 be the Pauli spin matrices. Show that

cos(ωt)σ1 − sin(ωt)σ2 = eiωtσ+ + e−iωtσ−, e±ωtσ±eiωtσ3/2 = eiωtσ3/2σ±

where σ± := (σ1 ± iσ2)/2.

Problem 11. Consider the Pauli spin matrices σ1, σ2, σ3. Can one finda 2× 2 invertible matrix K with K = K−1 and

Kσ1K = σ1, Kσ2K = −σ2, Kσ3K = σ3 ?

Problem 12. Let σ1, σ2, σ3 be the Pauli spin matrices and α ∈ R.(i) Calculate the 2× 2 matrices

exp(−iασ1/2), exp(−iασ2/2), exp(−iασ3/2).

Are the matrices unitary?(ii) Let

|ψ〉 =(

10

).

Find the state exp(−iασ1/2)|ψ〉 and calculate

〈ψ| exp(−iασ1/2)|ψ〉 and |〈ψ| exp(−iασ1/2)|ψ〉|2.

Problem 13. Let |0〉, |1〉 be an orthonormal basis in a two-dimensionalHilbert space. Consider the Hamilton operator

H = −12

~ω(e−iφ|1〉〈0|+ eiφ|0〉〈1|).

Qubits 5

Find exp(−iHt/~).

Problem 14. Consider the Hamilton operator

H(λ) =(

~ω λa12

λa12 −~ω

)where a12 ∈ R. Let I2 be the 2×2 identity matrix and E a real parameter.Solve the system of equations

det(H(λ)− I2E) = 0d

dEdet(H(λ)− I2E) = 0

with respect to E and λ.


H =(

~ω ∆∆ −~ω

).

Consider the unitary matrix

U =(

cos(φ) −e−iθ sin(φ)eiθ sin(φ) cos(φ)

).

Can one find φ, θ such that U∗HU is a diagonal matrix?

Problem 16. Consider the Pauli spin matrices σ1, σ2 and σ3. Can onefind an α ∈ R such that

exp(iασ3)σ1 exp(−iασ3) = σ2 ?

Problem 17. Let σ1, σ2, σ3 be the Pauli spin matrices. Let α1, α2, α3 ∈C. Find the conditions on α1, α2, α3 such that

U = α1σ1 + α2σ2 + α3σ3

is a unitary matrix.

Problem 18. Consider the map f : C2 → R3 defined by

f :(

cos(θ)eiφ sin(θ)

)7→

sin(2θ) cos(φ)sin(2θ) sin(φ)

cos(2θ)

.


Are the vectors in C2 and R3 normalized? Consider the four normalizedvectors in C2

1√2

(11

),

1√2

(1−1

),

1√2

(1i

),

1√2

(1−i

).

Find the vectors in R3.

Problem 19. Let σ1, σ2 and σ3 be the Pauli spin matrices. Calculate

U(α, β, γ) = e−iασ3/2e−iβσ2/2e−iγσ3/2

where α, β, γ are the Euler angles with the range 0 ≤ α < 2π, 0 ≤ β ≤ πand 0 ≤ γ < 2π.

Problem 20. Let H0 and H1 be a pair of real symmetric n×n matrices,where H0 is a diagonal matrix. Let

H(ε) := H0 + εH1. (1)

When ε is real, H(ε) is diagonalizable with eigenvalues E1(ε), . . ., En(ε).The eigenvalues are given by the characteristic polynomial

P (E, ε) := det(H(ε)− EIn) = 0 (2)

where In is the n × n unit matrix. When ε is complex, the eigenvaluesmay be viewed as the n values of a single function E(ε) of ε, analytic on aRiemann surface with N sheets joined at branch point singularities in thecomplex plane. The exceptional points in the complex ε plane are definedby the solution of (2) together with

d

dEdet(H(ε)− EIn) = 0. (3)

(i) Consider the two-level system

H(ε) =(

0 00 1

)+ ε

(0 11 0

).

Find the exceptional points of H(ε).(ii) Let ε1 and ε2 be the two exceptional points. Find the eigenvalues andeigenvectors of the matrices H(ε1) and H(ε2). Discuss.

Problem 21. Study the eigenvalue problem for the matrix

σ3 + eiφσ1

Qubits 7

for φ ∈ [0, π/2].

Problem 22. (i) Let φ ∈ R. Is the matrix

A =(

0 e−iφ

eiφ 0

)hermitian, unitary?(ii) Find the rank of the matrix.(iii) Find the eigenvalues and eigenvectors of A.(iv) Let I2 be the 2× 2 unit matrix. Find the eigenvalues of A⊗ I2.

Problem 23. Let |φ1〉, |φ2〉 be two normalized vectors in the Hilbertspace R2. Assume that

〈φ1|φ2〉 =12.

Give a geometric interpretation of this equation.

Problem 24. Consider the vectors

|ψ1〉 =(ii

eiπ

), |ψ2〉 =

(ii

sin(i)

)in the Hilbert space C2. Are the vectors normalized? If not normalize thevectors.

Problem 25. Let H be an arbitrary Hilbert space. Let |ψ〉 and |φ〉 bearbitrary normalized states in H. Find all the solutions of the equation

〈φ|ψ〉〈ψ|φ〉 = i.

Problem 26. What is the condition on φ1, φ2, φ3 (all real) such that

V =1√2

(1 eiφ1

eiφ2 eiφ3

)is a unitary matrix?

Problem 27. Consider the matrices

A =(

1 01 0

), B =

(1 10 0

), C =

(0 10 1

), D =

(0 01 1

).

Find unitary matrices U1, U2, U3, U4 such that

U−11 AU1 = B, U−1

2 BU2 = C, U−13 CU3 = D, U−1

4 DU4 = A.


Problem 28. (i) Consider the normalized state

|ψ〉 =(e−iφ/2 cos(θ/2)eiφ/2 sin(θ/2)

).

in the Hilbert space C2. Let σ1, σ2, σ3 be the Pauli spin matrices. Calculate

nj := 〈ψ|σj |ψ〉, j = 1, 2, 3

Is the vector

n =

n1

n2

n3

in R3 normalized?(ii) Consider the Hamilton operator

H(t) = −µ~2

B(t) · σ ≡ −µ~2

(B1(t)σ1 +B2(t)σ2 +B3(t)σ3)

where B(t) is a time-dependent homogeneous magnetic field. Show thatthe Schrodinger equation

i~d

dt|ψ(t)〉 = H(t)|ψ(t)〉

can be written asd

dtn(t) = −µB(t)× n

where × denotes the vector product.

Problem 29. Find the square roots of the Pauli spin matrices

σ0 =(

1 00 1

), σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)i.e. find the matrices Rj such that R2

j = σj (j = 0, 1, 2, 3).

Problem 30. Consider a d-dimensional Hilbert space with two orthonor-mal bases

|b11〉, |b12〉, . . . |b1d〉 ∈ B1

|b21〉, |b22〉, . . . |b2d〉 ∈ B2.

The two bases are said to be mutually unbiased bases if

|〈b2j |b1k〉| =1√d

Qubits 9

for all j, k = 1, . . . , d and 〈 | 〉 denotes the scalar product in the Hilbertspace. Consider the Hilbert space M2(C) of 2 × 2 matrices over C, wherethe scalar product is defined as

〈A|B〉 = tr(AB∗), A,B ∈M2(C)

Thus d = dim(M2(C)) = 4. The standard basis in this Hilbert space isgiven by

E11 =(

1 00 0

), E12 =

(0 10 0

), E21 =

(0 01 0

), E22 =

(0 00 1

).

Let UH be the Hadamard matrix

UH =1√2

(1 11 −1

)U∗H = UH .

(i) Show that the matrices Ejk (j, k = 1, 2)

Ejk = UHEjkU∗H , j, k = 1, 2

and the standard basis form mutually unbiased bases.(ii) Apply the vec-operator to the matrices Ejk and Ejk (j, k = 1, 2) to findmutually unbiased bases in the Hilbert space C4.

Problem 31. Let σ1, σ2, σ3 be the Pauli spin matrices. Find all 2 × 2matrices A such that

[σ1, A] = [σ2, A] = [σ3, A] = 02.

Problem 32. Let φ1, φ2, φ3, φ4 ∈ R. The 2 × 2 matrix U = (v1 v2)contains the two column vectors

v1 =1√2

(eiφ1

eiφ2

), v2 =

1√2

(eiφ3

eiφ4

).

Find the conditions on φ1, φ2, φ3, φ4 such that

〈v1|v2〉 = 0.

Is the matrix U unitary if this condition is satisfied.

Problem 33. (i) Find the norms of the vectors in the Hilbert space C2

|ψ〉 =(eiα

e−iα

), |φ〉 =

(sin(i)cos(i)

)


where α ∈ R.(ii) Normalize the vectors |ψ〉 and |φ〉.(iii) After normalizing the vectors calculate the probability

p(α) = |〈ψ(α)|φ〉|2.

Discuss p as a function of α.

Problem 34. Let α ∈ R. Consider the vector in C2

v =(

cosh(α)sinh(α)

).

Normalize the vector and then study the cases α → +∞ and α → −∞.Can one find a non-zero (column) vector u in C2 such that

u∗v = 0 ?

Problem 35. Is the state

|ψ〉 =(

cos(θ/2)eiφ/2

sin(θ/2)e−iφ/2

)normalized? Find a normalized vector which is orthogonal to this vector.If so calculate the density matrix ρ = |ψ〉〈ψ|.

Problem 36. Let σ1, σ2, σ3 be the Pauli spin matrices.(i) Consider the normalized state |ψ〉 in the Hilbert space C2

|ψ〉 =(

cos(θ/2)eiφ sin(θ/2)

)and the Pauli spin matrices σ1, σ2, σ3. Find

〈ψ|σ1|ψ〉, 〈ψ|σ2|ψ〉, 〈ψ|σ3|ψ〉.

(ii) Consider the normalized state in C2

ψ =(eiφ cos(θ)

sin(θ)

)⇒ ψ∗ = ( e−iφ cos(θ) sin(θ) ) .

Find the vector v = (v1 v2 v3)T in R3 with

v1 = ψ∗σ1ψ, v2 = ψ∗σ2ψ, v3 = ψ∗σ3ψ.

Is the vector v normalized?

Qubits 11

Problem 37. Let φ, θ ∈ R and

v1 =1√2

( √1 + sin(θ)eiφ/2

−√

1− sin(θ)e−iφ/2

)be the eigenvector of a 2× 2 matrix with eigenvalue λ1 = +1 and

v2 =1√2

( √1− sin(θ)eiφ/2√

1 + sin(θ)e−iφ/2

)the second eigenvector with eigenvalue λ2 = −1. Find v∗1v

∗1, v∗2v2 and

v∗1v2. Discuss. Findλ1v1v∗1 + λ2v2v∗2.

Discuss.

Chapter 2

Kronecker and TensorProduct

Problem 1. Let ε := e2πi/3 ≡ (−1 + i√

3)/2. Consider the eight statesin C8

|ψ1〉 =(

10

)⊗(

10

)⊗(

10

), |ψ2〉 =

(01

)⊗(

01

)⊗(

01

)

|ψ3〉 =1√3

((10

)⊗(

10

)⊗(

01

)+(

10

)⊗(

01

)⊗(

10

)+(

01

)⊗(

10

)⊗(

10

))|ψ4〉 =

1√3

((01

)⊗(

01

)⊗(

10

)+(

01

)⊗(

10

)⊗(

01

)+(

10

)⊗(

01

)⊗(

01

))|ψ5〉 =

1√3

((10

)⊗(

10

)⊗(

01

)+ ε

(10

)⊗(

01

)⊗(

10

)+ ε

(01

)⊗(

10

)⊗(

10

))|ψ6〉 =

1√3

((10

)⊗(

10

)⊗(

01

)+ ε

(10

)⊗(

01

)⊗(

10

)+ ε

(01

)⊗(

10

)⊗(

10

))|ψ7〉 =

1√3

((01

)⊗(

01

)⊗(

10

)+ ε

(01

)⊗(

10

)⊗(

01

)+ ε

(10

)⊗(

01

)⊗(

01

))|ψ8〉 =

1√3

((01

)⊗(

01

)⊗(

10

)+ ε

(01

)⊗(

10

)⊗(

01

)+ ε

(10

)⊗(

01

)⊗(

01

))(i) Calculate the scalar product 〈ψj |ψk〉 for j, k = 1, 2, . . . , 8.(ii) Which of the vectors are entangled?

12

Kronecker and Tensor Product 13

Problem 2. (i) Can we find 2 × 2 matrices A and B with detA =a11a22 − a12a21 6= 0, detB = b11b22 − b12b21 6= 0 (i.e. we assume that Aand B are invertible) such that

1√2

1001

= (A⊗B)1√2

0110

?

On the left-hand side we have the Bell state |Φ+〉 and on the right-handside we have the Bell state |Ψ+〉. Since A and B are invertible we find thatA⊗B is also invertible with (A⊗B)−1 = A−1 ⊗B−1.(ii) Can we also find 2 × 2 matrices A, B such that det(A) = det(B) = 1,i.e. A,B ∈ SL(2,R)?

Problem 3. Can we find 2 × 2 matrices A, B, C with det(A) = 1,det(B) = 1 and det(C) = 1 such that

1√3

01101000

= (A⊗B ⊗ C)

1√2

10000001

?

On the left-hand side we have the W-state and on the right-hand side wehave the GHZ-state.

Problem 4. Let A, B be n× n hermitian matrices over C. Let K be ann×n hermitian matrix over C and H = ~ωK be a Hamilton operator, where~ is the Planck constant and ω the frequency. The Heisenberg equation ofmotion for the operators A and B are given by

i~dA

dt= [A,H](t), i~

dB

dt= [B,H](t).

The solutions can be given as

A(t) = eitH/~Ae−itH/~, B(t) = eitH/~Be−itH/~.

(i) Find the time evolution of A⊗B, B ⊗A, A⊗A and B ⊗B.(ii) Assume that [A,H] = 2i~ωB and [B,H] = −2i~ωA. Simplify theHeisenberg equation of motion with these conditions.


Problem 5. Let A be an m×m hermitian matrix and let B be an n×nhermitian matrix. Then A⊗B, A⊗In, Im⊗B are also hermitian matrices,where Im is the m×m identity matrix. Let ε1, ε2 and ε3 be real parameters.Consider the Hamilton operator

H = ~ω(ε1A⊗B + ε2A⊗ In + ε3Im ⊗B).

The partition function Z(β) is given by Z(β) = tr(exp(−βH)), where His the (hermitian) Hamilton operator and tr denotes the trace. From thepartition function we obtain the Helmholtz free energy, entropy and specificheat.(i) Calculate Z(β) for the Hamilton operator given above.(ii) Consider the special case that n = m = 2 and A, B are any of the Paulispin matrices σ1, σ2, σ3.

Problem 6. Let A, B be n× n matrices over C. Is

tr(eA ⊗ eB) = tr(eA⊗B) ?

Prove or disprove.

Problem 7. (i) Let A, B be n× n matrices and In be the n× n identitymatrix. Show that

(A⊗ In)(In ⊗B)eA⊗In+In⊗B = (AeA)⊗ (BeB).

(ii) Let λ be an eigenvalue of A and µ be an eigenvalue of B. Provide aneigenvalue of (AeA)⊗ (BeB).

Problem 8. (i) Let A be an invertible n × n matrix. Find the inversematrix of

(A−1 ⊗ In)(In ⊗A).

(ii) Let B be an invertible n× n matrix. Calculate

(A−1 ⊗ In)(In ⊗A)(B−1 ⊗ In)(In ⊗B).

Problem 9. The two-qubit Pauli group P2 can be generated as

P2 = 〈σ1 ⊗ σ1, σ3 ⊗ σ3, σ1 ⊗ σ2, σ2 ⊗ σ3, σ3 ⊗ σ1〉.

It is of order 64. Generate the element i(I2 ⊗ I2).


Problem 10. Consider the hermitian matrices of the three dipole oper-ators

L1 =1√2

0 1 01 0 10 1 0

, L2 =1√2

0 −i 0i 0 −i0 i 0

, L3 =

1 0 00 0 00 0 −1

and the hermitian matrices of five quadrupole operators

U1 =

0 0 10 0 01 0 0

, U2 =

0 0 −i0 0 0i 0 0

,

V1 =1√2

0 1 01 0 −10 −1 0

, V2 =1√2

0 −i 0i 0 i0 −i 0

, Q0 =1√3

1 0 00 −2 00 0 1

.

Multiplying these eight hermitian matrices by i we obtain a basis for thesemi-simple Lie algebra su(3). Consider the Hamilton operator

H = κ0Q0 ⊗Q0 + κ1(V1 ⊗ V1 + V2 ⊗ V2) + κ2(U1 ⊗ U1 + U2 ⊗ U2).

Find the eigenvalues and eigenvectors of H.

Problem 11. Consider the Pauli spin matrices σ3, σ1, σ2. The eigenval-ues are given by +1 and −1 with the corresponding normalized eigenvectors(

10

),

(01

),

1√2

(11

),

1√2

(1−1

),

1√2

(i1

),

1√2

(−i1

).

Consider the three 4× 4 matrices

σ1 ⊗ σ1, σ2 ⊗ σ2, σ3 ⊗ σ3.

(i) Find the eigenvalues.(ii) Show that the eigenvectors can be given as product states (unentangledstates), but also as entangled states (i.e. they cannot be written as productstates). Explain.

Problem 12. (i) Consider the two 4× 4 matrices σ1 ⊗ σ3, σ3 ⊗ σ1. Findthe eigenvalues.(ii) Show that the eigenvectors can be given as product states (unentangledstates), but also as entangled states (i.e. they cannot be written as productstates). Explain.

Problem 13. Consider the Pauli spin matrix σ2. Find the eigenvaluesand eigenvectors for σ2 and σ2⊗σ2. For σ2⊗σ2 show that we find two sets


of entangled states for the eigenvectors and set of unentangled eigenvectors(product states).

Problem 14. Find the eigenvalues and eigenvectors of the Hamiltonoperator

H = ~ω1σ3 ⊗ I2 + ~ω2I2 ⊗ σ3 + εσ2 ⊗ σ2.

Problem 15. (i) Find the eigenvalues and eigenvectors of

σ1 ⊗ σ2 ⊗ σ3.

Can one find entangled eigenvectors?(ii) Find the eigenvalues and eigenvectors of the Hamilton operator

H = ε1(σ1⊗ I2⊗ I2) + ε2(I2⊗ σ2⊗ I2) + ε3(I2⊗ I2⊗ σ3) + γ(σ1⊗ σ2⊗ σ3)

where ε1, ε2, ε3, γ ∈ R.

Problem 16. (i) Let U1, U2 be unitary 2 × 2 matrices and Π1, Π2 be2× 2 projection matrices with Π1Π2 = 0 and Π1 + Π2 = I2. Show that

U1 ⊗Π1 + U2 ⊗Π2

is unitary.(ii) Let U1 = σ1, U2 = σ3 and

Π1 =12

(1 11 1

), Π2 =

12

(1 −1−1 1

).

Find the normalized state

(U1 ⊗Π1 + U2 ⊗Π2)((

10

)⊗(

01

)).

Show that this state is entangled, i.e. it can not be written as a productstate.

Problem 17. Consider the n × n unitary matrices U1, . . . , Un andthe n × n projection matrices Π1, . . . , Πn such that ΠjΠk = δjkIn andΠ1 + · · ·+ Πn = In. Show that the n2 × n2 matrix

n∑j=1

(Uj ⊗Πj)

is unitary.


Problem 18. (i) Let A be an n × n matrix over C and Π be an m ×mprojection matrix. Let z ∈ C. Calculate

exp(z(A⊗Π)).

(ii) Let A1, A2 be n× n matrices over C. Let Π1, Π2 be m×m projectionmatrices with Π1Π2 = 0. Calculate

exp(z(A1 ⊗Π1 +A2 ⊗Π2)).

(iii) Use the result from (ii) to find the unitary matrix

U(t) = exp(−iHt/~)

where H = ~ω(A1 ⊗ Π1 + A2 ⊗ Π2) and we assume that A1 and A2 arehermitian matrices.(iv) Apply the result of (iii) to

A1 = σ1, Π1 =12

(1 11 1

), A2 = σ3, Π2 =

12

(1 −1−1 1

).

Problem 19. Every 4× 4 unitary matrix U can be written as

U = (U1 ⊗ U2) exp(i(ασ1 ⊗ σ1 + βσ2 ⊗ σ2 + γσ3 ⊗ σ3))(U3 ⊗ U4)

where Uj ∈ U(2) (j = 1, 2, 3, 4) and α, β, γ ∈ R. Calculate

exp(i(ασ1 ⊗ σ1 + βσ2 ⊗ σ2 + γσ3 ⊗ σ3)).

Problem 20. Consider the Hilbert space C16 and the normalized state

|ψ〉 =1√2

(| ↑〉 ⊗ | ↑〉 ⊗ | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉 ⊗ | ↓〉 ⊗ | ↓〉

where

| ↑〉 =(

10

), | ↓〉 =

(01

).

Give a computer algebra implementation that calculates the 256 expecta-tion values

Tjk`m = 〈ψ|σj ⊗ σk ⊗ σ` ⊗ σn|ψ〉, j, k, `,m = 0, 1, 2, 3

where σ0, σ1, σ2, σ3 are the Pauli spin matrices with σ0 = I2 (2×2) identitymatrix.


Problem 21. Consider the unitary matrices

H =1√2

(1 11 −1

), A =

1√2

(1 ii 1

),

B =1√2

(1 1i −i

), C =

12

(1− i 1 + i1− i −1− i

)and

R =

1 0 0 00 0 1 00 1 0 00 0 0 −1

.

Find (B ⊗ C)(R(I2 ⊗A)R)(H ⊗H).

Problem 22. Consider the spin matrix for spin- 12

s1 =12σ1 =

12

(0 11 0

)with the eigenvalues 1/2 and −1/2 and the corresponding normalized eigen-vectors

e1/2 =1√2

(11

), e−1/2 =

1√2

(1−1

).

Do the four vectors

1√2

(e1/2 ⊗ e1/2 + e−1/2 ⊗ e−1/2),1√2

(e1/2 ⊗ e1/2 − e−1/2 ⊗ e−1/2),

1√2

(e1/2 ⊗ e−1/2 + e−1/2 ⊗ e−1/2),1√2

(e1/2 ⊗ e−1/2 − e−1/2 ⊗ e−1/2),

form a basis in C4? Prove or disprove.

Problem 23. Let N ≥ 1. Consider the Hilbert space C2N

. The (N + 1)Dicke states are defined by∣∣∣∣∣∣∣

N

2, `− N

2〉 :=

1√NC`

(|0〉 ⊗ · · · ⊗ |0〉︸︷︷︸`

⊗ |1〉 ⊗ · · · ⊗ |1〉︸︷︷︸N−`

+permutations)

where ` = 0, 1, . . . , N and NC` = N !/(`!(N − `)!). Write down the Dickestates for N = 2 and N = 3. Which of the states are entangled?


Problem 24. Consider the 2× 2 permutation matrices

P1 = I2 =(

1 00 1

), P2 =

(0 11 0

).

(i) Show that

Π1 =12

(P1 + P2), Π2 =12

(P1 − P2)

are projection matrices. Find Π1Π2. Discuss.(ii) Show that

Π1 =12

(P1 ⊗ P1 + P2 ⊗ P2), Π2 =12

(P1 ⊗ P1 − P2 ⊗ P2)

are projection matrices. Find Π1Π2. Discuss.

Problem 25. Consider the six 3× 3 permutation matrices

P1 = I3 =

1 0 00 1 00 0 1

, P2 =

1 0 00 0 10 1 0

, P3 =

0 1 01 0 00 0 1

,

P4 =

0 1 00 0 11 0 0

, P5 =

0 0 11 0 00 1 0

, P6 =

0 0 10 1 01 0 0

with the signatures of the permutation P1 → +1, P2 → −1, P3 → −1,P4 → +1, P5 → +1, P6 → −1.(i) Is

Π1 =16

(P1 + P2 + P3 + P4 + P5 + P6)

a projection matrix?(ii) Is

Π2 =16

(P1 − P2 − P3 + P4 + P5 − P6)

a projection matrix? Calculate Π1Π2. Discuss.(iii) Is

Π1 =16

(P1 ⊗ P1 + P2 ⊗ P2 + P3 ⊗ P3 + P4 ⊗ P4 + P5 ⊗ P5 + P6 ⊗ P6)

a projection matrix?(iv) Is

Π2 =16

(P1 ⊗ P1 − P2 ⊗ P2 − P3 ⊗ P3 + P4 ⊗ P4 + P5 ⊗ P5 − P6 ⊗ P6)


a projection matrix? Find Π1Π2. Discuss.

Problem 26. Consider the Hilbert space C9 and the three normalizedstates

|ψ12〉 =1√2

100

⊗ 0

10

− 0

10

⊗ 1

00

|ψ23〉 =

1√2

010

⊗ 0

01

− 0

01

⊗ 0

10

|ψ31〉 =

1√2

001

⊗ 1

00

− 1

00

⊗ 0

01

.

(i) Are the states entangled?(ii) Find the density matrices.(iii) Form a mixed state from the three density matrices.

Problem 27. Consider the two Hilbert spaces H1 = H2 = Cd and theproduct Hilbert space H = H1 ⊗H2. A state |ψ〉 ∈ H is called maximallyentangled if

trH1(|ψ〉〈ψ|) = trH2(|ψ〉〈ψ|) =1d.

Apply this definition to the Bell states in H = C4, i.e. d = 2

|ψ1〉 =1√2

1001

, |ψ2〉 =1√2

100−1

,

|ψ3〉 =1√2

0110

, |ψ4〉 =1√2

01−10

.

Problem 28. (i) Let

|1〉 =(

10

), |2〉 =

(01

)be the standard basis in C2. Calculate the 4× 4 matrix

P :=2∑j=1

2∑k=1

|j〉〈k| ⊗ |k〉〈j|.


What type of matrix is this?(ii) Calculate P 2. Discuss.(iii) Let

|1〉 =1√2

(11

), |2〉 =

1√2

(1−1

)be the Hadamard basis in C2. Calculate the 4× 4 matrix

Q :=2∑j=1

2∑k=1

|j〉〈k| ⊗ |k〉〈j|.

What type of matrix is this?(iv) Calculate Q2. Discuss.

Problem 29. Can the normalized state

1√2

( 1 1 0 0 0 0 0 0 )T

in the Hilbert space C8 be written as a product state of three normalizedvectors in C2?

Problem 30. (i) Let σ1, σ2, σ3 be the Pauli spin matrices. Find thecommutators and anticommutators

[σ1, σ2], [σ2, σ3], [σ3, σ1], [σ1, σ2]+, [σ2, σ3]+, [σ3, σ1]+.

(ii) Consider the 4 × 4 matrices σ1 ⊗ σ2, σ2 ⊗ σ3, σ3 ⊗ σ1. Find the com-mutators and anticommutators

[σ1 ⊗ σ2, σ2 ⊗ σ3], [σ2 ⊗ σ3 ⊗ σ3 ⊗ σ1], [σ3 ⊗ σ1, σ1 ⊗ σ2]

[σ1 ⊗ σ2, σ2 ⊗ σ3]+, [σ2 ⊗ σ3 ⊗ σ3 ⊗ σ1]+, [σ3 ⊗ σ1, σ1 ⊗ σ2]+

(iii) Consider the 8× 8 matrices σ1 ⊗ σ2 ⊗ σ3, σ3 ⊗ σ1 ⊗ σ2, σ2 ⊗ σ3 ⊗ σ1.Find the commutators and anticommutators

[σ1⊗σ2⊗σ3, σ3⊗σ1⊗σ2], [σ3⊗σ1⊗σ2, σ2⊗σ3⊗σ1], [σ2⊗σ3⊗σ1, σ1⊗σ2⊗σ3]

[σ1⊗σ2⊗σ3, σ3⊗σ1⊗σ2]+, [σ3⊗σ1⊗σ2, σ2⊗σ3⊗σ1]+, [σ2⊗σ3⊗σ1, σ1⊗σ2⊗σ3]+.

Problem 31. Let σ0, σ1, σ2, σ3 be the Pauli spin matrices, where σ0 = I2is the 2× 2 unit matrix. Is

P =12

3∑j=0

(σj ⊗ σj)


a permutation matrix?

Problem 32. (i) Let σ0, σ1, σ2, σ3 be the Pauli spin matrices, whereσ0 = I2 is the 2× 2 unit matrix. Let

v =

v1v2v3

be a vector in R3 with ‖v‖ ≤ 1. Show that

ρv =12

(σ0 + v1σ1 + v2σ2 + v3σ3)

is a density matrix.(ii) Is

ρ =14

(σ0 ⊗ σ0 +3∑j=1

vjσj ⊗ σj)

a density matrix?(iii) Is

ρ =123

(σ0 ⊗ σ0 ⊗ σ0 +3∑j=1

vjσj ⊗ σj ⊗ σj)

a density matrix? Extend the result to n Kronecker products.

Problem 33. Consider the invertible matrix

U =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

.

Can the matrix be written as the Kronecker product of two 2×2 matrices?

Problem 34. Are the two state in C9

|ψ1〉 =1√6

100

⊗ 1

00

+

010

⊗ 0

10

− 2

001

⊗ 0

01

|ψ2〉 = − 1√3

100

⊗ 1

00

+

010

⊗ 0

10

+

001

⊗ 0

01

orthogonal to each other?


Problem 35. Let σ1, σ3 be the Pauli spin matrices. Find the 4 × 4permutation matrix P such that

P (σ1 ⊗ σ3)P−1 = σ3 ⊗ σ1.

Problem 36. Consider the two normalized states

|ψ〉 =1√2

1001

, |φ〉 =1√2

(eiα cos(β)

sin(β)

)⊗ 1√

2

(eiδ cos(γ)

sin(γ)

)

with α, β, γ, δ ∈ [0, 2π). Find

maxα,β,γ,δ|〈ψ|φ〉|2.

Problem 37. Let|0〉, |1〉, . . . , |n〉

be an orthonormal basis in Cn+1. Are the states

|ψ0〉=1√2|0〉 ⊗ |0〉+

1√2n

n∑j=1

|j〉 ⊗ |j〉

|ψ1〉=1√2|0〉 ⊗ |0〉 − 1√

2n

n∑j=1

|j〉 ⊗ |j〉

normalized? Are the state orthogonal to each other? Is

ρ = (|ψ0〉〈ψ0|)⊗ (|ψ1〉〈ψ1|)

a density matrix?

Problem 38. Let σ1, σ2, σ3 be the Pauli spin matrices. Show that the4× 4 matrix

R =(

1√2

(1 11 −1

)⊗ 1√

2

(1 −i1 i

))(σ1⊗σ2)

(1√2

(1 11 −1

)⊗ 1√

2

(1 1i −i

))can be written as direct sum of two 2× 2 matrices. Discuss.

Problem 39. Let σ2 be the second Pauli spin matrix. Then

σ2 ⊗ σ2 =(

0 −ii 0

)⊗(

0 −ii 0

)=

0 0 0 −10 0 1 00 1 0 0−1 0 0 0

.


Find the normalized state (γ ∈ R)

eiγσ2⊗σ2

1000

≡ eiγσ2⊗σ2

((10

)⊗(

10

)).

Is the state entangled? Discuss.

Problem 40. Let ω = exp(iπ). Consider the 4× 4 matrices

S1 =(

0 11 0

)⊗ I2, S2 = I2 ⊗

(1 00 ω

)(i) Show that S2

1 = I4, S22 = I4 and S1S2S

−11 S−1

2 = ωI4.(ii) Find the commutator [S1, S2].

Problem 41. Let γ, αj , βj ∈ R. Can any vector in C4 be written as

(U1(α)⊗ U2(β))

cos(γ/2)

00

sin(γ/2)

?

Here U1(α), U2(β) are the unitary matrices

U1(α) = exp(iα3σ3/2) exp(iα1σ2/2) exp(iα2σ3/2)

U2(β) = exp(iβ3σ3/2) exp(iβ1σ2/2) exp(iβ2σ3/2)

and σ1, σ2, σ3 are the Pauli spin matrices.

Chapter 3

Matrix Properties

Problem 1. The vectors

v1 =1√2

101

, v2 =

010

, v3 =1√2

10−1

form an orthonormal basis in the Hilbert space C3. Find the unitary ma-trices U12, U23, U31 such that

U12v1 = v2, U23v2 = v3, U31v3 = v1.

Then calculate U31U23U12 and the matrix

V = λ1v1v∗1 + λ2v2v∗2 + λ3v3v∗3

where the complex numbers λ1, λ2, λ3 satisfy λ1λ1 = 1, λ2λ2 = 1, λ3λ3 = 1.Is the matrix unitary?

Problem 2. Can the unitary matrix (permutation matrix)

U =

1 0 0 00 1 0 00 0 0 10 0 1 0

be written as the Kronecker product of two 2×2 matrices, i.e. U = A⊗B?

25


Problem 3. Let A, B, C be n×n matrices. Let In be the n×n identitymatrix.(i) What can be said about the eigenvalues and eigenvectors of

A⊗ In ⊗ In + In ⊗B ⊗ In + In ⊗ In ⊗ C

if we know the eigenvalues and eigenvectors of A, B, C?(ii) Is

eA⊗In⊗In+In⊗B⊗In+In⊗In⊗C = eA ⊗ eB ⊗ eC ?

Problem 4. Let σ1, σ3 be the Pauli spin matrices. Calculate (θ ∈ R)

R(θ) = exp(−i(θ/2)(σ1 + σ3)/√

2)

Is the matrix R(θ) unitary?

Problem 5. Let σ1, σ2, σ3 be the Pauli spin matrices. Does the set of4× 4 matrices

I2 ⊗ I2, σ1 ⊗ σ1, −σ2 ⊗ σ2, σ3 ⊗ σ3

form a group under matrix multiplication?

Problem 6. The spin matrices for spin- 32 particles are given by

J1 =~2

0√

3 0 0√3 0 2 0

0 2 0√

30 0

√3 0

J2 =~2

0 −i

√3 0 0

i√

3 0 −2i 00 2i 0 −i

√3

0 0 i√

3 0

J3 =~2

3 0 0 00 1 0 00 0 −1 00 0 0 −3

.

(i) Show that the matrices are hermitian.(ii) Find the eigenvalues and eigenvectors of these matrices.(iii) Calculate the commutation relations, i.e. [J1, J2], [J2, J3], [J3, J1].(iv) Are the matrices unitary?

Matrix Properties 27

Problem 7. Two orthonormal bases in an n-dimensional complex Hilbertspace

|uj〉 : j = 1, 2, . . . , n , |vj〉 : j = 1, 2, . . . , n

are called mutually unbiased if inner products (scalar products) between allpossible pairs of vectors taken from distinct bases have the same magnitude1/√n, i.e.

|〈uj |vk〉| =1√n

for all j, k ∈ 1, 2, . . . , n .

(i) Find such bases for the Hilbert space C2. Start of with the standardbasis

u1 =(

10

), u2 =

(01

).

(ii) Find such bases for the Hilbert space C3. Start of with the standardbasis

u1 =

100

, u2 =

010

, u3 =

001

.

(iii) Find such bases for the Hilbert space C4 using the result from C2 andthe Kronecker product.

Problem 8. (i) Let A, B be n × n matrices over C such that A2 = Inand B2 = In. Furthermore assume that

[A,B]+ ≡ AB +BA = 0n

i.e. the anticommutator vanishes. Let α, β ∈ C. Calculate eαA+βB using

eαA+βB =∞∑j=0

(αA+ βB)j

j!.

(ii) Consider the case that n = 2 and

α = −iωt, A = σ3 =(

1 00 −1

)

β = −i∆t/~, B = σ1 =(

0 11 0

).

(iii) Consider the case that n = 8 and

α = −iωt, A = σ3 ⊗ σ3 ⊗ σ3


β = −i∆t/~, B = σ1 ⊗ σ1 ⊗ σ1.

Problem 9. Let A, B be n × n matrices with A2 = In and B2 = In.Assume that the commutator of A and B vanishes, i.e.

[A,B] = AB −BA = 0n.

Let a, b ∈ C. CalculateeaA+bB .

(ii) Let a = −iωt, b = −i∆t/~ (∆ real) and

A = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3, B = σ1 ⊗ σ1 ⊗ · · · ⊗ σ1

with n (even) factors of the Kronecker products. Then the conditions givenabove are satisfied. Simplify the result from (i) with this assumption.

Problem 10. Let A, B be n × n matrices with A2 = In and B2 = In.Assume that the anticommutator of A and B vanishes, i.e.

[A,B]+ = AB +BA = 0n.

(i) Let a, b ∈ C. CalculateeaA+bB .

(ii) Let a = −iωt, b = −i∆t/~ (∆ real) and

A = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3, B = σ1 ⊗ σ1 ⊗ · · · ⊗ σ1

with n (odd) factors of the Kronecker products. Then the conditions givenabove are satisfied. Simplify the result from (i) with this assumption.

Problem 11. Consider the Hilbert space C3 and the standard basis

|0〉 =

100

, |1〉 =

010

, |2〉 =

001

.

Consider the unitary matrices

R =

1 0 00 ω 00 0 ω2

, T =

0 0 11 0 00 1 0

where ω = e2πi/3.(i) Calculate the state R|j〉, T |j〉, where j = 0, 1, 2.


(ii) Find the commutator [R, T ].(iii) Consider the normalized state

|ψ〉 =1√3

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉+ |2〉 ⊗ 2〉).

Calculate the state (R⊗ T )|ψ〉 and discuss.


[σm ⊗ σn, σk ⊗ I2]≡ [σm, σk]⊗ σn[σm ⊗ σn, I2 ⊗ σk]≡ σm ⊗ [σn, σk]

where k,m, n ∈ 1, 2, 3 .

Problem 13. Given two arbitrary normalized states |ψ〉 and |φ〉 in C2.Find a 2 × 2 unitary matrix U such that |ψ〉 = U |φ〉, i.e. U must beexpressed in terms of the compenents of the states |ψ〉 and φ〉.

Problem 14. Consider the Hamilton operator in C4

H = −t(|00〉〈11|+ |11〉〈00|) + v(|00〉〈00|+ |11〉〈11|.

The kinetic parameter is t ≥ 0 and v is the potential parameter. Findthe eigenvalues and eigenvectors of H. Keep t = 1 fixed and disucss thedependence of the eigenvalues of H as a function of v.

Problem 15. Let H be an n × n hermitian matrix. Let λ1, . . . , λn bethe eigenvalues with the pairwise orthogonal normalized eigenvectors v1,. . . , vn. Then we can write

H =n∑`=1

λ`v`v∗` .

LetP = In − vjv∗j − vkv∗k + vjv∗k + vkv∗j , j 6= k.

(i) What is the condition on the eigenvalues of H such that PHP ∗ = H.(ii) Find P 2.

Problem 16. Let B be an n× n matrix with B2 = In. Show that

exp(−1

2iπ(B − In)

)≡ B.


Problem 17. Consider the vector

|ψ〉 =

sin(φ1) sin(φ2) sin(θ)sin(φ1) sin(φ2) cos(θ)

sin(φ1) cos(φ2)cos(φ1)

in the Hilbert space R4 with φ1, φ2, θ ∈ R. Find the norm of this vector.For which values of φ1, φ2, θ is the norm a mimimum? What is the use ofthis vector?

Problem 18. Let R be a nonsingular n × n matrix. Let A and B ben × n matrices. Assume that R−1AR and R−1BR are diagonal matrices.Calculate the commutator [A,B].

Problem 19. Let A, B be two n× n matrices. Assume that

tr(A) = 0, tr(B) = 0.

Can we conclude that tr(AB) = 0? Prove or disprove.

Problem 20. We know that any n × n hermitian matrix has only realeigenvalues. Assume that a given n × n matrix has only real eigenvalues.Can we conclude that the matrix is hermitian? Prove or disprove.

Problem 21. Consider the Hilbert space Cn. Let e1, e2, . . . , en be thestandard basis in Cn, Sn be the symmetric group of order n! and Uσ be theunitary matrix on ⊗nCn such that

Uσ(e1 ⊗ · · · ⊗ en) := eσ(1) ⊗ · · · ⊗ eσ(n)

where σ ∈ Sn. We define the matrix (“antisymmetrization operator”) inthe Hilbert space ⊗nCn by

Πn :=1n!

∑σ∈Sn

sgn(σ)Uσ

where sgn is the signature of the permutation σ ∈ Sn. The matrices Πn

are projection matrices. Find Π2 and Π3.

Problem 22. (i) The four-dimensional face-centered hypercubic latticeplays a central role in simulating three-dimensional hydrodynamics on acellular automata machine. Consider the four-dimensional face-centeredhypercubic lattice in connection with entanglement. This lattice is gener-ated from the four basis vectors

(±1,±1, 0, 0). (1)


Permuting the components of these four vectors in R4 we find 20 additionalvectors. Show that the 24 vectors can be classified as follows. In class Awe have eight vectors

1001

,

100−1

,

−1001

,

−100−1

,

0110

,

01−10

,

0−110

,

0−1−10

.

The normalization factor would be 1/√

2. In class B we have the eightvectors

0101

,

010−1

,

0−101

,

0−10−1

,

1010

,

−1010

,

10−10

,

−10−10

.

The normalization factor is also 1/√

2. In class C we have the eight vectors0011

,

001−1

,

00−11

,

00−1−1

,

1100

,

1−100

,

−1100

,

−1−100

.

Again the normalization factor is 1/√

2. Show that if α(nA,m) is the anglebetween the nth vector of class A and the mth vector of class B, then

α(nA,mB) = α(nB ,mC) = α(nC ,mA)

andα(nA,mA) = α(nB ,mB) = α(nC ,mC).

Each class contains four oppositely oriented pairs of vectors. This meansthat the ordering of the vectors is such that class B is related to class A inexactly the same way as class C is related to B and A is related to C.(ii) Show that the normalized vectors in class A are maximally entangled.(iii) Show that the vectors in class B and class C can be written as theKronecker product of two vectors from R2.(iv) The Hadamard gate given by the unitary matrix

H =1√2

(1 11 −1

)plays a central role in quantum computing. Consider now the 4× 4 matrix

R = I2 ⊗H


where ⊗ denotes the Kronecker product and I2 is 2×2 unit matrix. Thus Ritself is a unitary matrix. Applying this matrix to the 24 vectors. Discuss.(v) Show that the construction given above can be extended to higherdimensional cases. For example in R8 we would start with

1√2

(±1,±1, 0, 0, 0, 0, 0, 0)T

and all permutations. Show that this provides us with the GHZ-state

1√2

(1 0 0 0 0 0 0 1)T

which is fully entangled when we use the three tangle (based on the hy-perdeterminant) as measure of entanglement. Show that we find a set ofunentangled states, for example

(10

)⊗(

10

)⊗ 1√

2

(11

)=

1√2

11000000

.

Show that using the three-tangle as measure for entanglement we find 0 forthese vectors.

Problem 23. The associative algebra Md(C) of d × d matrices can beconsidered as a C∗ algebra with the square of the norm ‖ . ‖ defined by(A ∈Md(C))

‖A‖2 := largest eigenvalue of the (normal) matrix A∗A.

Let d = 2 and

A =(

1 ii −1

).

Find the norm.

Problem 24. Consider the Hilbert space Cd. Let |j〉 (j = 1, . . . , d) bean orthonormal basis in Cd. Then a d × d matrix A acting in Cd can bewritten as

A =d∑

j,k=1

ajk|j〉〈k|


with ajk ∈ C. Obviously A depends on the underlying orthonormal basis.If we have the standard basis, then A reduces to the matrix A = (ajk). Wecan associate a vector |ψA〉 in the Hilbert space Cd2 with the matrix A via

|ψA〉 =d∑

j,k=1

ajk|j〉 ⊗ |k〉.

(i) Let d = 2 and consider the standard basis

|1〉 =(

10

), |2〉 =

(01

).

Find A and |ψA〉.(ii) Let d = 2 and consider the Hadamard basis

|1〉 =1√2

(11

), |2〉 =

1√2

(1−1

).

Find A and |ψA〉.(iii) Let d = 3 and consider the basis

|1〉 =1√2

101

, |2〉 =

010

, |3〉 =1√2

10−1

.

Find A and |ψA〉.(iv) Describe the connection of the map A 7→ |ψA〉 with the vec-operator.

Problem 25. Let φ1, φ2 ∈ R. From the Bell basis

1√2

eiφ1

00eiφ1

,1√2

0eiφ2

eiφ2

0

,1√2

0eiφ2

−eiφ2

0

,1√2

eiφ1

00−eiφ1

we form the matrix

M(φ1, φ2) =1√2

eiφ1 0 0 eiφ1

0 eiφ2 eiφ2 00 eiφ2 −eiφ2 0eiφ1 0 0 −eiφ1

.

Is M(φ1, φ2) an element of the Lie group SU(4)?


Problem 26. (i) Let x1, x2, x3 ∈ R. Let σ1, σ2, σ3 be the Pauli spinmatrices. Show that

ei(x1σ1+x2σ2+x3σ3) = cos(r)I2 +sin(r)r

i(x1σ1 + x2σ2 + x3σ3)

=(

cos(r) + ix3 sin(r)/r i(x1 − ix2) sin(r)/ri(x1 + ix2) sin(r)/r cos(r)− ix3 sin(r)/r

)where r :=

√x2

1 + x22 + x2

3.(ii) Let y1, y2, y3 ∈ R and

X := x1σ1 + x2σ2 + x3σ3, Y := y1σ1 + y2σ2 + y3σ3.

Consider the maps

X ↔ x =

x1

x2

x3

, Y ↔ y =

y1y2y3

.

Let x · y := x1y1 + x2y2 + x3y3 (scalar product). Show that

x · y =12

tr(XY ).

(iii) Show that

− i2

[X,Y ]↔ x× y =

x2y3 − x3y2x3y1 − x1y3x1y2 − x2y1

.

Problem 27. Let s be a spin with a fixed total angular momentumquantum number

s ∈ 1/2, 1, 3/2, 2, . . ..

The (normalized) eigenstates of x3-angular momentum |s,m〉 form a ladderwith

m = −s,−s+ 1, . . . , s− 1, s.

The eigenstates |s,m〉 form an orthonormal basis in a 2s + 1 dimensionalHilbert space. For example if s = 1/2 we have the two states |1/2,−1/2〉,|1/2, 1/2〉 and can identify

|1/2, 1/2〉 7→(

10

), |1/2,−1/2〉 7→

(01

).


Thus we have the Hilbert space C2. For s = 1 we have the three states|1,−1〉, |1, 0〉, |1, 1〉 and can identify

|1,−1〉 7→

001

, |1, 0〉 7→

010

, |1, 1〉 7→

100

.

A spin coherent state |s, θ, φ〉 for s = 1/2, 1, 3/2, . . . can be given by

|s, θ, φ〉 =m=s∑m=−s

√(2s)!

(s+m)!(s−m)!(cos(θ/2))s+m(sin(θ/2))s−me−imφ|s,m〉.

(i) Find |1/2, θ, φ〉 and write it as a vector in C2.(ii) Find |1, θ, φ〉 and write it as a vector in C3.(iii) For a given s find the scalar product 〈s,m|s, θ, φ〉.

Problem 28. (i) Consider the Pauli spin matrix σ2 and the Lie groupSL(2,C). Let S ∈ SL(2,C). Show that

Sσ2ST = σ2

where T denotes the transpose.(ii) Show that

(S ⊗ S)(σ2 ⊗ σ2)(ST ⊗ ST ) = σ2 ⊗ σ2.

Problem 29. Let |1〉, |2〉, . . . , |d〉 be an orthonormal basis in the Hilbertspace Cd. Consider the matrix

S =d∑

j,k=1

(|j〉〈k| ⊗ |k〉〈j|).

Is S independent of the chosen orthonormal basis?

Problem 30. (i) Let φ1, φ2 ∈ R. Show that

U(φ1, φ2) =1√2

(eiφ1 −e−iφ2

eiφ2 e−iφ1

)is unitary. Is U(φ1, φ2) an element of SU(2)? Find the eigenvalues ofU(φ1, φ2).


(ii) Let φ1, φ2, φ3, φ4 ∈ R. Show that

U(φ1, φ2, φ3, φ4) =1√2

eiφ1 0 0 −e−iφ2

0 eiφ3 −e−iφ4 00 eiφ4 e−iφ3 0eiφ2 0 0 e−iφ1

is unitary. Is U(φ1, φ2, φ3, φ4) an element of SU(4)? Find the eigenvaluesof U(φ1, φ2, φ3, φ4).(iii) Let φ1, φ2 ∈ R. Show that

U(φ1, φ2) =1√2

eiφ1 0 −e−iφ2

0√

2 0eiφ2 0 e−iφ1

is unitary. Is U(φ1, φ2) an element of SU(2)? Find the eigenvalues ofU(φ1, φ2).

Problem 31. Consider the Hadamard matrix

U =1√2

(1 11 −1

).

The eigenvalues of the Hadamard matrix are given by +1 and −1 with thecorresponding normalized eigenvectors

1√8

(√4 + 2

√2√

4− 2√

2

),

1√8

( √4− 2

√2

−√

4 + 2√

2

).

How can this information be used to find the eigenvalues and eigenvectorsof the Bell matrix

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

.

Problem 32. (i) Consider the Hilbert space C4. Do the vectors

v1 =12

1111

, v2 =1√2

10−10

, v3 =1√2

010−1

, v4 =12

1−11−1

form an orthonormal basis in C4. Prove or disprove.


(ii) Can the vectors v1, v2, v3, v4 be written as Kronecker products ofvectors in C2. Prove or disprove.(iii) Consider the 4× 4 matrices

S =

0 0 1 00 1 0 01 0 0 00 0 0 1

, T =

1 0 0 00 0 0 10 0 1 00 1 0 0

.

Find the eigenvalues and normalized eigenvectors of the two matrices. Com-pare to (i). Disucss.(iv) Find the commutator of S and T , i.e. [T, S]. What can be said abouteigenvectors of such a pair of matrices? Discuss. Hint. Look at your resultfrom (iii).

Problem 33. Consider the matrix

U =

1/√

2 0 1/√

20 1 0

1/√

2 0 −1/√

2

.

(i) Is the matrix unitary?(ii) Find the eigenvalues and nonnormalized eigenvectors of U . Use thisinformation to write down the spectral decomposition of U .(iii) Find a skew-hermitian matrix K such that U = exp(K). One canutilize the results from (ii).(iv) Apply the unitary matrix to the normalized state

|ψ〉 =1√3

111

.

Find the state U |ψ〉 and calculate the probability |〈ψ|K|ψ〉|2.

Problem 34. Let φ1, φ2, φ3, φ4 ∈ R. Consider the 2× 2 matrix

U(φ1, φ2, φ3, φ4) =1√2

(eiφ1 eiφ2

eiφ3 eiφ4

).

The matrix contains the two column vector

v1 =1√2

(eiφ1

eiφ2

), v2 =

1√2

(eiφ3

eiφ4

).

Find the conditions on φ1, φ2, φ3, φ4 such that

〈v1|v2〉 = 0.


Is the matrix unitary if this condition is satisfied?

Problem 35. (i) An n× n matrix H = (hjk) over C is called a complexHadamard matrix if |hjk| = 1 for j, k = 1, . . . , n and HH∗ = nIn. Notethat 1√

nH is then a unitary matrix. Let φ ∈ [0, π). Let n = 4. Show that

H(φ) =

1 1 1 11 ieiφ −1 −ieiφ1 −1 1 −11 −ieiφ −1 ieiφ

is a complex Hadamard matrix.(ii) Given two complex Hadamard matrices H1 and H2. Is H1 ⊗ H2 acomplex Hadamard matrix?


γ3 = i~ωA, A =

0 0 1 00 0 0 −1−1 0 0 00 1 0 0

where ~ and ω (frequency) are constants.(i) Find

exp(−iγ3t/~).

(ii) Let

|ψ(t = 0)〉 =1√2

1001

be the initial state in the Hilbert space C4. Calculate

|ψ(t)〉 = exp(−iγ3t/~)|ψ(t = 0)〉

and thus solve the Schrodinger equation.(iii) If we know the eigenvalues of γ3 what can be said about the eigenvaluesof exp(−iγ3t/~)?

Problem 37. Let α, θ, φ ∈ R. Consider the vector in C4

v =

sinh(α) sin(θ) cos(φ)sinh(α) sin(θ) sin(φ)

cosh(α) cos(θ)cosh(α) sin(θ)

.


(i) Normalize the vector.(ii) Apply the Bell matrix

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

to the normalized vector. Calculate v∗Bv. Discuss.

Problem 38. Let σ1, σ2, σ3 be the Pauli spin matrices. For the Diracequation the following 4× 4 matrices play a central role. We define

β :=(I2 02

02 −I2

), αk =

(02 σkσk 02

)k = 1, 2, 3.

Let γk = iβαk for k = 1, 2, 3, γ0 = −iβ and γ5 = iγ1γ2γ3γ0. Find thegamma matrices and calculate their anticommutators.

Problem 39. (i) Consider the finite-dimensional Hilbert space Cd. Asymmetric informatially complete positive operator valued measure (SIC-POVM) consists of d2 outcomes that are subnormalized projection matricesΠj onto pure states

Πj =1d|ψj〉〈ψj |

for j, k = 1, . . . , d2 such that

|〈ψk|ψk〉|2 =1 + dδjkd+ 1

.

Consider the case d = 2. Show that the normalized vectors

|ψ1〉=

√(3 +

√3)/6

eiπ/4√

(3−√

3)/6

|ψ2〉=

√(3 +

√3)/6

−eiπ/4√

(3−√

3)/6

|ψ3〉=

eiπ/4√

(3−√

3)/6√(3 +

√3)/6

|ψ4〉=

−eiπ/4√(3−√

3)/6√(3 +

√3)/6


satisfy this condition.(ii) Consider the matrices σ1, −iσ2, σ3. Find

σ1|ψ1〉, −iσ2|ψ1〉, σ3|ψ1〉.

(iii) Let d = 2. Let

Sd :=d∑j=1

|j〉⊗|j〉⊗〈j|⊗〈j|+∑k>j=1

1√2

(|j〉⊗|k〉+|k〉⊗|j〉)⊗ 1√2

(〈j|⊗〈k|+〈k|⊗〈j|)

where |1〉, |2〉 denotes the standard basis in C2, i.e.

|1〉 =(

10

), |2〉 =

(01

).

Show thatd2∑j=1

|ψj〉 ⊗ |ψj〉〈ψj | ⊗ 〈ψj | =2dd+ 1

.

(iv) Can one find a SIC-POVM in C4 using the states from (i) and theKronecker product?

Problem 40. Let a1, a2, b1, b2 be real. Find the normalization factorsfor the vector in C4

|ψ〉 =

a1 cos(φ/2) + b1 sin(φ/2)a2 cos(φ/2) + b2 sin(φ/2)ia1 sin(φ/2)− ib1 cos(φ/2)ia2 sin(φ/2)− ib2 cos(φ/2)

.

Problem 41. Any 2 × 2 matrix can be written as a linear combinationof the Pauli spin matrices and the 2× 2 identity matrix

A = aI2 + bσ1 + cσ2 + dσ3

where a, b, c, d ∈ C.(i) Find A2 and A3.(ii) Use the result from (i) to find all matrices A such that A3 = σ1.

Problem 42. Let r, s, θ ∈ R. Consider the Hamilton operator given bythe 2× 2 matrix

K =H

~ω=(reiθ ss re−iθ

).


(i) Is the matrix a normal matrix?(ii) Is the matrix hermitian?(iii) Find the eigenvalues and eigenvectors of K.

Problem 43. Consider the vector space of n × n matrices over C. LetB1, B2, . . . , Bn2 be a basis. Assume that all B’s are invertible. Is B−1

1 ,B−1

2 , . . . , B−1n2 also a basis for the vector space?

Problem 44. What can be said about the eigenvalues of an n×n matrixwhich is unitary and skew-hermitian? Give an example of such a matrix.

Problem 45. Let φ11, φ12, φ21, φ22 ∈ R. Consider the matrix

V =1√2

(eiφ11 eiφ12

eiφ21 eiφ22

).

(i) What are the conditions on φ11, φ12, φ21, φ22 such that the matrix isunitary?(ii) What are the conditions on φ11, φ12, φ21, φ22 such that the matrix ishermitian?What are the conditions on φ11, φ12, φ21, φ22 such that V = V −1?

Problem 46. Is the 3× 3 matrix

V =

1/√

2 −1/√

6 1/√

31/2 −1/(2

√3) −

√2/3

1/2√

3/2 0

unitary?

Problem 47. Let A be an n × n matrix over C. An n × n matrix B iscalled a square root of A if B2 = A. Find the square roots of the 2 × 2identity matrix applying the spectral theorem. The eigenvalues of I2 areλ1 = 1 and λ2 = 1. As normalized eigenvectors choose(

eiφ1 cos(θ)eiφ2 sin(θ)

),

(eiφ1 sin(θ)−eiφ2 cos(θ)

)which form an orthonormal basis in C2. Four cases (

√λ1,√λ2) = (1, 1),

(√λ1,√λ2) = (1,−1), (

√λ1,√λ2) = (−1, 1), (

√λ1,√λ2) = (−1,−1) have

to be studied. The first and last case are trivial. So study the second case(√λ1,√λ2) = (1,−1). The second case and the third case are “equivalent”.


Problem 48. Let |j〉 (j = 1, . . . , d) be an orthonormal basis in Cd and〈k| (k = 1, . . . , d) be the dual basis. We define

Rjk = |j〉〈k|, j, k = 1, . . . , d.

Show that

RjkR`m = Rjmδ`k, [Rjk, R`m] = Rjmδ`k −R`kδjm,d∑j=1

Rjj = Id.

Hint. Utilize

Problem 49. Let |0〉, |1〉, . . . |d− 1〉 be an orthonormal basis in Cd. LetTjk ∈ C with j, k = 0, 1, . . . , d− 1. Consider the linear operator

T =d−1∑j=0

d−1∑k=0

Tjk|j〉〈k|.

(i) Let d = 2 and

|0〉 =(

10

), |1〉 =

(01

).

Find T .(ii) Let d = 2 and

|0〉 =1√2

(11

), |1〉 =

1√2

(1−1

).

Find T .

Problem 50. Let H be a hermitian n×n matrix describing the Hamiltonoperator and acting in the Hilbert space Cn. Let A, B be n× n hermitianmatrices and |ψ〉 ∈ Cn. One defines (quantum correlation function)

Q(|ψ〉) :=12〈ψ|(A(t)B −AB(t) +BA(t)−B(t)A)|ψ〉

whereA(t) = eiHt/~Ae−iHt/~, B(t) = eiHt/~Be−iHt/~.

(i) Let

H = ~ωσ2, A = σ1, B = σ3, |ψ〉 =(

cos(θ)sin(θ)

).

(ii) Let

H = ~ωσ2⊗σ2, A = σ1⊗σ1, B = σ3⊗σ3, |ψ〉 =

cos(φ1)

sin(φ1) cos(φ2)sin(φ1) sin(φ2) cos(φ3)sin(φ1) sin(φ2) sin(φ3)


Problem 51. Let H be a hermitian matrix. Then all unitary matricesU0 = In, U1, . . . , Uk with UjHU

∗j = H form a group under matrix multi-

plication, where j = 0, 1, . . . , k. Note that depending on H is group couldconsist only of U0 = In. From UjHU

∗j = H it follows that [H,Uj ] = 0n. If

v is an (normalized) eigenvector of H with eigenvalue λ, then Ujv is alsoan eigenvector of H with the same eigenvalue since

HUjv = UjHv = λUjv.

The eigenvectors of H are bases of the irreducible representations of thegroup of H and can be classified according to them. Let

H =

0 1 01 0 10 1 0

.

(i) Find all 3× 3 permutation matrices P such that PHP−1 = H.(ii) Find the eigenvalues λ1, λ2, λ3 and the corresponding normalized eigen-vectors v1, v2, v3 of H.(iii) Find Pvj for j = 1, 2, 3 and the permutation matrices found in (i)

Problem 52. Consider the Bell basis

v1 =1√2

1001

, v2 =1√2

0110

, v3 =1√2

01−10

, v4 =1√2

100−1

which form an orthonormal basis in C4. We can form 24 = 4! unitarymatrices

(vj1 vj2 vj3 vj4) jk 6= j` (pairwise)

with the lexigographical order (v1 v2 v3 v4), (v1 v2 v4 v3), . . . , (v4 v3 v2 v1).Apply the 24 unitary matrices to the normalized vector

w =12

1111

≡ 1√2

(11

)⊗ 1√

2

(11

).

Discuss.

Problem 53. Can the Bell matrix be written as

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

= c0σ0⊗σ0 +c1σ1⊗σ1 +c2σ2⊗σ2 +c3σ3⊗σ3


where σ0 = I2 and σ1, σ2, σ3 are the Pauli spin matrices.

Problem 54. Let S1, S2, S3 be the spin matrices for spin

s =12, 1,

32, 2, . . .

These matrices are (2s+ 1)× (2s+ 1) hermitian matrices with trace equalto 0 satisfying the commutation relations

[S1, S2] = iS3, [S2, S3] = iS1, [S3, S1] = iS2.

The eigenvalues of these three matrices are s, s − 1, . . . ,−s for a given s.Furthermore one has

S21 + S2

2 + S23 = s(s+ 1)I2s+1

where I2s+1 is the (2s+ 1)× (2s+ 1) identity matrix.(i) Show that tr(S2

j ) = 13s(s+ 1)(2s+ 1).

(ii) Show that tr(SjSk) = 0 for j 6= k and j, k = 1, 2, 3.(iii) Show that the hermitian (2s+ 1)2 × (2s+ 1)2 matrices

H =H

~ω= S1 ⊗ S1 + S2 ⊗ S2 + S3 ⊗ S3

K =K

~ω= S1 ⊗ S2 + S2 ⊗ S3 + S3 ⊗ S1

admit the same spectrum for all s.(iv) Find the commutator

[S1 ⊗ S1 + S2 ⊗ S2 + S3 ⊗ S3, S1 ⊗ S2 + S2 ⊗ S3 + S3 ⊗ S1]

and anticommutator

[S1 ⊗ S1 + S2 ⊗ S2 + S3 ⊗ S3, S1 ⊗ S2 + S2 ⊗ S3 + S3 ⊗ S1]+.

(v) For the case s = 12 , s = 1, s = 3

2 find the eigenvalues and the normalizedeigenvectors.(vi) Calculate exp(zH) and exp(zK).

Problem 55. (i) Consider the hermitian 3 × 3 matrices to describe aparticle with spin-1

S1 :=~√2

0 1 01 0 10 1 0

, S2 :=~√2

0 −i 0i 0 −i0 i 0

, S3 := ~

1 0 00 0 00 0 −1

.


With S+ := S1 + iS2, S− := S1 − iS2 we find

S+ =√

2~

0 1 00 0 10 0 0

, S− =√

2~

0 0 01 0 00 1 0

. (3)

An example of a spin-1 particle is the photon. Let m, n be normalizedvectors in R3 which are orthogonal, i.e. mTn = 0. Find the eigenvalues ofthe 3× 3 matrix

K = (m · S)2 − (n · S)2

where m · S = m1S1 +m2S2 +m3S3.(ii) Show that

Pm = I3 − (m · S)2

is a projection operator.

Problem 56. Let A, B be n×n matrices over C. Let v be a normalized(column) vector in Cn. Let 〈A〉 := v∗Av and 〈B〉 := v∗Bv. We have theidentity

AB ≡ (A− 〈A〉In)(B − 〈B〉In) +A〈B〉+B〈A〉 − 〈A〉〈B〉In.

We approximate AB as AB ≈ A〈B〉+B〈A〉 − 〈A〉〈B〉In.(i) Let

A = σ1, B = σ2, u =1√2

(11

).

Find AB and A〈B〉+B〈A〉− 〈A〉〈B〉In and the distance (Frobenius norm)between the two matrices.(ii) Apply the result to the case n = 2 and

A = σ1, B = σ2, v =1√2

(1−1

).

(iii) Consider the case

A = σ1, B = σ2, v =(

cos(θ)sin(θ)

).

Problem 57. Let a,b ∈ R3 and σ1, σ2, σ3 be the Pauli spin matrices.We define

a · σ := a1σ1 + a2σ2 + a3σ3.

What is the condition on a, b such that

(a · σ)(b · σ) ≡ (a · b)I2 + i(a× b) · σ?


Here × denotes the vector product and I2 is the 2× 2 identity matrix.

Problem 58. Let σ1, σ2, σ3 be the Pauli spin matrices. Consider the2× 2 matrix over the complex numbers

Π(n) :=12

I2 +3∑j=1

njσj

where n := (n1, n2, n3) (nj ∈ R) is a unit vector, i.e. n2

1 + n22 + n2

3 = 1.(i) Describe the property of Π(n), i.e. find Π∗(n), tr(Π(n)) and Π2(n),where tr denotes the trace. The trace is the sum of the diagonal elementsof a square matrix.(ii) Find the vector

Π(n)(eiφ cos(θ)

sin(θ)

).

Discuss.

Problem 59. Let σ1, σ2, σ3 be the Pauli spin matrices. Let A, B be twoarbitrary 2× 2 matrices. Is

12

tr(AB) ≡3∑j=1

(12

tr(σjA))(

12

tr(σjB))

?

Problem 60. Let σj (j = 0, 1, 2, 3) be the Pauli spin matrices, where σ0

is the 2× 2 identity matrix. Form the four 4× 4 matrices

γk =(

02 σk−σk 02

), k = 0, 1, 2, 3

where 02 is the 2× 2 identity matrix.(i) Are the matrices γk linearly independent?(ii) Find the eigenvalues and eigenvectors of the γk’s.(iii) Are the matrices γk invertible. Use the result from (ii). If so, find theinverse.(iv) Find the commutators [γk, γ`] for k, ` = 0, 1, 2, 3. Find the anticommu-tators [γk, γ`]+ for k, ` = 0, 1, 2, 3.(v) Can the matrices γk be written as the Kronecker product of two 2× 2matrices?


Problem 61. Consider the 4× 4 matrices

α1 =

0 0 0 10 0 1 00 1 0 01 0 0 0

= σ1 ⊗ σ1

α2 =

0 0 0 −i0 0 i 00 −i 0 0i 0 0 0

= σ1 ⊗ σ2

α3 =

0 0 1 00 0 0 −11 0 0 00 −1 0 0

= σ1 ⊗ σ3.

Let a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3), d = (d1, d2, d3) beelements in R3 and

a ·α := a1α1 + a2α2 + a3α3.

Calculate the traces

tr((a ·α)(b ·α)), tr((a ·α)(b ·α)(c ·α)(d ·α)).

Problem 62. Let σ1, σ2, σ3 be the Pauli spin matrices. Consider the4× 4 gamma matrices

γ1 =(

02 σ1

−σ1 02

), γ2 =

(02 σ2

−σ2 02

), γ3 =

(02 σ3

−σ3 02

)and

γ0 =(I2 02

02 −I2

).

Find γ1γ2γ3γ0 and tr(γ1γ2γ3γ0).

Problem 63. Find the eigenvalues and eigenvectors of σ1σ2σ3.

Problem 64. Consider the spin-1 matrices

S1 =1√2

0 1 01 0 10 1 0

, S2 =1√2

0 −i 0i 0 −i0 i 0

, S3 =

1 0 00 0 00 0 −1

and

Sα,1 = Sα ⊗ I3 ⊗ I3, Sα,2 = I3 ⊗ Sα ⊗ I3, Sα,3 = I3 ⊗ I3 ⊗ Sα


with α = 1, 2, 3. Let

S1 =

S1 ⊗ I3 ⊗ I3S2 ⊗ I3 ⊗ I3S3 ⊗ I3 ⊗ I3

, S2 =

I3 ⊗ S1 ⊗ I3I3 ⊗ S2 ⊗ I3I3 ⊗ S3 ⊗ I3

, S3 =

I3 ⊗ I3 ⊗ S1

I3 ⊗ I3 ⊗ S2

I3 ⊗ I3 ⊗ S3

and

S2 × S3 =

I3 ⊗ S2 ⊗ S3 − I3 ⊗ S3 ⊗ S2

I3 ⊗ S3 ⊗ S1 − I3 ⊗ S1 ⊗ S3

I3 ⊗ S1 ⊗ S2 − I3 ⊗ S2 ⊗ S1

Thus

S1·(S2×S3) = S1⊗S2⊗S3−S1⊗S3⊗S2+S2⊗S3⊗S1−S2⊗S1⊗S3+S3⊗S1⊗S2−S3⊗S2⊗S1.

Find the eigenvalues of S1 · (S2 × S3).

Problem 65. (i) Find the eigenvalues and eigenvectors of

σ1 ⊗ σ2 ⊗ σ3.

Can one find entangled eigenvectors?

(ii) Find the eigenvalues and eigenvectors of the Hamilton operator

H = ε1(σ1⊗ I2⊗ I2) + ε2(I2⊗ σ2⊗ I2) + ε3(I2⊗ I2⊗ σ3) + γ(σ1⊗ σ2⊗ σ3)

where ε1, ε2, ε3, γ ∈ R.

Problem 66. Find the nonzero (column) vectors u ∈ C16 such that

(σ1 ⊗ σ3 ⊗ I2 ⊗ σ3)u = u

(σ3 ⊗ σ1 ⊗ σ3 ⊗ I2)u = u

(I2 ⊗ σ3 ⊗ σ1 ⊗ σ3)u = u

(σ3 ⊗ I2 ⊗ σ3 ⊗ σ1)u = u.

Problem 67. Consider the Pauli spin matrices σ1, σ2, σ3. Find theskew-hermitian matrices Σ1, Σ2, Σ3 such that

σ1 = exp(Σ1), σ2 = exp(Σ2), σ3 = exp(Σ3).

Find the commutators [Σ1,Σ2], [Σ2,Σ3], [Σ3,Σ1] and compare with thecommutators [σ1, σ2], [σ2, σ3], [σ3, σ1].


Problem 68. (i) Consider the three (hermitian) spin-1 matrices

S1 =1√2

0 1 01 0 10 1 0

, S2 =i√2

0 −1 01 0 −10 1 0

, S3 =

1 0 00 0 00 0 −1

all with the eigenvalues +1, 0 and −1. Show that S3

j = Sj .(ii) Let φ ∈ R. Show that

exp(iφSj) = I3 + i sin(φ)Sj − (1− cos(φ))S2j

which is a unitary matrix.

Problem 69. (i) Let σ1, σ2, σ3 be the Pauli spin matrices and z1, z2, z3 ∈C. Calculate

exp(z1σ1 + z2σ2 + z3σ3).

(ii) Calculate the matrix

exp(z1σ1 ⊗ σ1 + z2σ2 ⊗ σ2 + z3σ3 ⊗ σ3).

Problem 70. Find the unitary matrix

U(t) = eiφ sin(ωt)σ1

with the Pauli spin matrix σ1.

Problem 71. Let z ∈ C. Calculate

exp(−z(σ1 ⊗ σ1))(σ2 ⊗ σ2) exp(z(σ1 ⊗ σ1))

exp(−z(σ2 ⊗ σ2))(σ3 ⊗ σ3) exp(z(σ2 ⊗ σ2))

exp(−z(σ3 ⊗ σ3))(σ1 ⊗ σ1) exp(z(σ3 ⊗ σ3)).

Problem 72. Let σ1, σ2, σ3 be the Pauli spin matrices. Calculate thecommutator

[σj ⊗ σk, σ` ⊗ σm]

where j 6= ` and k 6= m.

Problem 73. Let σ3, σ1 be the Pauli spin matrices. Find the commuta-tors

[σ1 ⊗ σ1 ⊗ σ1, σ3 ⊗ σ3 ⊗ σ3]


and[σ1 ⊗ σ1 ⊗ σ1 ⊗ σ1, σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3].

Discuss the general case with n Kronecker products.

Problem 74. Consider the spin-1 matrices S1, S2, S3. Find the eigen-values of the hermitian matrix

K =H

~ω= cos(θ)(S1⊗S1+S2⊗S2+S3⊗S3)+sin(θ)(S1⊗S1+S2⊗S2+S3⊗S3)2.

Problem 75. Consider the Pauli spin matrices σ1, σ2, σ3 and the 4× 4matrices

σ1 ⊗ σ2, σ2 ⊗ σ3, σ3 ⊗ σ2.

Find the commutators

[σ1 ⊗ σ2, σ2 ⊗ σ3], [σ2 ⊗ σ3, σ3 ⊗ σ1], [σ3 ⊗ σ2, σ1 ⊗ σ2].

Discuss

Problem 76. Let σ1, σ2, σ3 be the Pauli spin matrices. Calculate thecommutators

[σ1 ⊗ I2 + I2 ⊗ σ2, σ2 ⊗ I2 + I2 ⊗ σ3]

[σ2 ⊗ I2 + I2 ⊗ σ3, σ3 ⊗ I2 + I2 ⊗ σ1]

[σ3 ⊗ I2 + I2 ⊗ σ1, σ1 ⊗ I2 + I2 ⊗ σ2].

Problem 77. Let σ1, σ2, σ3 be the Pauli spin matrices. Consider the4× 4 matrices

σ1 ⊗ I2 + I2 ⊗ σ1, σ2 ⊗ I2 + I2 ⊗ σ2, σ3 ⊗ I2 + I2 ⊗ σ3.

Find the commutators. Discuss.

Problem 78. Consider the 26 × 26 unitary and hermitian matrices

X = σ1 ⊗ σ2 ⊗ σ3 ⊗ σ1 ⊗ σ2 ⊗ σ3, S = σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3.

Find the commutator [X,S] and anticommutator [X,S]+.

Problem 79. Consider the Bell matrix

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1


which is a unitary matrix. Each column vector of the matrix is a fully en-tangled state. Are the normalized eigenvectors of B are also fully entangledstates?

Problem 80. Consider the Pauli spin matrices σ1, σ2, σ3. Let x1, x2, x3 ∈R. Show that

ei(x1σ1+x2σ2+x3σ3) = I2 cos(r) +sin(r)r

i(x1σ1 + x2σ2 + x3σ3)

where r2 = x21 + x2

2 + x23.

Problem 81. Consider the Pauli spin matrices to describe a spin- 12 par-

ticle. In the square array of 4× 4 matrices

I2 ⊗ σ3 σ3 ⊗ I2 σ3 ⊗ σ3

σ1 ⊗ I2 I2 ⊗ σ1 σ1 ⊗ σ1

σ1 ⊗ σ3 σ3 ⊗ σ1 σ2 ⊗ σ2

each row and each column is a triad of commuting operators. Consider thehermitian 3× 3 matrices to describe a particle with spin-1

S1 :=~√2

0 1 01 0 10 1 0

, S2 :=~√2

0 −i 0i 0 −i0 i 0

, S3 := ~

1 0 00 0 00 0 −1

.

Is in the square array of 9× 9 matrices

I3 ⊗ S3 S3 ⊗ I3 S3 ⊗ S3

S1 ⊗ I3 I3 ⊗ S1 S1 ⊗ S1

S1 ⊗ S3 S3 ⊗ S1 S2 ⊗ S2

each row and each column a triad of commuting operators?

Problem 82. Let σ1, σ2, σ3 be the Pauli spin matrices and

a =

a1

a2

a3

∈ R3, b =

b1b2b3

∈ R3, σ =

σ1

σ2

σ3

.

Calculate (a×σ)T ) · (b×σ), where × denotes the vector product and · thescalar product.

Problem 83. Consider the Pauli spin matrices σ = (σ1, σ2, σ3). Let q,r, s, t be unit vectors in R3. We define

Q := q · σ, R := r · σ, S := s · σ, T := t · σ


where q · σ := q1σ1 + q2σ2 + q3σ3. Calculate

(Q⊗ S +R⊗ S +R⊗ T −Q⊗ T )2.

Express the result using commutators.

Problem 84. Let σ1, σ2, σ3 be the Pauli spin matrices.(i) Find

R1x(α) := exp(−iα(σ1 ⊗ I2)), R1y(α) := exp(−iα(σ2 ⊗ I2))

where α ∈ R and I2 denotes the 2× 2 unit matrix.(ii) Consider the caseR1x(α = π/2) andR1y(α = π/4). CalculateR1x(π/2)R1y(π/4).Discuss.

Problem 85. Let σ1, σ2 and σ3 be the Pauli spin matrices. We defineσ+ := σ1 + iσ2 and σ− := σ1 − iσ2. Let

c∗k := σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗(

12σ+

)⊗ I2 ⊗ I2 ⊗ · · · ⊗ I2

where σ+ is on the kth position and we have N − 1 Kronecker products.Thus c∗k is a 2N × 2N matrix.(i) Find ck.(ii) Find the anticommutators [ck, cj ]+ and [c∗k, cj ]+.(iii) Find ckck and c∗kc

∗k.

Problem 86. Using the definitions from the previous problem we define

s−,j :=12

(σx,j − iσy,j) =12σ−,j , s+,j :=

12

(σx,j + iσy,j) =12σ+,j

and

c1 = s−,1

cj = exp

(iπ

j−1∑`=1

s+,`s−,`

)s−,j for j = 2, 3, . . .

(i) Find c∗j .(ii) Find the inverse transformation.(iii) Calculate c∗jcj .

Problem 87. Find the conditions on c1, c2, c3 ∈ C such that

(c1σ1 ⊗ σ1 + c2σ2 ⊗ σ2 + c3σ3 ⊗ σ3)((

10

)⊗(

10

))=(

10

)⊗(

10

).


Problem 88. Let σ1, σ2, σ3 be the Pauli spin matrices. Find the 3 × 3matrix R defined by

Rjk = tr(σj ⊗ σk), j, k = 1, 2, 3.

Problem 89. Let σ1, σ2, σ3 be the Pauli spin matrices. Calculate thetrace of σ1, σ2, σ3, σ1σ2, σ1σ3, σ2σ3, σ1σ2σ3.

Problem 90. Consider

exp

ε n∑j=1

s−j s−j+1

with s−n+1 = s−1 (cyclic boundary condition) and the term

Y =1m!

n∑j=1

s−j s−j+1

m

.

Let

u =(

10

), v =

(01

).

CalculateY (u⊗ u⊗ · · · ⊗ u), Y (v ⊗ v ⊗ · · · ⊗ v).

Problem 91. Let z ∈ C. Calculate the commutator

[σ2 ⊗ σ2 ⊗ σ2, σ1 ⊗ σ1 ⊗ σ1]

andexp(−zσ2 ⊗ σ2 ⊗ σ2)(σ1 ⊗ σ1 ⊗ σ1) exp(zσ2 ⊗ σ2 ⊗ σ2).

Problem 92. Find the eigenvalues of the unitary operator

U = exp(−iπ

4b†b⊗ σ3

).

Note that eiπ/4 = (1 + i)/√

2, e−iπ/4 = (1− i)/√

2.


B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

.


Write the matrix B in the form

B =3∑

j1,j2=0

cj1,j2σj1 ⊗ σj2 .

Problem 94. Consider the 4× 4 matrix

M =

0 0 −a −b0 0 b −a−a b 0 0−b −a 0 0

where a, b ∈ R and the Bell matrix

B =1√2

1 0 0 10 −1 −1 00 −1 1 01 0 0 −1

Show that BTMB can be written as the direct sum of two 2× 2 matrices.


B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

.

(i) Find all matrices A such that BAB∗ = A.(ii) Find all matrices A such that BAB∗ is a diagonal matrix.

Problem 96. Consider the standard basis in the vector space of 2 × 2matrices

E00 =(

1 00 0

), E01 =

(0 10 0

), E10 =

(0 01 0

), E11 =

(0 00 1

).

and the mutually unbiased basis

µ0 =1√2

(1 00 1

), µ1 =

1√2

(0 11 0

), µ2 =

1√2

(0 −ii 0

), µ3 =

1√2

(1 00 −1

),

Express the Bell matrix

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1


with the basis given by µj ⊗ µk (j, k = 0, 1, 2, 3).

Problem 97. Let σ0 = I2, σ1, σ2, σ3 be the Pauli spin matrices. Find

(σ1 ⊗ σ1 + σ2 ⊗ σ2)2, (σ1 ⊗ σ1 + σ2 ⊗ σ2)3.

Problem 98. Let σ1, σ2, σ3 be the Pauli spin matrices. Let

R := σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3.

(i) Find tr(R). Using this result what can be said about the eigenvalues ofR.(ii) Find R2. Using this result and the result from (i) derive the eigenvaluesof the matrix R.(iii) Find 1

4 (I4 +R)2.

Problem 99. Given the 4× 4 matrix

A =1√2

1 0 1 00 1 0 11 0 −1 00 1 0 −1

.

Are the four column vectors

1√2

1010

,1√2

0101

,1√2

10−10

,1√2

010−1

eigenvectors of A? Are the vectors entangled?

Problem 100. Find 4×4 matrices A, B consisting of Kronecker productsof σ1, σ2, σ3, I2 such that

[A,B] = σ2 ⊗ σ3 − σ3 ⊗ σ2.

Problem 101. Let A, B be hermitian n × n matrices. Consider theHamilton operator

H = A⊗ In + In ⊗B + ε(A⊗B)

where ε ∈ R. Let H0 = A⊗ In + In ⊗B. Find the Moller operator

Ω± := limT→∓∞

exp(−iHT/~) exp(iH0T/~).


Problem 102. Calculate

exp(−iπ(σ3⊗I2⊗I2 +I2⊗I2⊗I2)(I2⊗σ3⊗I2 +I2⊗I2⊗I2)(I2⊗I2⊗σ1/8).

This is the Toffoli gate.

Problem 103. Let σ1, σ2, σ3 be the Pauli spin matrices. Let α ∈ R.Calculate

exp(α(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3))

and the trace of this expression

tr exp(α(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3)).

Problem 104. Every 4× 4 unitary matrix U can be written as

U = (U1 ⊗ U2) exp(i(ασ1 ⊗ σ1 + βσ2 ⊗ σ2 + γσ3 ⊗ σ3))(U3 ⊗ U4)

where Uj ∈ U(2) (j = 1, 2, 3, 4) and α, β, γ ∈ R. Calculate

exp(i(ασ1 ⊗ σ1 + βσ2 ⊗ σ2 + γσ3 ⊗ σ3)).

Problem 105. Let ω = exp(2πi/4) ≡ exp(πi/2). Consider the four64× 64 invertible matrices

S1 =

1 0 0 00 ω 0 00 0 ω2 00 0 0 ω3

⊗

1 0 0 00 ω3 0 00 0 ω2 00 0 0 ω

⊗ I4

S2 = I4 ⊗

0 1 0 00 0 1 00 0 0 11 0 0 0

⊗ I4

S3 = I4 ⊗

1 0 0 00 ω 0 00 0 ω2 00 0 0 ω3

⊗

1 0 0 00 ω3 0 00 0 ω2 00 0 0 ω

S4 = I4 ⊗ I4 ⊗

0 1 0 00 0 1 00 0 0 11 0 0 0

.

Find S4j for j = 1, 2, 3, 4. Find

S1S2S−11 S−1

2 , S2S3S−12 S−1

3 , S3S4S−13 S−1

4 , S4S1S−14 S−1

1 .


Find the commutators [S1, S2], [S2, S3], [S3, S4], [S4, S1].

Problem 106. Consider the Pauli spin matrices σ1, σ2 and σ3. Can onefind an α ∈ R such that

exp(iασ3)σ1 exp(−iασ3) = σ2 ?


S2 =i√2

0 −1 01 0 −10 1 0

.

Is the matrix hermitian? Find the eigenvalues and eigenvectors of S2.(ii) Show that S3

2 = S2.(iii) Let φ ∈ R. Find exp(iφS2).

Problem 108. (i) Let τ = (√

5 − 1)/2 be the golden mean number.Consider the 2× 2 matrices

B1 =(e−i7π/10 0

0 −e−i3π/10), B2 =

(−τe−iπ/10 −i

√τ

−i√τ −τeiπ/10

).

The matrices are invertible. Are the matrices unitary? Is B1B2B1 =B2B1B2?(ii) Show that using computer algebra

B−22 B4

1B−12 B1B

−12 B1B2B

−21 B2B

−11 B−5

2 B1B−12 ≈

(0 ii 0

).

Problem 109. Consider the Hamilton operator H = H0 + H1, where

H0 = ~ωσ3, H1 = ~ωσ1.

Let U and U0 be the unitary matrices

U = exp(−iHt/~), U0 = exp(−iH0t/~).

Let n be a positive integer. The Moller wave operators

Ω± := limn→±∞

UnU−n0 .

Owing to their intertwining property the Moller wave operators transformthe eigenvectors of the free dynamics U0 = exp(−iH0t/~) into eigenvectorsof the interacting dynamics U = exp(−iHt/~). Find Ω±.



U = α1σ1 + α2σ2 + α3σ3

is a unitary matrix. Note that

UU∗ = (α1σ1 + α2σ2 + α3σ3)(α∗1σ1 + α∗2σ2 + α∗3σ3)= (α1α

∗1 + α2α

∗2α3α3)I2 + (α1α

∗2 − α2α

∗1)σ1σ2 + (α3α

∗1 − α1α

∗3)σ3σ1 + (α2α

∗3 − α3α

∗2)σ2σ3.


U =1√3

(I2 ⊗ I2 ⊗ I2 + iσ1 ⊗ σ1 ⊗ σ1 + iσ3 ⊗ σ3 ⊗ σ3)

unitary?

Problem 112. Consider the Pauli spin matrices σ0 = I2, σ1, σ2, σ3. Thematrices are unitary and hermitian.(i) Is the 4× 4 matrix

1√2

(σ0 σ1

σ2 σ3

)unitary?(ii) Is the 4× 4 matrix

1√2

(σ0 σ1

−iσ2 σ3

)unitary?

Problem 113. Consider the two spin-1 matrices

L2 =

0 0 i0 0 0−i 0 0

, L3 =

0 −i 0i 0 00 0 0

.

Let θ, φ ∈ R. Calculate T (θ, φ) = exp(−iφL3) exp(−iθL2). Is T (θ, φ) anelements of SO(3,R)?

Problem 114. The following states form an orthonormal basis in theHilbert space C3

|π+〉 =1√2

101

, |π0〉 =

010

, |π−〉 =1√2

10−1

.


These states play a role for the π-mesons. Show that the states

|π+〉 ⊗ |π+〉, |π−〉 ⊗ |π−〉

1√2

(|π+〉 ⊗ |π0〉+ |π0〉 ⊗ |π+〉), 1√2

(|π0〉 ⊗ |π−〉+ |π−〉 ⊗ |π0〉)

1√2

(|π+〉⊗|π0〉−|π0〉⊗|π+〉), 1√2

(|π+〉⊗|π−〉−|π−〉⊗|π+〉), 1√2

(|π0〉⊗|π−〉−|π−〉⊗|π0〉)

1√6

(2|π0〉⊗|π0〉+|π+〉⊗|π−〉+|π−〉⊗|π+〉), 1√3

(|π+〉⊗|π−〉+|π−〉⊗|π+〉−|π0〉⊗|π0〉)

form an orthonormal basis in the Hilbert space C9. Which of these statesare entangled?

Problem 115. (i) The electronic scattering matrix has the form

S(φ1, φ2, φ3, γ) = eiφ1σ0eiφ2σ3eiγσ2eiφ3σ3

where φ1, φ2, φ3 ∈ [0, 2π), γ ∈ [0, π/2). Find S(φ1, φ2, φ3, γ).(ii) Find

T (φ1, φ2, φ3, γ) = eiφ1σ0 ⊗ eiφ2σ3 ⊗ eiγσ2 ⊗ eiφ3σ3 .

Problem 116. Consider the 27 × 27 matrices

−3 − 2 − 1 0 1 2 3X02 = I2 ⊗ I2 ⊗ I2 ⊗ σz ⊗ I2 ⊗ σz ⊗ I2X−20 = I2 ⊗ σz ⊗ I2 ⊗ σz ⊗ I2 ⊗ I2 ⊗ I2X12 = I2 ⊗ I2 ⊗ I2 ⊗ I2 ⊗ σz ⊗ σz ⊗ I2

X−2−1 = I2 ⊗ σz ⊗ σz ⊗ I2 ⊗ I2 ⊗ I2 ⊗ I2X23 = I2 ⊗ I2 ⊗ I2 ⊗ I2 ⊗ I2 ⊗ σz ⊗ σz

X−3−2 = σz ⊗ σz ⊗ I2 ⊗ I2 ⊗ I2 ⊗ I2 ⊗ I2.

LetK = λ(X02 +X−20 +X12 +X−2−1 +X23 +X−3−2)

where λ ∈ R. Calculate tr(exp(K)) and discuss the behaviour on λ.

Problem 117. Consider the two 4× 4 matrices σ1 ⊗ σ3, σ3 ⊗ σ1.(i) Find the eigenvalues.(ii) Show that the eigenvectors can be given as product states (unentangledstates), but also as entangled states (i.e. they cannot be written as productstates). Explain.


Problem 118. Find a unitary matrix U which can be written as a directsum of two 2× 2 matrices and

U12

1111

≡ U 1√2

(11

)⊗ 1√

2

(11

)=

1√2

1001

.

Problem 119. A wave-scattering problem can be described by its scat-tering matrix U . In a stationary problem, U relates the outgoing-wave tothe ingoing-wave amplitudes. The condition of flux conservation impliesunitary of U , i.e.

UU† = I

where I is the identity operator. If, additionally, the scattering problem isinvariant under the operation of time reversal, we also have U = UT , i.e.U is symmetric. Find all 2× 2 unitary matrices that also satisfy U = UT .Do these matrices form a subgroup of the Lie group U(2)?

Problem 120. Is

(σ3 ⊗ σ2 ⊗ σ1)(σ1 ⊗ σ2 ⊗ σ3)(σ3 ⊗ σ2 ⊗ σ1) = σ1 ⊗ σ2 ⊗ σ3 ?

Is(σ3 ⊗ I2 ⊗ σ1)(σ1 ⊗ σ2 ⊗ σ3)(σ3 ⊗ I2 ⊗ σ1) = σ1 ⊗ σ2 ⊗ σ3 ?

Problem 121. Let n be odd and n ≥ 3. Consider the matrices

A3 =

1/√

2 0 1/√

20 1 0

1/√

2 0 −1/√

2

, A5 =

1/√

2 0 0 0 1/√

20 1/

√2 0 1/

√2 0

0 0 1 0 00 1/

√2 0 −1/

√2 0

1/√

2 0 0 0 −1/√

2

and generally

An =

1/√

2 0 . . . 0 0 0 . . . 0 1/√

20 1/

√2 . . . 0 0 0 . . . 1/

√2 0

...... . . .

......

... . . ....

...0 0 . . . 1/

√2 0 1/

√2 . . . 0 0

0 0 . . . 0 1 0 . . . 0 00 0 . . . 1/

√2 0 −1/

√2 . . . 0 0

...... . . .

......

... . . ....

...0 1/

√2 . . . 0 0 0 . . . −1/

√2 0

1/√

2 0 . . . 0 0 0 . . . 0 −1/√

2

.


Find the eigenvalues and eigenvectors of A3, A5. Then solve the generalcase. The rank of the matrix An is n. Thus the matrices are invertible.The determinant of An is −1. Since tr(An) = 1 the sum of the eigenvaluesis 1.

Problem 122. Let σ1, σ2, σ3 be the Pauli spin matrices.(i) Let z ∈ C. Calculate cosh(zσj), sinh(zσj), j = 1, 2, 3. Note thatσ2j = I2.

(ii) Show that sin(θσj) = sin(θ)σj .(iii) Find the matrix

U = exp(i

1√2

(σ1 + σ3)).

Is the matrix U unitary? Prove or disprove. If so find the group generatedby U .

Problem 123. Let v be a normalized vector in Cn. Do all n×n matricesM which satisfy

Mv = v, det(M) = 1

form a group under matrix multiplication?

Problem 124. Let c†1, c†2, . . . , c

†n be Fermi creation operators and c1, c2, . . . , cn

be Fermi annihilation operators with the anticommutation relations

[c†j , ck]+ = δjkI.

(i) Consider the unitary matrix

U =(

0 11 0

).

Is the operator

KU =(c†1 c†2

)( 0 11 0

)(c1c2

)= c†1c2 + c†2c1

unitary?(ii) Consider the hermitian matrix

U =(

0 −ii 0

).

Is the operator

KH =(c†1 c†2

)( 0 −ii 0

)(c1c2

)= −ic†1c2 + ic†2c1


hermitian?(iii) Consider the nonnormal matrix

N =(

0 10 0

).

Is the operator

KN =(c†1 c†2

)( 0 10 0

)(c1c2

)= c†1c2 + c†2c1

nonnormal?

Problem 125. Let Γ = (Γjk) (j, k = 1, . . . , n), Γ1, Γ2 be n × n skew-hermitian matrices. Then V = exp(Γ), V1 = exp(Γ1), V2 = exp(Γ2) areunitary matrices. Let c†1, . . . , c†n be Fermi creation operators and c1, . . . ,cn be Fermi annihilation operators. Then

U(V ) = exp(n∑j=1

n∑k=1

Γjkc†jck)

is a unitary operator with V = exp(Γ). The commutation relation for theoperators are c†jck are

[c†jck, c†`cm] = δk`c

†jcm − δjmc

†`ck.

Show that owing to these commutation relations we have

U(V1)U(V2) = U(V1V2), U(V −1) = U−1(V ) = U†(V ), U(In) = I

where I is the identity operator.

Problem 126. Consider the 2× 2 matrices

A =(

0 10 0

), B =

(0 00 1

).

Then [A,B] = A and A, B form a basis of a two-dimensional non-abelianLie algebra. Let c†1, c†2 be Fermi creation operators and c1, c2 be Fermiannihilation operators. We define

A :=(c†1 c†2

)( 0 10 0

)(c1c2

)= c†1c2

B :=(c†1 c†2

)( 0 00 1

)(c1c2

)= c†2c2.


Find the commutator [A, B]. Discuss.

Problem 127. Let A, B be n × n matrices over C and [A,B] be thecommutator of A and B. Let c†1, . . . , c†n be Fermi creation operators andc1, . . . , cn be Fermi annihilation operators. We define the operators

A =(c†1 . . . c†n

)A

c1...cn

, B =(c†1 . . . c†n

)B

c1...cn

.

Show that[A, B] = [A,B].

Utilize that[c†i cj , c

†kc`] = c†i c`δjk − c

†kcjδi`.

Problem 128. Let c†1, c†2 be Fermi creation operators and annihilationand c1, c2 be Fermi annihilation operators. Consider the operators

F1 = c†1c†2, F2 = c2c1, F3 = c†1c2, F4 = c†2c1

i.e. F2 = F †1 and F4 = F †1 . Find the commutators [Fj , Fk] and anti-commutators [Fj , Fk]+.

Problem 129. Consider a vector a in C4 and the corresponding 2 × 2matrix A via the vec−1 operator

a =

a1

a2

a3

a4

⇒(a1 a3

a2 a4

)

and analogously

b =

b1b2b3b4

⇒(b1 b3b2 b4

).

Show thata∗b = tr(A∗B).

Problem 130. Let v1, v2, v3 be elements of C2. Find the conditions onv1, v2, v3 such that

v1 ⊗ v2 ⊗ v3 = v3 ⊗ v2 ⊗ v1.


Problem 131. Let n ≥ 1 and |0〉〉, |1〉, . . . , |n〉 be an orthonormal basisin Cn+1. Consider the linear operators ((n+ 1)× (n+ 1) matrices)

an =n∑j=1

√j|j − 1〉〈j|, a†n =

n∑k=1

√k|k〉〈k − 1|.

Find the commutator [an, a†n]. Note that

n∑`=0

|`〉〈`| = In+1.

Problem 132. Consider the vector v = 12 ( 1 0 1 1 0 1 )T ∈ C6.

Find a Schmidt decomposition of v over C6 = C2 ⊗ C3 and over C6 =C3 ⊗ C2.

Problem 133. Consider the Pauli spin matrices σ1, σ2, σ3. Let u0, u1, u2, u3 ∈R and

u20 + u2

1 + u22 + u2

3 = 1.

Is

U = u0I2 +3∑j=1

ujσj

a unitary matrix?

Problem 134. Consider the spin-1 matrices

S1 =1√2

0 1 01 0 10 1 0

, S2 =1√2

0 i 0−i 0 i0 −i 0

, S3 =

1 0 00 0 00 0 −1

.

Show that the eigenvalues of the matrix

M(k) = S23 + k2S2

2

are given by 1, k2, 1 + k2.

Problem 135. Find all 2 × 2 matrices A and B over C such that the4× 4 matrix

U = A⊗ I2 + I2 ⊗Bis unitary. Start of with

A =(r11e

iα11 r12eiα12

r21eiα21 r22e

iα22

), A =

(s11e

iβ11 s12eiβ12

s21eiβ21 s22e

iβ22

),


Problem 136. Let A be a 2 × 2 matrix which admit the eigenvaluesλ1 = +1 and λ2 = −1 with the corresponding normalized eigenvectors

v1 =1√2

(e−iφ/2(cos(θ/2)− sin(θ/2))eiφ/2(cos(θ/2) + sin(θ/2))

), v2 =

1√2

(−e−iφ/2(cos(θ/2) + sin(θ/2))eiφ/2(cos(θ/2)− sin(θ/2))

).

The eigenvectors form an orthonormal basis in C2. Reconstruct the matrixA from this imformation applying the spectral theorem.

Problem 137. Let S1, S2, S3 be the spin matrices for spin s = 12 , 1, 3

2 ,2, 5

2 , . . . . Thus the size of the matrices is (2s + 1) × (2s + 1). Find theeigenvalues and eigenvectors of the (2s+ 1)3 × (2s+ 1)3 matrix

S1 ⊗ S3 ⊗ S2.

Problem 138. Let σ1, σ2, σ3 be the Pauli spin matrices. Consider thefive gamma matrices

Γ1 =(σ1 02

02 σ1

), Γ2 =

(σ2 02

02 σ2

), Γ3 =

(02 σ3

σ3 02

),

Γ4 =(

02 −iσ3

iσ3 02

), Γ5 =

(−σ3 02

02 σ3

)Note that

ΓµΓν + ΓνΓµ = δµνI4.

Find the ten matrices Γ[µ,ν] defined by

Γ[µ,ν] :=12i(ΓµΓν − ΓνΓµ).

Do these fifteen matrices together with the 4 × 4 identity matrix form anorthogonal basis in the vector space of 4× 4 matrices over C?


12

tr(eiπσ1/2eiπσ2eiπσ3) = 1.

Problem 140. Let a, b ∈ R and a 6= b. Find all unitary matrices U suchthat

U

(a 00 b

)U−1 =

(b 00 a

).


Problem 141. Let J2 be the 2 × 2 matrix with all entries 1 and I2 the2× 2 identity matrix. Find the eigenvalues and normalized eigenvectors of

I2 ⊗ J2 + J2 ⊗ I2.

Extend to Jn and In.

Chapter 4

Density Operators


ρ =(

3/4√

2e−iφ/4√2eiφ/4 1/4

).

(i) Is the matrix a density matrix?(ii) If so do we have a pure state or a mixed state?(iii) Find the eigenvalues of ρ.(iv) Find tr(σ1ρ), where σ1 is the first Pauli spin matrix.

Problem 2. Let ε ∈ [0, 1]. Is

ρε =(

ε√ε(1− ε)e−iφ√

ε(1− ε)eiφ 1− ε

)with 0 ≤ φ < 2π a density matrix?

Problem 3. (i) Find a normalized state |φ〉 in the Hilbert space C2 suchthat we have the density matrix

|φ〉〈φ| = 12

(I2 +

1√2

(σ1 + σ3)).

(ii) Find a normalized state |ψ〉 in the Hilbert space C2 such that we havethe density matrix

|ψ〉〈ψ| = 12

(I2 +

1√3

(σ1 + σ2 + σ3)).

67


Problem 4. Let σ1, σ2, σ3 be the Pauli spin matrices. Find the conditionson the coefficients aj , bj and cjk such that ρ

ρ =14

(I4 + (3∑j=1

ajσj)⊗ I2 + I2 ⊗ (3∑j=1

bjσj) +3∑

j,k=1

cjkσj ⊗ σk)

is a density matrix.

Problem 5. Let m,n ∈ R3 and ‖m‖ = ‖n‖ = 1. Is the 4× 4 matrix

ρ(m,n) =14

(I4 + (n · σ)⊗ I2 + I2 ⊗ (m · σ) + (n · σ)⊗ (m · σ))

a density matrix?


ρ =

1/2 0 1/40 1/4 0

1/4 0 1/4

.

(i) Find the eigenvalues of ρ.(ii) Is ρ a density matrix? Prove or disprove. If so, is ρ a mixed or purestate?

Problem 7. Consider the normalized state

|ψ〉 = e−iφ

ei(α+γ) cos(β) sin(θ)e−i(α−γ) sin(β) sin(θ)

cos(θ)

.

Find the density matrix ρ = |ψ〉〈ψ| and the eigenvalues of ρ.

Problem 8. Let ε ∈ R and |ε| < 1. Is the 4× 4 matrix

ρ(ε) =12

1 0 0 1− ε0 0 0 00 0 0 0

1− ε 0 0 1

a density matrix?

Problem 9. Show that the 4× 4 matrices

ρ− =14

(I2 ⊗ I2 − σ1 ⊗ σ1 − σ2 ⊗ σ2 − σ3 ⊗ σ3)

Density Operators 69

ω− =14

(I2 ⊗ I2 − σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3)

ω+ =14

(I2 ⊗ I2 + σ1 ⊗ σ1 − σ2 ⊗ σ2 + σ3 ⊗ σ3)

ρ+ =14

(I2 ⊗ I2 + σ1 ⊗ σ1 + σ2 ⊗ σ2 − σ3 ⊗ σ3)

are density matrices. How they are related to the 4 Bell states

|ψ−〉 =1√2

01−10

, |φ−〉 =1√2

100−1

, |φ+〉 =1√2

1001

, |ψ+〉 =1√2

0110

?

Problem 10. Let ρ1 and ρ2 be density matrices is a finite-dimensionalHilbert space. Let λ ∈ [0, 1]. Is

λρ1 + (1− λ)ρ2

a density matrix?

Problem 11. Show that

ρ =

ε1 0 0

√ε1ε2

0 0 0 00 0 1− ε1 − ε2 0√ε1ε2 0 0ε2

where 0 ≤ ε1, ε2 ≤ 1 and ε1 + ε2 ≤ 1 is a density matrix.

Problem 12. Consider the density matrix

ρ =4∑j=1

pj |ψj〉〈ψj |, 0 ≤ pj ≤ 1,4∑j=1

pj = 1

where the |ψj〉 are the Bell states

|ψ1〉 =1√2

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉), |ψ2〉 =1√2

(|0〉 ⊗ |0〉 − |1〉 ⊗ |1〉)

|ψ3〉 =1√2

(|0〉 ⊗ |1〉+ |1〉 ⊗ |0〉), |ψ2〉 =1√2

(|0〉 ⊗ |1〉 − |1〉 ⊗ |0〉).

Write ρ using the Pauli spin matrices σ1, σ2, σ3, the 2× 2 identity matrixI2 and the Kronecker product.


Problem 13. Consider the Hilbert space Cn. Let ρ be a density matrix,i.e. ρ ≥ 0 and tr(ρ) = 1. The mean value of an observable A (hermitiann× n matrix) is given by

〈A〉 = tr(ρA).

If the density ρ is unkown, then it may be determined using n2 mean values〈A(k)〉 (k = 1, 2, . . . , n2) obtained from measurement if the set A(k) is abasis in the space of all hermitian n× n matrices.(i) Let n = 2,

A = σ2 =(

0 −ii 0

)and

tr(ρA) = 0, tr(ρA2) = 1, tr(ρA3) = 0, tr(ρA4) = 1.

Find the density matrix.(ii) Let n = 2 and

tr(ρI2) = 1, tr(ρσ1) = −1, tr(ρσ2) = 0, tr(ρσ3) = 0.

Find ρ.

Problem 14. (i) Let x1, x2, x3 ∈ R. Consider the hermitian matrix

ρ =12

(1 + x3 x1 − ix2

x1 + ix2 1− x3

).

Find the condition on x1, x2, x3 such that ρ2 = ρ. Is this matrix then adensity matrix?(ii) Let ε ∈ [0, 1]. Consider the hermitian matrix

ρ =12

ε+ x3 0 x1 − ix2

0 2− 2ε 0x1 + ix2 0 ε− x3

.

Find the condition on x1, x2, x3 and ε such that ρ2 = ρ.


ρ =4∑j=1

pj |ψj〉〈ψj |, 0 ≤ pj ≤ 1,4∑j=1

pj = 1

where the |ψj〉 are the Bell states

|ψ1〉 =1√2

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉), |ψ2〉 =1√2

(|0〉 ⊗ |0〉 − |1〉 ⊗ |1〉)


|ψ3〉 =1√2

(|0〉 ⊗ |1〉+ |1〉 ⊗ |0〉), |ψ2〉 =1√2

(|0〉 ⊗ |1〉 − |1〉 ⊗ |0〉).

Write ρ using the Pauli spin matrices and the 2× 2 identity matrix I2.

Problem 16. Let A be a nonzero n×n matrix over C. Consider the map

A→ ρ =AA∗

tr(AA∗).

(i) Show that ρ is a density matrix.(ii) Show that ρ is invariant under the map A→ AU , where U is an n× nunitary matrix.(iii) Is AA∗ = A∗A in general?(iv) Consider the map

A→ σ =A∗A

tr(A∗A).

Is σ = ρ? Prove or disprove.

Problem 17. Consider the state

|ψ〉 =(

cos(θ)eiφ sin(θ)

)and the density matrix

ρ(0) = |ψ〉〈ψ|.

Given the Hamilton operator

H = ~ωσ1.

Solve the von Neumann equation for the given ρ(0) and the given H. Thevon Neumann equation is given by

i~dρ

dt= [H, ρ](t)

with the solutionρ(t) = e−iHt/~ρ(0)eiHt/~.

Problem 18. Consider the Bell state

|ψ〉 =1√2

1001


and the density matrixρ = |ψ〉〈ψ|.

Given the Hamilton operator

H = ~ωσ1 ⊗ σ1.

Solve the von Neumann equation for given ρ and the given H. The vonNeumann equation is given by

i~dρ

dt= [H, ρ](t)

with the solutionρ(t) = e−iHt/~ρ(0)eiHt/~.

Problem 19. (i) Is the 2× 2 matrix

ρ =(

1/2 −i/2i/2 1/2

)a density matrix?(ii) Can one find a state |ψ〉 in C2 such that

ρ = |ψ〉〈ψ| ?

(iii) Are the 4× 4 matrices

ρ⊗ ρ, ρ⊕ ρ, ρ ? ρ

density matrices? Here ⊗ denotes the Kronecker product, ⊕ the direct sumand ? operation which is defined for two 2× 2 matrices A and B as

A ? B =

a11 0 0 a12

0 b11 b12 00 b21 b22 0a21 0 0 a22

.

Problem 20. Let |0〉, |1〉 be the standard basis in C2. Consider theentangled state

|ψ〉 =1√2

(|0〉 ⊗ |1〉 − |1〉 ⊗ |0〉)

with the density matrix ρ = |ψ〉〈ψ|. Find the reduced density matrix ρ1.Discuss.



ρ =12

(1 + r cos(θ) r sin(θ)e−iφ

r sin(θ)eiφ 1− r cos(θ)

)a density matrix? What are the conditions on r, θ, φ?

Problem 22. Consider a finite dimensional Hilbert space of dimensiond on which the density matrix ρ acts. A quantum operation is representedby a completely positive and trace preserving map Λ which takes the form

Λ(ρ) =d2∑j=1

VjρV∗j .

Show that the trace preserving condition tr(Λ(ρ)) = tr(ρ) is equivalent tothe equality

d2∑j=1

V ∗j Vj = I.

Problem 23. Let S be the set of unit vectors in the Hilbert space Cn.Let u ∈ S. A function µ(u) from S to R is called a generalized probabilitymeasure if the following two conditions hold: (i) for u ∈ S, 0 ≤ µ(u) ≤ 1,(ii) if u1, . . . , un form an orthonormal basis in the Hilbert space Cn, then∑nj=1 µ(uj) = 1.

Let n ≥ 3. Then any generalized probability measure µ on Cn has the form

µ(ρ) = tr(ρuu∗)

for a uniquely defined density matrix ρ. (Gleason 1957)

(i) Consider the Hilbert space C3, the orthonormal basis

u1 =1√2

101

, u2 =

010

, u3 =1√2

10−1

and the density matrix

ρ =13

1 1 11 1 11 1 1

.


Find µ(u1), µ(u2), µ(u3).

(ii) Consider the Hilbert space C4, the orthonormal basis

u1 =1√2

eiφ

00eiφ

, u2 =1√2

eiφ

00−eiφ

, u3 =1√2

0eiφ

eiφ

0

, u4 =1√2

0eiφ

−eiφ0

and the density matrix

ρ =14

1 1 1 11 1 1 11 1 1 11 1 1 1

.

Find µ(u1), µ(u2), µ(u3), µ(u4).

Problem 24. Consider the two 2× 2 density matrices

ρ =(ρ11 ρ12

ρ21 ρ22

), σ =

(σ11 σ12

σ21 σ22

).

Is the 4× 4 matrix

ρ ? σ =12

ρ11 0 0 ρ12

0 σ11 σ12 00 σ21 σ22 0ρ21 0 0 ρ22

a density matrix?


ρ =12

(1 11 1

)and let σ1, σ2, σ3 be the Pauli spin matrices. Calculate the commutators[ρ, σ1], [ρ, σ2], [ρ, σ3] and discuss.


ρ =12

(1 11 1

).

Find the Cayley transform

U = (ρ− iI2)(ρ+ iI2)−1


and then the commutator [ρ, U ]. Discuss

Problem 27. Consider the Pauli spin matrices σ1, σ2, σ3. Find thenormalized eigenvectors

v11, v12, v21, v22, v31, v32

and construct the six density matrices (pure states)

ρjk = vjkv∗jk

where j = 1, 2, 3 and k = 1, 2. Calculate commutators [ρjk, ρj′k′ ] and anti-commutators [ρjk, ρj′k′ ]+ and compare to the commutators [σj , σk] andanti-commutators [σj , σk]+.

Problem 28. Does the density matrix

ρ =14

1 0 0 10 1 1 00 1 1 01 0 0 1

represent a pure or mixed state?

Problem 29. Consider the Hamilton operator acting in the Hilbert spaceC4

H = ~ω1(σ3 ⊗ I2 + I2 ⊗ σ3) + ~ω2(σ1 ⊗ σ1)

where ω1, ω2 > 0.(i) Find the (real) eigenvalues (the matrix H is hermitian) E0, E1, E2, E3

with the ordering E0 ≤ E1 ≤ E2 ≤ E3.(ii) Find the corresponding normalized eigenvectors |E0〉, |E1〉, |E2〉, |E3〉.Are the eigenvectors separable?(iii) Calculate the partition function Z(β) (β = 1/(kBT )) defined by

Z(β) :=3∑j=0

exp (−βEj) .

(iv) We define

pj(β) :=e−βEj

Z(β), j = 0, 1, 2, 3.

Calculate the density matrix

ρ(β) =3∑j=0

pj(β)|Ej〉〈Ej |.


Do we have a mixed or pure state? Study the cases ρ(∞) and ρ(0).

Problem 30. Let ε1, ε2, ε3 ∈ R. Consider the hermitian matrix

ρ(ε1, ε2, ε3) =14

1 + ε3 0 0 ε1 + ε2

0 1− ε3 ε1 − ε2 00 ε1 − ε2 1− ε3 0

ε1 + ε2 0 0 1 + ε3

.

What is the condition such that ρ(ε1, ε2, ε3) is a density matrix? For theeigenvalues of the matrix ρ(ε1, ε2, ε3) we find

λ1 =14

(1 + ε1 + ε2 + ε3), λ2 =14

(1 + ε1 − ε2 − ε3),

λ3 =14

(1− ε1 + ε2 − ε3), λ4 =14

(1− ε1 − ε2 + ε3).

Problem 31. Consider the Hilbert space Cn. Let ρ be a density matrixin this Hilbert space and H and K be two hermitian n× n matrices. Onedefines

〈H〉 := tr(ρH), 〈H2〉 := tr(ρH2)

and analogously for K. Let

∆H :=√〈H2〉 − 〈H〉2, ∆K :=

√〈K2〉 − 〈K〉2.

Then we have the uncertainty relation

(∆H)(∆K) ≥ 12|〈i[H,K]〉| .

Let

ρ =12

1 0 00 0 00 0 1

and

H =

0 1 01 2 00 0 0

, K =

0 i 0−i 0 00 0 0

.

Show that the uncertainty relation becomes an equality for the given ρ, Hand K.

Problem 32. Let Id be the d× d identity matrix. Consider the matrix

ρ =1d

(Id +K)


where K is a hermitian d × d matrix with all diagonal entries equal to 0.What is the condition on such a K such that ρ is a density matrix?

Problem 33. Let α ∈ [0, 1]. Show that

ρ(α) =14

1− α 0 0 0

0 1 + α −2α 00 −2α 1 + α 00 0 0 1− α

is a density matrix (so-called Werner state). Find the eigenvalues andeigenvectors of ρ.

Problem 34. Are the matrices

ρ(θ) =12

2 sin2 θ 0 0 0

0 cos2 θ cos2 θ 00 cos2 θ cos2 θ 00 0 0 0

and

ρ(θ) =12

2 cos2 θ 0 0 0

0 sin2 θ sin2 θ 00 sin2 θ sin2 θ 00 0 0 0

density matrices? Prove or disprove. If so, do we have a mixed or purestate?

Problem 35. Is the matrix

ρ =14

1 0 0 10 1 1 00 1 1 01 0 0 1

≡ 14

(I2 ⊗ I2 + σ1 ⊗ σ1)

a density matrix?

Problem 36. Can the density matrix

ρ =14

1 0 1 00 1 0 11 0 1 00 1 0 1

be written as a Kronecker product of two 2× 2 density matrices?


Problem 37. Let ρ be a density matrix given as an n× n matrix and Ube an n× n unitary matrix. Then UρU−1 is again a density matrix. Let

ρ =14

1 0 0 10 1 1 00 1 1 01 0 0 1

, U =1√2

1 0 0 10√

2 0 00 0

√2 0

−1 0 0 1

.

Find the density matrix UρU−1.

Problem 38. Let

ρ∓1 =12

0 0 0 00 1 ∓1 00 ∓1 1 00 0 0 0

, ρ∓2 =12

1 0 0 ∓10 0 0 00 0 0 0∓1 0 0 1

be the four density matrices for the Bell states.(i) Let t ∈ [0, 1]. Is the convex combination

ρ = tρ1 + (1− t)ρ2

a density matrix?(ii) The Hilbert-Schmidt distance d(ρ1, ρ2) is given by

d(ρ1, ρ2) :=√

tr((ρ1 − ρ2)2).

Find d(ρ1, ρ2) for the given density matrices.

Problem 39. Let σ1, σ2, σ3 be the Pauli spin matrices. Find the condi-tions on the real coefficients rj , uj , tjk (j, k = 1, 2, 3) such that

ρ =14

(I2 ⊗ I2 +3∑j=1

rjσj ⊗ I2 +3∑j=1

ujI2 ⊗ σj +3∑j=1

3∑k=1

tjkσj ⊗ σk)

is a density matrix. Note that since tr(σj) = 0 for j = 1, 2, 3 we havetr(ρ) = 1.

Problem 40. The variance of an observable A and a density operator ρin a Hilbert space H is defined as

V (ρ,A) := tr(ρA2)− (tr(ρA))2.

Let |ψ〉 be a normalized state in the Hilbert space H. Show that if ρ =|ψ〉〈ψ| (pure state) we obtain

V (|ψ〉〈ψ|, A) = 〈ψ|A2|ψ〉 − 〈ψ|A|ψ〉2.


Problem 41. (i) Consider the spin-1 matrices

S1 =1√2

0 1 01 0 10 1 0

, S2 =i√2

0 −1 01 0 −10 1 0

, S3 =

1 0 00 0 00 0 −1

which are hermitian and traceless. Let I3 be the 3× 3 unit matrix. Let

v =

v1v2v3

be a vector in R3 with ‖v‖ ≤ 1. Is the matrix

ρ =13

(I3 +3∑j=1

vjSj)

a density matrix. Obviously this matrix is hemitian and has trace 1, butare all the eigenvalues are non-zero?(ii) Is the matrix

ρ =19

(I3 ⊗ I3 +3∑j=1

vjSj ⊗ Sj)

a density matrix?

Problem 42. (i) Consider the three 3× 3 matrices

ρ1 =

1 0 00 0 00 0 0

, ρ2 =13

1 1 11 1 11 1 1

, ρ3 =13

1 0 00 1 00 0 1

Which of these matrices are density matrices?(ii) For the matrices which represent density matrices found out whether itrepresents of pure state or mixed state. If it is pure state find the state |ψ〉in the Hilbert space C3 such that ρ = |ψ〉〈ψ|.

Problem 43. Consider a mixture of 25% of the pure state (1, 0)T , 25%of the pure state (0, 1)T and 50% of the pure state 1√

2(1, 1)T described by

the density matrix

ρ =14

(10

)( 1 0 ) +

14

(01

)( 0 1 ) +

12

1√2

(11

)1√2

( 1 1 ) .

Find the spectral representation of ρ. Use the spectral representation of ρ tofind another mixture of pure states with the same (measurement) statisticalproperties as ρ.



|ψ〉 =(

cos(θ)eiφ sin(θ)

)in the Hilbert space C2, where φ, θ ∈ R. Let ρ(t = 0) = ρ(0) = |ψ〉〈ψ| bea density matrix at time t = 0. Given the Hamilton operator H = ~ωσ1.Solve the von Neumann equation to find ρ(t).

Problem 45. Let H1 and H2 be two Hilbert spaces and H1 ⊗H2 be theproduct Hilbert space. Let ρ be a density operators of the Hilbert spaceH1 ⊗H2. Show that if one of the reduced density operators trH2(ρ) = ρ1

or trH1(ρ) = ρ2 is pure, then ρ = ρ1 ⊗ ρ2. If both ρ1 and ρ2 are pure, thenρ is pure too.

Problem 46. Let α ∈ [0, 1] and φ ∈ R. Is

ρ(α, φ) =12

α 0 e−iφ

0 2− 2α 0eiφ 0 α

a density matrix?

Problem 47. Let |φj〉 (j = 1, . . . , d) be an orthonormal basis in theHilbert space Cd. Is

ρ =1d

d∑j,k=1

|φj〉〈φk|

a density matrix.

Problem 48. (i) Consider the density matrix (pure state)

ρ =(

1 00 0

)=(

10

)( 1 0 ) .

Apply the Cayley transform to find the corresponding unitary matrix. Dis-cuss.(ii) Consider the density matrix (pure state)

ρ =12

(1 −ii 1

)=

1√2

(1i

)1√2

( 1 −i ) .

Apply the Cayley transform to find the corresponding unitary matrix. Dis-cuss.


(iii) Consider the n× n density matrix (pure state)

ρ =1n

1 1 · · · 11 1 · · · 1...

.... . .

...1 1 · · · 1

.

Apply the Cayley transform to find the corresponding unitary matrix. Dis-cuss.(iv) Consider the mixed state

ρ =(

1/2 00 1/2

).

Apply the Cayley transform to find the corresponding unitary matrix. Dis-cuss.

Problem 49. Let |n〉 (n = 0, 1, . . . , N) be the standard basis in CN+1.Consider the states

|θ, φ〉 =N∑n=0

(N

n

)1/2

(cos(θ/2))N−n(sin(θ/2))ne−inφ|n〉.

Consider the density matrix

ρ(t) =N∑n=0

N∑m=0

C∗m(t)Cn(t)|n〉〈m|.

Show that

Q(θ, φ, t) :=N + 1

4π〈θ, φ|ρ(t)|θ, φ〉

=N + 1

4π

N∑m=0

N∑n=0

(N

m

)1/2(N

n

)1/2

C∗m(t)Cn(t)

×(cos(θ/2))2N−m−n(sin(θ/2))m+ne−i(m−n)φ.

Problem 50. A quantum system is described by the density matrix ρ apositive semi-definite operator with tr(ρ) = 1. The observable is describedby self-adjoint operators A and their expectation values are given by tr(Aρ).Consider the Hilbert space C2, the density matrices

ρ1 =(

1/2 00 1/2

), ρ2 =

(1/2 1/21/2 1/2

)


and the hermitian 2× 2 matrix

σ2 =(

0 −ii 0

).

Findtr(ρ1σ2), tr(ρ2σ2), tr((ρ1 ⊗ ρ2)(σ2 ⊗ σ2)).

Problem 51. Let t ∈ [0, 1]. Let ρ1, ρ2 be two density matrices.(i) Is the convex combination

ρ = tρ1 + (1− t)ρ2

a density matrix.(ii) If so apply it to the density matrices which are related to the Bell states

ρ1 =12

0 0 0 00 1 1 00 1 1 00 0 0 0

, ρ2 =12

1 0 0 10 0 0 00 0 0 01 0 0 1

.

(iii) The Hilbert-Schmidt distance d(ρ1, ρ2) is given by

d(ρ1, ρ2) =√

tr((ρ1 − ρ2)2).

Find the distance for the two density matrices given in (ii).

Problem 52. Let σ1, σ2, σ3 be the Pauli spin matrices and I2 the 2× 2identity matrix.(i) Show that the four matrices

ρ1 =12

(I2 + σ3), ρ2 =12

(I2 − σ3), ρ3 =12

(I2 + σ1), ρ4 =12

(I2 + σ2)

are density matrices.(ii) Show that the four matrices ρ1, ρ2, ρ3, ρ4 form a basis in the Hilbertspace M2(C) with the scalar product 〈A,B〉 := tr(AB∗).

Problem 53. Consider the Hilbert space C2 and the projection matrices

Π1 =12

(1 11 1

), Π2 =

12

(1 −1−1 1

).

Find Π1Π2 and Π1 + Π2. Let

ρ(θ) =(

cos(θ)sin(θ)

)( cos(θ) sin(θ) ) .


Findtr(ρΠ1), tr(ρΠ2).

Problem 54. Let σ1, σ2, σ3 be the Pauli spin matrices and I2 be the2× 2 identity matrix.(i) Show that the four 2× 2 matrices

ρ1 =12

(I2 + σ3), ρ2 =12

(I2 − σ3), ρ3 =12

(I2 + σ1), ρ4 =12

(I2 + σ2)

are density matrices in the Hilbert space C2.(ii) Show that the four matrices form a basis in the Hilbert space M2(C)with the scalar product 〈A,B〉 := tr(AB∗).(iii) Are the matrices ρ1⊗ρ1, ρ2⊗ρ2, ρ3⊗ρ3 density matrices in the Hilbertspace M2(C).

Problem 55. Let A, B be hermitian n × n matrices and 〈A〉 = tr(Aρ)with ρ an n × n density matrix or 〈A〉 = 〈ψ|A|ψ〉 with |ψ〉 a normalizedstate in Cn. Let

σ2A := 〈A2〉 − 〈A〉2, σ2

B := 〈B2〉 − 〈B〉2.

Then

σ2Aσ

2B ≥

∣∣∣∣12 〈[A,B]+〉 − 〈A〉〈B〉∣∣∣∣2 +

∣∣∣∣12 〈[A,B]〉∣∣∣∣2 .

(i) Let n = 2 and A = σ1, B = σ2 and

ρ =12

(1 11 1

).

Find the left and right hand side of the inequality.(ii) Let n = 2 and A = σ1, σ2 and

|ψ〉 =1√2

(11

).

Find the left and right hand side of the inequality.

Problem 56. Let

|ψ0〉 =1√2

(11

), |ψ1〉 =

1√2

(1i

).

Calculateρ = (|ψ0〉〈ψ0|)⊗ (|ψ1〉〈ψ1|).


Is ρ a density matrix?

Problem 57. Let A, B be positive semidefinite n× n matrices. Then

det(A+B) ≥ det(A) + det(B), tr(AB) ≤ tr(A)tr(B).

Let

A =13

1 1 11 1 11 1 1

, B =13

1 0 00 1 00 0 1

.

Both matrices are density matrices, with A a pure state and B a mixedstate. Calculate the left- and right-hand side of the two inequality. Discuss.

Problem 58. Let v1, v2, . . . , vn be an orthonormal basis in Cn and µ1,µ2, . . . , µn be nonnegative numbers such that

∑j=1 µj = 1. Is

ρ =n∑j=1

µjvjv∗1

a density matrix? If so would it cover pure and mixed states?

Problem 59. Let

ρ =(ρ11 ρ12

ρ21 ρ22

)be a density matrix. Is

ρ =

ρ11 0 ρ12

0 0 0ρ21 0 ρ22

a density matrix?

Problem 60. Let v1, . . . , vn (column vectors) be an orthonormal basisin Cn and let λ1, . . . , λn be nonnegative real numbers with

n∑j=1

λj = 1.

Is

ρ =n∑j=1

λjvjv∗j

a density matrix?


Problem 61. Consider the Hilbert space C4 and the states

|ψ1〉=1√2

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉 =1√2

1001

|ψ2〉=1√2

(|0〉 ⊗ |1〉+ |1〉 ⊗ |0〉) =1√2

0110

.

Let γ ∈ [−1, 1]. Consider the density matrix

ρ =12

(1 + γ)|ψ1〉〈ψ2|.

Let

M =(

0 e−iθ1

eiθ1 0

)⊗(

0 e−iθ2

eiθ2 0

).

Problem 62. Consider the Hilbert space Cn. Let ρ be a density matrix.Then the diagonal part σ of ρ is also a density matrix. Let f be a convexfunction on the interval [0, 1]. Then (Klein inequality)

tr(f(ρ)) ≥ tr(f(σ)).

Let

ρ =12

(1 11 1

)f(x) = x2.

Calculate the right-hand side and left-hand side of the inequality.

Problem 63. Let ρ1, ρ2 be density matrices. Then one has (Kleininequality)

tr(f(ρ1)− f(ρ2)− (ρ1 − ρ2)f ′(ρ2)) ≥ 0

where f : (0,∞) is a convex function. Consider

ρ1 =12

(1 11 1

), ρ2 =

12

(1 −1−1 1

)and f(x) = x2. Calculate the left-hand side of the inequality.


ρ =13

1 1 −11 1 −1−1 −1 1


is a density matrix (pure state). Find the normalized vector v in C3 suchthat

ρ = vv∗.


ρ =12

(1 −1−1 1

)and let A be an 2× 2 real symmetric matrix. Assume that

tr(ρA) = −1, tr(ρA2) = 1.

Reconstruct the matrix from this information.

Problem 66. Let p = (p1, p2, . . . , pN ) be a probability vector, i.e. pj ≥ 0and

∑Nj=1 pj = 1. Let U be a unitary N ×N matrix. Then

ρ(U,p) = U

p1 0 · · · 00 p2 0 · · ·· · · · · · · · · · · ·0 0 · · · pN

U∗

is a density matrix. Find the density matrix for the case N = 2 and withp = (1/4, 3/4) and

U =1√2

(1 11 −1

).

Problem 67. Let x1, x2, x3, x4 ∈ [−1, 1]. Is

ρ =12

(1 + x3 x1 − ix2

x1 − ix2 1− x3

)a density matrix?

Chapter 5

Partial Trace

Problem 1. Consider the finite-dimensional Hilbert spaces H1 = Cn1

and H2 = Cn2 . Let H1⊗H2 be the product Hilbert space. Let |ψ〉 and |φ〉be states in the product Hilbert space H1 ⊗H2. Show that if

trH2(|ψ〉〈ψ|) = trH2(|φ〉〈φ|)

then there exists a unitary matrix U acting in the Hilbert space H2 suchthat

|ψ〉 = (In1 ⊗ U)|φ〉where In1 is the identity matrix in the Hilbert space H1.

Problem 2. Consider the GHZ-state in the Hilbert space C8 (C8 ∼=C2 ⊗ C2 ⊗ C2)

|GHZ〉 =1√2

((10

)⊗(

10

)⊗(

10

)+(

01

)⊗(

01

)⊗(

01

)).

Then the density matrix is given by the 8× 8 matrix

ρ = |GHZ〉〈GHZ| = 12

1 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 0 0 0 0 0 0 1

.

87


(i) Calculate the partial trace ρAB = trC(ρ) with the basis

I4 ⊗(

10

), I4 ⊗

(01

).

(ii) Calculate the partial trace ρA = trB(ρAB) with the basis

I2 ⊗(

10

), I2 ⊗

(01

).

Problem 3. Consider the product Hilbert space `2(N0) ⊗ C2s+1, wheres = 1/2, 1, 3/2, 2, . . . is the spin. Find the partial trace over C2s+1.

Chapter 6

Reversible Logic Gates

Problem 1. Find the truth table for the boolean function

f(a, a′, b, b′) = (a · b′)⊕ (a′ · b).

Problem 2. The Feynman gate is a 2 input/2 output gate given by

x′1 = x1

x′2 = x1 ⊕ x2

(i) Give the truth table for the Feynman gate.(ii) Show that copying can be implemented using the Feynman gate.(iii) Show that the complement can be implemented using the Feynmangate.(iv) Is the Feynman gate invertible?

Problem 3. Consider the 3-input/3-output gate given by

x′1 = x1

x′2 = x1 ⊕ x2

x′3 = x1 ⊕ x2 ⊕ x3.

(i) Give the truth table.(ii) Is the transformation invertible.

89



x′1 = x1

x′2 = x1 ⊕ x2

x′3 = x3 ⊕ (x1 · x2).

(i) Give the truth table.(ii) Is the gate invertible?


x′1 = x1 ⊕ x3

x′2 = x1 ⊕ x2

x′3 = (x1 · x2)⊕ (x1 · x3)⊕ (x2 · x3).


Problem 6. Consider the Toffoli gate

T : 0, 13 → 0, 13, T (a, b, c) := (a, b, (a · b)⊕ c)

where a is the NOT operation, + is the OR operation, · is the AND oper-ation and ⊕ is the XOR operation.

1. Express NOT (a) exclusively in terms of the TOFFOLI gate.

2. Express AND(a, b) exclusively in terms of the TOFFOLI gate.

3. Express OR(a, b) exclusively in terms of the TOFFOLI gate.

4. Show that the TOFFOLI gate is invertible.

Thus the TOFFOLI gate is universal and reversible (invertible).

Problem 7. Consider the Fredkin gate

F : 0, 13 → 0, 13, F (a, b, c) := (a, a · b+ a · c, a · c+ a · b)

where a is the NOT operation, + is the OR operation, · is the AND oper-ation and ⊕ is the XOR operation.

1. Express NOT (a) exclusively in terms of the FREDKIN gate.

2. Express AND(a, b) exclusively in terms of the FREDKIN gate.

Reversible Logic Gates 91

3. Express OR(a, b) exclusively in terms of the FREDKIN gate.

4. Show that the FREDKIN gate is invertible.

Thus the FREDKIN gate is universal and reversible (invertible).

Problem 8. The Toffoli gate T(x1, x2;x3) has 3 inputs (x1, x2, x3) andthree outputs (y1, y2, y3) and is given by

(x1, x2, x3)→ (x1, x2, x3 ⊕ (x1 · x2))

where x1, x2, x3 ∈ 0, 1 , ⊕ is the XOR-operation and · the AND-operation.Give the truth table.

Problem 9. A generalized Toffoli gate T(x1, x2, . . . , xn;xn+1) is a gatethat maps a boolean pattern (x1, x2, . . . , xn, xn+1) to

(x1, x2, . . . , xn, xn+1 ⊕ (x1 · x2 · . . . · xn))

where ⊕ is the XOR-operation and · the AND-operation. Show that thegeneralized Toffoli gate includes the NOT-gate, CNOT-gate and the originalToffoli gate.

Problem 10. The Fredkin gate F(x1;x2, x3) has 3 inputs (x1, x2, x3) andthree outputs (y1, y2, y3). It maps boolean patterns

(x1, x2, x3)→ (x1, x3, x2)

if and only if x1 = 1, otherwise it passes the boolean pattern unchanged.Give the truth table.

Problem 11. The generalized Fredkin gate F(x1, x2, . . . , xn;xn+1, xn+2)is a gate is the mapping of the boolean pattern

(x1, x2, . . . , xn, xn+1, xn+2)→ (x1, x2, . . . , xn, xn+2, xn+1)

if and only if the boolean product x1 ·x2 · . . . ·xn = 1 (· is the bitwise ANDoperation), otherwise the boolean pattern passes unchanged. Let n = 2and (x1, x2, x3, x4) = (1, 1, 0, 1). Find the output.

Problem 12. Is the gate (a, b, c ∈ 0, 1 )

(a, b, c)→ (a, a · b⊕ c, a · c⊕ b)

reversible?


Problem 13. Prove that the Fredkin gate is universal. A set of gates iscalled universal if we can build any logic circuits using these gates assumingbit setting gates are given.

Problem 14. The half-adder is given by

S =A⊕BC =A ·B.

Construct a half-adder using two Toffoli gates.


x′1 = x1 ⊕ x3

x′2 = x1 ⊕ x2

x′3 = (x1 + x2)⊕ (x1 + x3)⊕ (x2 + x3).



x′1 = x1

x′2 = x2

x′3 = x3

x′4 = x4 ⊕ x1 ⊕ x2 ⊕ x3.



x′1 = x1 ⊕ x3

x′2 = x2 ⊕ x3 ⊕ (x1 · x2)⊕ (x2 · x3)x′3 = x1 ⊕ x2 ⊕ x3

x′4 = x4 ⊕ x3 ⊕ (x1 · x2)⊕ (x2 · x3) .


Problem 18. Show that one Fredkin gate

(a, b, c)→ (a, a · b+ a · c, a · c+ a · b)

Reversible Logic Gates 93

is sufficient to implement the XOR gate. Assume that either b or c areavailable.

Problem 19. Show that the map f : 0, 13 → 0, 13

abc xyz000 -> 000100 -> 100010 -> 101110 -> 011001 -> 001101 -> 010011 -> 110111 -> 111

is invertible. The map describes a reversible half-adder. If c = 0, then x isthe first digit of the sum a+ b and y is the carry bit. If c = 1, then z is thefirst digit of the sum a+ b+ c and y is the carry bit.

Problem 20. Show that the Toffoli gate which maps

|a〉 ⊗ |b〉 ⊗ |c〉 7→ |a〉 ⊗ |b〉 ⊗ |c⊕ (a · b)〉

can simulate the FANOUT and the NAND gate.

Problem 21. (i) Let x1, x2 ∈ 0, 1. Let ⊕ be the XOR operation. Showthat

(x1, x2) 7→ (x1 ⊕ 1, x1 ⊕ x2)

is a 2-bit reversible gate.(ii) Let

|0〉 =(

10

), |1〉 =

(01

).

Find the 4× 4 permutation matrix P such that

P (|x1〉 ⊗ |x2〉) = |x1 ⊕ 1〉 ⊗ |x1 ⊕ x2〉 .

(iii) Show that(x1, x2) 7→ (x1 ⊕ x2, x2 ⊕ 1)

is a 2-bit reversible gate.(iv) Find the 4× 4 permutation matrix P such that

P (|x1〉 ⊗ |x2〉) = |x1 ⊕ x2〉 ⊗ |x2 ⊕ 1〉.


(v) Given a 4 × 4 permutation matrix (as a quantum gate). How can oneconstruct a corresponding 2-bit reversible gate? Apply it to the permuta-tion matrix

P =

0 1 0 00 0 1 00 0 0 11 0 0 0

.

Problem 22. The NOT, AND and OR gate form a universal set of oper-ations (gates) for boolean algebra. The NAND operation is also universalfor boolean algebra. However these sets of operations are not reversible setsof operations. Consider the Toffoli and Fredkin gates

TOFFOLI : 0, 13 → 0, 13, TOFFOLI(a, b, c) = (a, b, (a · b)⊕ c)

FREDKIN : 0, 13 → 0, 13, FREDKIN(a, b, c) = (a, a·c+a·b, a·b+a·c)

where a is the NOT operation, + is the OR operation, · is the AND oper-ation and ⊕ is the XOR operations.

1. Express NOT(a) exclusively in terms of the TOFFOLI gate.2. Express NOT(a) exclusively in terms of the FREDKIN gate.3. Express AND(a,b) exclusively in terms of the TOFFOLI gate.4. Express AND(a,b) exclusively in terms of the FREDKIN gate.5. Express OR(a,b) exclusively in terms of the TOFFOLI gate.6. Express OR(a,b) exclusively in terms of the FREDKIN gate.7. Show that the TOFFOLI gate is reversible.8. Show that the FREDKIN gate is reversible.Thus the TOFFPLI and FREDKIN gates are eachuniversal and reversible(invertible).

Chapter 7

Unitary Transformationsand Quantum Gates

Problem 1. Consider the compact Lie group SU(4). Let U ∈ SU(4).Then the 4× 4 matrix U can be factorized as follows

U = (V1 ⊗ V2) exp

i

2

3∑j=1

θjσj ⊗ σj

(V3 ⊗ V4)

where V1, V2, V3, V4 ∈ SU(2) and θj ∈ R. Let

S =

0 0 0 10 0 1 00 1 0 01 0 0 0

.

Show that S ∈ SU(4). Find the factorization given above for S.

Hint. Since [σj ⊗ σj , σk ⊗ σk] = 04 we can write

exp

i

2

3∑j=1

θjσj ⊗ σj

≡ exp(iθ12σ1 ⊗ σ1

)exp

(iθ22σ2 ⊗ σ2

)exp

(iθ32σ3 ⊗ σ3

).

95



B =1√2

1 0 0 10 1 −1 00 1 1 0−1 0 0 1

.

(i) Show that B is invertible and find B−1. Is B unitary?(ii) Express B2 using the Pauli spin matrices, an overall phase and theKronecker product.(iii) Find a 4× 4 matrix A such that B = exp(iA).(iv) Can one find a positive integer n such that Bn = I4?(v) Show that

B =1√2

(I4 +B2).

Problem 3. Consider the state |ψ〉 in the Hilbert space C9

|ψ〉 =1√3

100

⊗ 1

00

+

010

⊗ 0

10

+

001

⊗ 0

01

.

Is the state invariant under U ⊗ U , where U is the 3× 3 unitary matrix

U =

0 0 10 1 01 0 0

.

Problem 4. (i) The Schrodinger equation is given by

i~dψ

dt= Hψ(t) (1)

with ψ(0) the initial value. The evolution of ψ(t) is determined by

ψ(t) = U(t)ψ(0) (2)

where U(t) is a unitary evolution operator and U(0) = I. Show that

i~dU(t)dt

ψ(0) = HU(t)ψ(0). (3)

(ii) Assume that H = ~ωσ3. Find U(t).

Unitary Transformations and Quantum Gates 97

Problem 5. Let I2 be the 2 × 2 identitity matrix and σ1 be the Paulispin matrix and |0〉, |1〉 be the standard basis. The CNOT-gate can berepresented as

UCNOT = |0〉〈0| ⊗ I2 + |1〉〈1| ⊗ σ1.

Is UCNOT hermitian? Is U2CNOT = I4?

Problem 6. Let U be an n×n unitary matrix. Show that if the bipartitestates |ψ〉, |φ〉 ∈ Cn ⊗ Cm satisfy

|φ〉 = (U ⊗ Im)|ψ〉

then the ranks of the corresponding reduced density matrices satisfy

r(ρψ1 ) ≥ r(ρφ1 ), r(ρψ2 ) ≥ r(ρφ2 ).


VH =1√2

(1 11 −1

)⊗ I2, VM =

(0 11 0

)⊗ I2

VC =(

0 00 1

)⊗ U2 +

(eiχ 00 0

)⊗ I2

where U2 is an arbitrary 2 × 2 unitary matrix and χ ∈ R. Consider the4× 4 density matrix

ρin =(

10

)( 1 0 )⊗ ρ2 ≡

(1 00 0

)⊗ ρ2

where ρ2 is an arbitrary 2× 2 density matrix. Find the density matrix

ρout = VHVMVCVHρinV∗HV∗CV∗MV

∗H .

Problem 8. The n-qubit Pauli group is defined by

Pn := I2, σ1, σ2, σ3 ⊗n ⊗ ±1, ±i

where σ1, σ2, σ3 are the 2×2 Pauli matrices and I2 is the 2×2 identity ma-trix. The dimension of the Hilbert space under consideration is dimH = 2n.Thus each element of the Pauli group Pn is (up to an overall phase ±1, ±i)a Kronecker product of Pauli matrices and 2 × 2 identity matrices actingon n qubits.


The n-qubit Clifford group Cn is the normalizer of the Pauli group. A2n × 2n unitary matrix U acting on n qubits is an element of the Cliffordgroup iff

UMU∗ ∈ Pn for each M ∈ Pn.

This means the unitary matrix U acting by conjugation takes a Kroneckerproduct of Pauli matrices to Kronecker product of Pauli matrices. Anelement of the Clifford group is defined as this action by conjugation, sothat the overall phase of the unitary matrix U is not relevant. In otherwords the Clifford group is the group of all matrices that leave the Pauligroup invariant.(i) What is order of the n-qubit Pauli group?(ii) Show that the single-qubit Hadamard gate

UH =1√2

(1 11 −1

)and the single-qubit phase gate

UP =(

1 00 i

)are elements of the Clifford group C1.(iii) Show that the CNOT-gate

UCNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

is an element of C2.(iv) Is the Fredkin gate an element of C3?

Problem 9. Find the 4× 4 matrix

U = e−iπ(σ1⊗I2)/4e−iπ(σ3⊗σ3)/4e−iπ(σ2⊗I2)/4.

Is the matrix U unitary?

Problem 10. Consider the four unary gates (2× 2 unitary matrices)

N =(

0 11 0

), H =

1√2

(1 11 −1

),

V =(

1 00 eiπ/2

), W =

(1 00 eiπ/4

)


and the state

|ψ〉 =1√2

(11

).

Calculate the state NHVW |ψ〉 and the expection value 〈ψ|NHVW |ψ〉.

Problem 11. (i) Let U be an n×n unitary matrix. Then the eigenvaluestake the form eiφ, where φ ∈ R. Let eiφ1 , . . . , eiφn be the eigenvalues of Uwith the corresponding normalized eigenvectors u1, . . . , un which form anorthonormal basis in Cn. Then one has (spectral decomposition)

U =n∑j=1

eiφjuju∗j .

Then the unitary matrix

V =n∑j=1

eiφj/2uju∗j

satisfies V 2 = U and can be viewed as the square root of U . Show that[U, V ] = 0n.(ii) Let U1, U2 be two unitary matrices with the spectral representation

U1 =n∑j=1

eiφ1ju1ju∗1j

U2 =n∑j=1

eiφ2ju2ju∗2j

where eiφ1j , eiφ2j (j = 1, . . . , n) are the eigenvalues of U1 and U2, re-spectively and u1j , u2j (j = 1, . . . , n) are the corresponding normalizedeigenvectors of U1 and U2, respectively. Let the unitary matrices

V1 =n∑j=1

eiφ1j/2u1ju∗1j

V2 =n∑j=1

eiφ2j/2u2ju∗2j

be the square roots of U1 and U2, respectively. Find the commutators[U1, U2] and [V1, V2].(iii) Study the question from (ii) under the condition that the bases u1j

and u2j (j = 1, . . . , n) are mutually unbiased bases, i.e.

|〈u1j |u2k〉|2 =1n, j, k = 1, . . . , n.


Problem 12. (i) Consider the Hadamard gate

UH =1√2

(1 11 −1

)≡ 1√

2(σ3 + σ1)

with the eigenvalues +1 and −1. Find a square root of the Hadamard gate.(ii) The star product of the Hadamard gate with itself provides the Bellmatrix

B = UH ? UH =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

.

Use the result from (i) to find a square root of the Bell matrix. Note thatthe eigenvalues of the Bell matrix are +1 (twice) and −1 (twice).

Problem 13. In the Hilbert space C4 the Bell states

1√2

1001

,1√2

0110

,1√2

01−10

,1√2

100−1

(i) Let ω = e2πi/4. Apply the Fourier transformation

UF =12

1 1 1 11 ω ω2 ω3

1 ω2 1 ω2

1 ω3 ω2 ω

to the Bell states and study the entanglement of these states.(ii) Apply the Haar wavelet transformation

UH =12

1 1 1 11 1 −1 −1√2 −

√2 0 0

0 0√

2 −√

2

to the Bell states and study the entanglement of these states.(iii) Apply the Walsh-Hadamard transformation

UW =12

1 1 1 1−1 −1 1 1−1 1 1 −11 −1 1 −1

to the Bell states and study the entanglement of these states.


Extend to the Hilbert space C2n

with the first Bell state given by

1√2

( 1 0 · · · 0 1 )T

Problem 14. Consider the Bell matrix B and the normalized vector v

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

, v =12

1111

≡ 1√2

(11

)⊗ 1√

2

(11

).

Is the normalized vector Bv entangled?

Problem 15. Find a 4× 4 unitary matrix U such that

U

1000

=1√2

1001

, U

0100

=1√2

0110

,

U

0010

=1√2

01−10

, U

0001

=1√2

100−1

.

Problem 16. Find a 4× 4 matrix U such that

U12

1111

=1√2

1001

.

Problem 17. Apply the quantum Fourier transform to the state

12

7∑j=0

cos(2πj/8)|j〉

where the quantum Fourier transform is given by

UQFT =1

2√

2

7∑j,k=0

e−i2πkj/8|k〉〈j|.


Is the operator UQFT unitary? Prove or disprove. Remember that

eiθ = cos(θ) + i sin(θ)

N−1∑k=0

ei2πk(n−m)/N = Nδnm.

We use|j〉, j = 0, 1, . . . , 7

as an orthonormal basis in C8.

Problem 18. Write the Bell matrix

UB =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

as a linear combination of Kronecker products of Pauli spin matrices.

Problem 19. Let α, β, γ ∈ R. Show that any U ∈ SU(2) can be writtenas

U = exp(iγσ3) exp(iβσ1) exp(iασ3).

Problem 20. (i) Let A, B be n× n matrices over R. Show that one canfind a 2n× 2n unitary matrix U such that

U

(A B−B A

)U∗ =

(A+ iB 0n

0n A− iB

).

Here 0n denotes the n× n zero matrix.(ii) Use the result from (i) to show that

det(

A B−B A

)= det(A+ iB)det(A+ iB) ≥ 0.

Problem 21. Let u be a column vector in Cn with u∗u = 1, i.e. thevector is normalized. Consider the matrix

U = In − 2uu∗.

(i) Show that U is hermitian.(ii) Show that U is unitary.


Problem 22. Can one find a 2× 2 unitary matrix such that

U

(−1 00 1

)U−1 =

(0 −1−1 0

).

Problem 23. (i) Do the 2× 2 unitary matrices

A =(e−iπ/4 0

0 ie−iπ/4

), B =

1√2

(1 ii 1

)satisfy the braid-like relation

ABA = BAB.

(ii) Find the smallest n ∈ N such that An = I2.(iii) Find the smallest m ∈ N such that Bm = I2.

Problem 24. Find all (n+ 1)× (n+ 1) matrices A such that

A∗UA = U

where U is the unitary matrix

U =

0 0 i0 In−1 0−i 0 0

and det(A) = 1. Consider first the case n = 2.

Problem 25. Consider the 2×2 hermitian matrices A and B with A 6= Bwith the eigenvalues λ1, λ2; µ1, µ2; and the corresponding normalized eigen-vectors u1, u2; v1, v2, respectively. Form from the normalized eigenvectorsthe 2× 2 matrix (

u∗1v1 u∗1v2

u∗2v1 u∗2v2

).

Is this matrix unitary? Find the eigenvalues of this matrix and the corre-sponding normalized eigenvectors of the 2 × 2 matrix. How are the eigen-values and eigenvectors are linked to the eigenvalues and eigenvectors of Aand B?

Problem 26. Let In be the n× n unit matrix. Is the 2n× 2n matrix

Ω =1√2

(In iInIn −iIn

)


unitary?

Problem 27. Consider the two 2× 2 unitary matrices

U1 =(

1 00 1

), U2 =

(0 11 0

).

Can one find a unitary 2× 2 matrix V such that

U1 = V U2V∗ ?

Problem 28. Let U be an n× n unitary matrix.(i) Is U + U∗ invertible?(ii) Is U + U∗ hermitian?(iii) Calculate exp(ε(U + U∗)), where ε ∈ R

Problem 29. Let U be an n × n unitary matrix. Then U + U∗ is ahermitian matrix. Can any hermitian matrix represented in this form?

Problem 30. (i) Find the condition on the n× n matrix A over C suchthat In +A is a unitary matrix.(ii) Let B be an 2× 2 matrix over C. Find all solutions of the equation

B +B∗ +BB∗ = 02.

Problem 31. Find all 2× 2 invertible matrices A such that

A+A−1 = I2.

Problem 32. Let z1, z2, w1, w2 ∈ C. Consider the 2× 2 matrices

U =(

0 z1z2 0

), V =

(0 w1

w2 0

)where z1z1 = 1, z2z2 = 1, w1w1 = 1, w2w2 = 1. This means the matricesU , V are unitary. Find the condition on z1, z2, w1, w2 such that thecommutator [U, V ] is again a unitary matrix.


U = α1σ1 + α2σ2 + α3σ3


is a unitary matrix.

Problem 34. Consider the unitary matrix

U =1√2

(I2 ⊗ I2 + iσ1 ⊗ σ2) =1√2

1 0 0 10 1 −1 00 1 1 0−1 0 0 1

.

Calculate

U(σ1 ⊗ I2)U−1, U(σ2 ⊗ I2)U−1, U(σ3 ⊗ I2)U−1,

U(σ1 ⊗ σ1)U−1, U(σ2 ⊗ σ2)U−1, U(σ3 ⊗ σ3)U−1.

Problem 35. (i) What are the conditions on φ11, φ12, φ21, φ22 ∈ R suchthat

U(φ11, φ12, φ21, φ22) =1√2

(eiφ11 eiφ12

eiφ21 eiφ22

)is a unitary matrix?(ii) What are the condition on φ11, φ12, φ21, φ22 ∈ R such that U(φ11, φ12, φ21, φ22)is an element of SU(2)?

Chapter 8

Entropy

Problem 1. An n× n density matrix ρ is a positive semidefinite matrixsuch that tr(ρ) = 1. The nonnegative eigenvalues of ρ are the probabilitiesof the physical states described by the corresponding eigenvectors. Theentropy of the statistical state described by the density matrix ρ is definedby

S(ρ) := −tr(ρ ln(ρ))

with the convention 0 ln(0) = 0. For the n×n hermitian matrix H (energyoperator) the statistical average of the energy E is defined by

E := tr(Hρ).

Letψ(ρ) := tr(Hρ)− tr(ρ ln(ρ)).

(i) Show thatln tr(eH) = max tr(Hρ) + S(ρ) .

(ii) Show that−S(ρ) = max tr(Hρ)− ln tr(eH) .

Problem 2. The von Neumann entropy, the standard measure of ran-domness of a statistical ensemble described by a n× n density matrix ρ, isdefined by

S(ρ) = −tr(ρ log(ρ)) = −n∑j=1

λj log(λj)

106

Entropy 107

where λj (j = 1, 2, . . . , n) are the eigenvalues of the density matrix ρ andthe log is taken to base n, the dimension of the Hilbert space Cn. Considerthe density matrix in C4

ρ =

1/3 0 0 1/60 1/6 0 00 0 1/6 0

1/6 0 0 1/3

.

Find the eigenvalues of ρ and then the von Neumann entropy S(ρ).

Problem 3. Consider the normalized states |ψk〉, k = 0, 1, . . . , N − 1in the Hilbert space CN . A positive operator valued measure is specifiedby a decomposition of the identity matrix IN into M positive semidefinitematrices Pm, i.e.

IN =M−1∑m=0

Pm.

The mutual information is defined by

I =N−1∑n=0

M−1∑m=0

pnm logN

(pnmpn·p·m

)where

pnm := 〈ψn|Pm|ψn〉

are the joint probabilities and

pn· :=M−1∑m=0

pnm, p·m :=N−1∑n=0

pnm

are their marginals. Let M = N = 2 and

P0 =12

(1 11 1

), P1 =

12

(1 −1−1 1

)

|ψ0〉 =1√2

(1−1

), |ψ1〉 =

(01

).

Find pnm, pn·, p·m and then I.

Problem 4. Let A, B be n× n hermitian matrices acting in the Hilbertspace Cn. Assume that the eigenvalues of A are pairwise different andanalogously for B. Then the normalized eigenvectors |αj〉 (j = 1, . . . , n) ofA form an orthonormal basis in Cn and analogously for B the normalized


eigenvectors |βj〉 (j = 1, . . . , n) form an orthonormal basis in Cn. Let|ψ〉 be a normalized state in Cn. Then there are n possible outcomes formeasurements of each observable and the probabilties pj(A, |ψ〉), pj(B, |ψ〉)(j = 1, . . . , n) are given by

pj(A, |ψ〉) := |〈ψ|αj〉|2, pj(B, |ψ〉) := |〈ψ|βj〉|2.

Let H|ψ〉(X) be the Shannon information entropy

H|ψ〉(X) := −n∑j=1

pj(X, |ψ〉) ln(pj(X, |ψ〉))

corresponding to the probability distribution pj(X, |ψ〉) (j = 1, . . . , n).The (Maassen-Uffink) entropic uncertainty relation is given by

H|ψ〉(A) +H|ψ〉(B) ≥ −2 ln( max1≤j,k≤n

|〈αj |βk〉|) > 0.

Note that the right-hand side does not involve the state |ψ〉.(i) Let

A = σ1 =(

0 11 0

), B = σ3 =

(1 00 −1

), |ψ〉 =

(cos(θ)sin(θ)

).

Calculate the left and right-hand side of the entropic uncertainty relation.Is the entropic uncertainty relation tight for this case?(ii) The (Landau-Pollak) uncertainty relation states that

arccos(√PA) + arccos(

√PB) ≥ arccos( max

1≤j,k≤n|〈αj |βk〉|)

wherePA := max

1≤j≤npj(A, |ψ〉), PB := max

1≤j≤npj(B, |ψ〉).

Calculate the left-hand and right-hand side of this uncertainty relation forA and B given in (i).

Problem 5. Consider the Hilbert space Cn. Let A, B be two hermitiann × n matrices (observable). Assume that A and B have non-degenerateeigenvalues with the corresponding normalized eigenvectors |a1〉, |a2〉, . . . ,|an〉 and |b1〉, |b2〉, . . . , |bn〉, respectively. The entropic uncertainty relationis an inequality given by

S(A) + S(B) ≥ S(AB)

where

S(A) = −n∑j=1

|〈ψ|aj〉|2 ln(|〈ψ|aj〉|2), S(B) = −n∑j=1

|〈ψ|bj〉|2 ln(|〈ψ|bj〉|2),

Entropy 109

and S(AB) is a positive constant which gives the lower bound of the right-hand side of the inequality. Consider the Hilbert space C2. Let

A = σ1, B = σ2, |ψ〉 =(

cos θsin θ

).

Find S(A), S(B) and S(A) + S(B).

Problem 6. Consider the Hilbert space Cn and |ψ〉 ∈ Cn. Let A andB n × n hermitian matrices (observable) with non-degenerate eigenvaluesand corresponding normalized eigenvectors |uj〉, |vj〉 (j = 1, . . . , n). Theentropic uncertainty relation is an inequality of the form

S(A) + S(B) ≥ SAB

where

S(A) = −n∑j=1

|〈ψ|uj〉|2 ln(|〈ψ|uj〉|2), S(A) = −n∑j=1

|〈ψ|vj〉|2 ln(|〈ψ|vj〉|2)

and SAB is a positive constant providing the lower bound of the right-handside of the inequality. Let

A = σ1 =(

0 11 0

), B = σ3 =

(1 00 −1

)and

|ψ〉 =(

cos(θ)sin(θ)

).

Calculate S(A) and S(B).

Chapter 9

Measurement

Problem 1. Consider the tripartite states

|GHZ〉 =1√2

(|000〉+ |111〉), |W 〉 =1√3

(|001〉+ |010〉+ |100〉).

Find the probabilityp = |〈W |GHZ〉|2.

Problem 2. Consider the W -state

|W 〉 =1√3

(|0〉 ⊗ |0〉 ⊗ |1〉+ |0〉 ⊗ |1〉 ⊗ |0〉+ |1〉 ⊗ |0〉 ⊗ |0〉).

Apply the invertible local operator

L =(√

a√d

0√c

)⊗(√

3 00√

3b/√a

)⊗(

1 00 1

)to the W -state, where a, b, c > 0 and d = 1− (a+ b+ c) ≥ 0. Calculate theprobability |〈W |LW 〉|2.

Problem 3. Consider the single qubit state

|ψ〉 := a|0〉+ b|1〉, |a|2 + |b|2 = 1.

Rewrite the first two qubits of the state

|ψ〉 ⊗ 1√2

(|01〉+ |10〉)

110

Measurement 111

in terms of the Bell basis

|Φ+〉= 1√2

(|00〉+ |11〉),

|Φ−〉= 1√2

(|00〉 − |11〉),

|Ψ+〉= 1√2

(|01〉+ |10〉),

|Ψ−〉= 1√2

(|01〉 − |10〉).

Describe how to obtain |ψ〉 as the state of the last qubit by measuring thefirst two qubits in the Bell basis. Suppose that the only errors which canoccur to three qubits are described by the transforms

I ⊗ I ⊗ I, I ⊗UNOT ⊗UNOT , I ⊗UP ⊗UP , I ⊗ (UPUNOT )⊗ (UPUNOT ).

Describe how an arbitrary error

αI⊗I⊗I+βI⊗UNOT⊗UNOT +δI⊗UP⊗UP +γI⊗(UPUNOT )⊗(UPUNOT )

on the state1√2

(|01〉+ |10〉)⊗ |ψ〉

can be corrected to obtain the correct |ψ〉 as the last qubit.


|ψ〉 =1√2

1001

≡ 1√2

((10

)⊗(

10

)+(

01

)⊗(

01

)).

and the 2× 2 unitary matrices

UA =(

cos(π/8) − sin(π/8)sin(π/8) cos(π/8)

), UB = U−1

A =(

cos(π/8) sin(π/8)− sin(π/8) cos(π/8)

),

Note that cos(π/8) = 12 (√

2 +√

2. sin(π/8) = 12 (√

2 +√

2. Let I2 be the2× 2 identity matrix. Find

〈ψ|(UA ⊗ I2)|ψ〉, 〈ψ|(I2 ⊗ UB)|ψ〉, 〈ψ|(UA ⊗ UB)|ψ〉

and the probabilities

|〈ψ|(UA ⊗ I2)|ψ〉|2, |〈ψ|(I2 ⊗ UB)|ψ〉|2, |〈ψ|(UA ⊗ UB)|ψ〉|2.


These probabilities play a role for the CHSH game.


|ψ〉 =1√2

((10

)⊗(

01

)−(

01

)⊗(

10

))=

1√2

01−10

.

Alice has the first qubit and Bob has the second qubit. Let I2 be the 2× 2identity matrix and UH be the Hadamard matrix

UH =1√2

(1 11 −1

).

If Alice receives the bit a = 0 and Bob receives the bit b = 0, then Aliceapplies I2 to her qubit and Bob applies I2 to his quibit, i.e. I2 ⊗ I2 to theBell states. If Alice receives the bit a = 1 and Bob receives the bit b = 0,then Alice applies UH to her qubit and Bob applies I2 to his quibit, i.e.UH⊗I2 is applied to the Bell states. If Alice receives the bit a = 0 and Bobreceives the bit b = 1, then Alice applies I2 to her qubit and Bob appliesUH to his quibit, i.e. I2⊗UH is applied to the Bell states. If Alice receivesthe bit a = 1 and Bob receives the bit b = 1, then Alice applies UH to herqubit and Bob applies UH to his quibit, i.e. UH ⊗UH is applied to the Bellstates. Find the states

|ψ1〉 = (I2⊗I2)|ψ〉, |ψ2〉 = (UH⊗I2)|ψ〉, |ψ3〉 = (I2⊗UH)|ψ〉, |ψ4〉 = (UH⊗UH)|ψ〉

and the probabilities |〈ψj |ψ〉|2 for j = 1, 2, 3, 4.

Chapter 10

Entanglement

Problem 1. Consider the singlet state (Bell state)

|ψ〉 =1√2

01−10

.

Let σ1, σ2, σ3 be the Pauli spin matrices. Show that the matrices I2 ⊗ I2,−σ1 ⊗ σ1, −σ2 ⊗ σ2, −σ3 ⊗ σ3 leave the state |ψ〉 invariant.

Problem 2. Let σ1, σ2, σ3 be the Pauli spin matrices. Calculate thecommutators

[σ1 ⊗ σ1, σ2 ⊗ σ2], [σ1 ⊗ σ1, σ3 ⊗ σ3], [σ2 ⊗ σ2, σ3 ⊗ σ3].

Problem 3. Let |0〉, |1〉 be the standard basis in the Hilbert space C2.Consider the GHZ-state

|ψ〉 =1√2

(|0〉 ⊗ |0|〉 ⊗ |0〉+ |1〉 ⊗ |1〉 ⊗ |1〉).

Find the expectation values

〈ψ|σ1 ⊗ σ2 ⊗ σ2|ψ〉, 〈ψ|σ2 ⊗ σ1 ⊗ σ2|ψ〉,

〈ψ|σ2 ⊗ σ2 ⊗ σ1|ψ〉, 〈ψ|σ1 ⊗ σ1 ⊗ σ1|ψ〉.

113


Problem 4. Consider the state in the Hilbert space C4n0(τ, φ, θ)n1(τ, φ, θ)n2(τ, φ, θ)n3(τ, φ, θ)

=

sin((τ − φ)/2) sin(θ/2)sin((τ + φ)/2) cos(θ/2)cos((τ − φ)/2) sin(θ/2)cos((τ + φ)/2) cos(θ/2)

.

The state is obviously normalized, i.e. n20 + n2

1 + n22 + n2

3 = 1. Find theconditions on φ, τ , θ such that n0n3 = n1n2 (separability condition). Showthat in this case the state can be written as product state.

Problem 5. Let |0〉, |1〉 be an arbitrary orthonormal basis. Can the state

|ψ〉 =1√2|0〉 ⊗ |0〉+

1√8|0〉 ⊗ |1〉+

1√8|1〉 ⊗ |0〉+

1√4|1〉 ⊗ |1〉

be written as a product state?


H = ~ω(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3).

(i) Is the 4 × 4 matrix H hermitian? Find the trace of H. What can besaid about the eigenvalues of H.(ii) Find the eigenvalues and normalized eigenvectors of H.(iii) Calculate exp(−iHt/~).


U1 = eiπσ1/4 ⊗ eiπσ1/4, U2 = eiπσ2/4 ⊗ eiπσ2/4.

CalculateU∗1 (σ3 ⊗ σ3)U1, U∗2 (σ3 ⊗ σ3)U2.


|ψ〉 =12

(|0〉 ⊗ |0〉+ eiφ1 |0〉 ⊗ |1〉+ eiφ2 |1〉 ⊗ |0〉+ eiφ3 |1〉 ⊗ |1〉).

(i) Let φ3 = φ1 + φ2. Is the state |ψ〉 a product state?(ii) Let φ3 = φ1 + φ2 + π. Is the state |ψ〉 a product state?

Problem 9. There are six different types of quark known as flavor: up,down, charm, strange, top, bottom. Consider the two equations for states

cos θ(

1√3

(|uu〉+ |dd〉+ |ss〉))

+sin θ(

1√6

(|uu〉+ |dd〉 − 2|ss〉))

=1√2

(|uu〉+|dd〉)

Entanglement 115

cos θ(

1√6

(|uu〉+ |dd〉 − 2|ss〉))− sin θ

(1√3

(|uu〉+ |dd〉+ |ss〉))

= −|ss〉

where |uu〉 ≡ |u〉 ⊗ |u〉 etc. Find cos θ and sin θ from this two equations.

Problem 10. Let

|0〉 =

100

, |1〉 =

010

, |2〉 =

001

.

Consider the normalized state (Aharonov state)

|ψ〉 =1√6

(|012〉 − |021〉+ |120〉 − |102〉+ |201〉 − |210〉)

where |012〉 = |0〉 ⊗ |1〉 ⊗ |2〉 etc and

S3 =

1 0 00 0 00 0 −1

.

Is |ψ〉 an eigenstate of Sz ⊗ Sz ⊗ Sz?


H =12

(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 + I2 ⊗ I2).

(i) Is H hermitian? Find the trace of H.(ii) Calculate H2 and tr(H2).(iii) Using the result from (ii) calculate exp(iθH), exp(−iπH/4) and exp(−iπH/2).(iv) Using the results from (i) and (ii) find the eigenvalues of H.(v) Find the normalized eigenstates of H.


|ψ〉 =1

2√

2(|0000〉−|0011〉−|0101〉+|0110〉+|1001〉+|1010〉+|1100〉+|1111〉)

where we used the notation |0000〉 ≡ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 etc. and

|0〉 =(

10

), |1〉 =

(01

).

Calculate the states

(σ1 ⊗ σ3 ⊗ σ3 ⊗ σ1)|ψ〉, (σ1 ⊗ σ1 ⊗ I2 ⊗ σ3)|ψ〉, (I2 ⊗ σ1 ⊗ σ1 ⊗ I2)|ψ〉


and

(I2 ⊗ σ2 ⊗ σ3 ⊗ σ2)|ψ〉, (σ1 ⊗ σ2 ⊗ σ2 ⊗ σ1)|ψ〉, (I2 ⊗ σ3 ⊗ σ2 ⊗ σ2)|ψ〉.

Problem 13. The hyperdeterminant of a 2×2×2 hypermatrix C = (cijk)(i, j, k ∈ 0, 1 ) is defined by

DetC := −12

1∑i,j,k,m,n,p=0

1∑i′,j′,k′,m′,n′,p′=0

εii′εjj′εkk′εmm′εnn′εpp′cijkai′j′mcnpk′cn′p′m′

where ε00 = ε11 = 0, ε01 = 1, ε10 = −1.(i) Calculate DetC.(ii) Consider the three qubit state

|ψ〉 =1∑

i,j,k=0

cijk|i〉 ⊗ |j〉 ⊗ |k〉.

The three tangle τ3 is a measure of entanglement and is defined for the threequbit state |ψ〉 as

τ123 := 4|DetC|where C = (cijk). Find the three tangle for the GHZ-state

|GHZ〉 =1√2

(|0〉 ⊗ |0〉 ⊗ |0〉+ |1〉 ⊗ |1〉 ⊗ |1〉)

and the W -state

|W 〉 =1√3

(|0〉 ⊗ |0〉 ⊗ |1〉+ |0〉 ⊗ |1〉 ⊗ |0〉+ |1〉 ⊗ |0〉 ⊗ |0〉).

Problem 14. Calculate the product of the unitary matrices

exp(iπ(σ2 ⊗ I2)/4) exp(−iπ(σ3 ⊗ σ3)/4) exp(−iπ(σ1 ⊗ I2)/4).

Problem 15. Let |0〉, |1〉 be an orthonormal basis in C2. Consider thenormalized state

|ψ〉 =1∑

j,k=0

cjk|j〉 ⊗ |k〉

in the Hilbert space C4 and the 2 × 2 matrix C = (cjk). Using the 4coefficients cjk, j, k ∈ 0, 1) we form a multilinear polynomial p in twovariables x1, x2

p(x1, x2) = c00 + c01x1 + c10x2 + c11x1x2. (1)

Entanglement 117

Show that determinant detC = c00c11 − c01c10 is the unique irreduciblepolynomial (up to sign) of content one in the 4 unkowns cjk that vanisheswhenever the system of equations

p =∂p

∂x1=

∂p

∂x2= 0 (2)

has a solution (x∗1, x∗2) in C2.


|ψ〉 =1∑

j,k,`=0

cjk`|j〉 ⊗ |k〉 ⊗ |`〉

in the Hilbert space C8 and the 2× 2× 2 array C = (cjk`) (j, k, ` ∈ 0, 1.Using the 8 coefficients cjk` we form a multilinear polynomial in threevariables x1, x2, x3

p(x1, x2, x3) = c000+c001x1+c010x2+c100x3+c011x1x2+c101x1x3+c110x2x3+c111x1x2x3.(1)

Show that the hyperdeterminant

DetC = c2000c2111 + c2001c

2110 + c2010c

2101 + c2100c

2011

−2(c000c001c110c111 + c000c010c101c111

+c000c100c011c111 + c001c010c101c110

+c001c100c011c110 + c010c100c011c101)+4(c000c011c101c110 + c001c010c100c111)

is the unique irreducible polynomial (up to sign) of content one in the 8unkowns cjk` that vanishes whenever the system of equations

p =∂p

∂x1=

∂p

∂x2=

∂p

∂x3= 0 (2)

has a solution (x∗1, x∗2, x∗3) in C3.


|ψ〉 =1√2

(|H〉 ⊗ |V 〉 − |V 〉 ⊗ |H〉).

We define a polarization state that is rotated by an angle α from the hori-zontal axis as

|α〉 = cos(α)|H〉+ sin(α)|V 〉


and analogously|β〉 = cos(β)|H〉+ sin(β)|V 〉.

Calculate the probability

p(α, β) = |(〈α| ⊗ 〈β|)|ψ〉|2 .


|ψ〉 =

1000

and the unitary operator (4× 4 matrix)

U = e−iπσ2/4 ⊗ I2.

Find the state U |ψ〉.

Problem 19. Let σ1, σ2, σ3 be the Pauli spin matrices. Consider theHamilton operator

H = J(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3).

(i) Let ε = Jβ ≡ J/(kBT ), where kB is the Boltzmann constant and T theabsolute temperature and J > 0. Calculate

ρ(ε) =1

Z(ε)exp(−βH) ≡ 1

Z(ε)exp(−ε(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3))

where Z(ε) is the partition function

Z(ε) = tr exp(−ε(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3)).

(ii) The concurrence C(ρ(ε)) is defined by

C(ρ(ε)) = max(0, µ1(ε)− µ2(ε)− µ3(ε)− µ4(ε))

where the µj ’s are the square roots of the eigenvalues of the 4× 4 matrix

ρ(σ2 ⊗ σ2)ρ∗(σ2 ⊗ σ2)

in decreasing order. Calculate C(ρ(ε)) and discuss the result as function ofε ≡ Jβ.

Entanglement 119

Problem 20. Can we find 2× 2 matrices S1 and S2 such that

(S1 ⊗ S2)((

10

)⊗(

10

))=

1√2

1001

. (1)

Problem 21. Let N be an integer larger than 5. Consider the followingstate

|ψ〉 =1√N

N−1∑j=0

|jmodN〉 ⊗ |3jmodN〉 ⊗ |5jmodN〉.

Let U be the quantum Fourier transform. Calculate (U ⊗U ⊗U)|ψ〉. Writethe answer in the basis |0〉, |1〉, . . . , |N − 1〉⊗3. Show that it is the su-perposition of equally probable states. Find the probability.


H = ~ωσ1 ⊗ σ3 ⊗ σ1

and the corresponding unitary operator

U(t) = e−iHt/~ = e−iωtσ1⊗σ3⊗σ1 .

(i) Calculate H2 and U(t).(ii) Show that U(t) can be written as, i.e. we decompose U(t) into elemen-tary gates of one qubit rotations and two qubits interactions,

U(t) = e−iπI⊗I⊗σ2/4eiπσ3⊗I⊗σ3/4eiπσ1⊗I⊗I/4e−iωtσ3⊗σ3⊗Ie−iπσ1⊗I⊗I/4e−iπσ3⊗I⊗σ3/4eiπI⊗I⊗σ2/4

where I is the 2× 2 unit matrix.

Problem 23. Let σ1, σ2, σ3 be the Pauli spin matrices. Find the 4 × 4matrix

U = e−iπ(σ1⊗I2)/4e−iπ(σ3⊗σ3)/4e−iπ(σ2⊗I2)/4.

Is the matrix unitary?


H =12

(−~ω1σ3 ⊗ I2 − ~ω2I2 ⊗ σ3 + ~γσ3 ⊗ σ3) .

FindU = e−iπ(σ1⊗I2)/2e−iHt/~e−iπ(σ1⊗I2)/2e−iHt/~.


Give an interpretation of the result.


|ψ〉 = cos(α)|00〉+ sin(α)|11〉, 0 < α < π/4

where α is called the Schmidt angle.(i) Find the eigenvalues of the density matrix |ψ〉〈ψ|.(ii) Find the partically traced density matrix (we find when we trace overone of the subsystems).(iii) Show that the partically traced has two unequal and non-zero eigen-values λ1 = cos2(α) and λ2 = sin2(α).(iv) Calculate the von Neumann entropy for the corresponding density ma-trix. Show that the entropy grows monotonically with the Schmidt angle.


UH =1√2

(1 11 −1

).

Is UH ∈ SU(2)? Is iUH ∈ SU(2)?

Problem 27. Consider the finite-dimensional Hilbert space

HN := span |n〉 : n = 0, 1, . . . , N − 1

i.e. dim(H)N = N and 〈n|m〉 = δnm with m = 0, 1, . . . , N − 1. Let

|φ`〉 :=1√N

N−1∑n=0

exp(inφ`)|n〉, φ` := φ0 + 2π`

N

for ` ∈ ZN . We define a self-adjoint phase operator as

φN :=N−1∑`=0

φ`|φ`〉〈φ`|.

Find the matrix elements of the phase operator φN in the occupation num-ber basis |n〉 with n = 0, 1, . . . , N − 1.

Problem 28. Calculate the three-tangle for the W -state

|W 〉 =1√3

((10

)⊗(

10

)⊗(

01

)+(

10

)⊗(

01

)⊗(

10

)+(

01

)⊗(

10

)⊗(

10

)).

Entanglement 121

Problem 29. Summarize the requirements for quantum computation.

Problem 30. Find the eigenvalues and normalized eigenvectors of theHamilton operator

H = ~ω(σ3 ⊗ σ3) + ∆(σ1 ⊗ σ1).

Calculate exp(−iHt/~).

Problem 31. Consider the XX-model described by the Hamilton oper-ator

HXX =N∑j=1

(J(σx,jσx,j+1) +Bσz,j)

with the periodic boundary conditions σ1,N+1 = σ1,1, σ3,N+1 = σ3,1. Wehave

σ1,j = I2 ⊗ · · · ⊗ I2 ⊗ σ1 ⊗ I2 ⊗ · · · ⊗ I2where σ1 is at the j-position with j = 1, 2, . . . , N . Let

Σ3 :=N∑j=1

σ3,j .

Calculate the commutator [HXX ,Σ3]. Discuss.

Problem 32. Consider the Hamilton operator (so-called transverse XY -model in one dimension)

H = −gL−1∑j=0

σzj −L−1∑j=0

(1 + γ

2σxj σ

xj+1 +

1− γ2

σyj σyj+1

)where σx, σy, σz are the Pauli spin matrices, 0 ≤ γ ≤ 1, g is a constantand we impose cyclic boundary conditions. This means σxL = σx0 , σyL = σy0 ,σzL = σz0 .(i) Find the commutator [H, C], where

C :=L−1∏j=0

σzj .

(ii) Calculate C2. Show that C and C2 form a group under matrix mul-tiplication. Give the character table. What are the eigenvalues of C? Wedefine

Q :=12

(I − C)


where I is the unit operator (2L × 2L identity matrix). Calculate theeigenvalues of Q.(iii) Let L = 4. Calculate the eigenvalues of H.(iv) Let γ = 0. Calculate [Z, H], where

Z =L−1∑j=0

σzj .

Discuss.


|ψ〉 =1∑

j,k=0

cjk|j〉 ⊗ |k〉.

Using the four coefficients cjk we form the polynomial p in the two variablesx1, x2

p(x1, x2) = c00 + c01x1 + c10x2 + c11x1x2.

Consider the three equations p = 0, ∂p/∂x1 = 0, ∂p/∂x2 = 0, i.e.

p(x1, x2) = c00 + c01x1 + c10x2 + c11x1x2 = 0

and

∂p

∂x1= c01 + c11x2 = 0

∂p

∂x2= c10 + c11x1 = 0.

Show that this system of three equations with two unkowns x1, x2 onlyadmits solutions if

det(C) ≡ c00c11 − c01c10 = 0

where C is the 2× 2 matrix

C =(c00 c01c10 c11

).

Problem 34. Consider the finite dimensional Hilbert space

HN := span |n〉 : n = 0, 1, . . . , N − 1

Entanglement 123

where 〈n′|n〉 = δnn′ . Thus dimHN = N . We define the state

|φ`〉 :=1√N

N−1∑n=0

exp(inφ`)|n〉, φ` := φ0 + 2π`

N

for ` ∈ ZN . We define the linear operator

φN :=N−1∑n=0

φ`|φ`〉〈φ`|.

Find the matrix elements of this linear operator in the occupation numberbasis |n〉.

Problem 35. We consider the finite-dimensional Hilbert space H = C2n

and the normalized state

|ψ〉 =1∑

j1,j2,...,jn=0

cj1,j2,...,jn |j1〉 ⊗ |j2〉 ⊗ · · · ⊗ |jn〉

in this Hilbert space. Here |0〉, |1〉 denotes the standard basis. Let εjk(j, k = 0, 1) be defined by ε00 = ε11 = 0, ε01 = 1, ε10 = −1. Let n be evenor n = 3. Then an n-tangle can be introduced by

τ1...n = 2

∣∣∣∣∣∣∣∣∣1∑

α1,...,αn=0...

δ1,...,δn=0

cα1...αncβ1...βncγ1...γncδ1...δn

×εα1β1εα2β2 · · · εαn−1βn−1εγ1δ1εγ2δ2 · · · εγn−1δn−1εαnγnεβnδn

∣∣ .(i) Consider the case n = 4 and a state |ψ〉 with c0000 = 1/

√2, c1111 = 1/

√2

and all other coefficients are 0. Find τ1234.(ii) Consider the case n = 4 and a state |ψ〉 with c0000 = 1/

√2, c1111 =

−1/√

2 and all other coefficients are 0. Find τ1234.(iii) Consider the case n = 4 and a state |ψ〉 with c0001 = 1/

√2, c1110 =

1/√

2 and all other coefficients are 0. Find τ1234.(iv) Consider the case n = 4 and a state |ψ〉 with c0001 = 1/

√2, c1110 =

−1/√

2 and all other coefficients are 0. Find τ1234.

Problem 36. The n-qubit Pauli group is defined by

Pn := I2, σ1, σ2, σ3 ⊗n ⊗ ±1, ±i

where σ1, σ2, σ3 are the 2 × 2 Pauli matrices and I2 is the 2 × 2 identitymatrix. The dimension of the Hilbert space under consideration is dimH =


2n. Thus each element of the Pauli group Pn is (up to an overall phase±1, ±i) a Kronecker product of Pauli matrices and 2× 2 identity matricesacting on n qubits. What is the order of the n-qubit Pauli group?


H = ~ω(σ3 ⊗ σ3) + ∆1σ1 ⊗ σ1 + ∆2σ2 ⊗ σ2.

(i) Find the eigenvalues. Discuss energy level crossing. Find the normalizedeigenvectors.(ii) Calculate the commutators

[σ1 ⊗ σ1, σ2 ⊗ σ2], [σ2 ⊗ σ2, σ3 ⊗ σ3], [σ3 ⊗ σ3, σ2 ⊗ σ2]

(iii) Use the result from (ii) to calculate exp(−iHt/~).

Problem 38. Let σ1 = σ1, σ2 = σ2, σ3 = σ3 be the Pauli spin matrices.We form the nine 4× 4 matrices

Σjk := σj ⊗ σk, j, k = 1, 2, 3.

Note that [Σjk,Σmn] = 0. The variance of an hermitian operator O and awave vector |φ〉 is defined by

VO(|φ〉) := 〈φ|(O)2|φ〉 − (〈φ|O|φ〉)2.

The remoteness for a given normalized state |ψ〉 in C4 is defined by

R(|ψ〉) =3∑j=1

3∑k=1

(〈ψ|(Σjk)2|ψ〉 − (〈ψ|Σjk|ψ〉)2

).

Find the remoteness for the Bell states

|φ+〉 =1√2

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉), |φ−〉 =1√2

(|0〉 ⊗ |0〉 − |1〉 ⊗ |1〉)

|ψ+〉 =1√2

(|0〉 ⊗ |1〉+ |1〉 ⊗ |0〉), |ψ−〉 =1√2

(|0〉 ⊗ |1〉 − |1〉 ⊗ |0〉).

Problem 39. Let e1, e2, e3 be the standard basis in the Hilbert spaceC3. Are the states in the Hilbert space C27 are entangled

1√6

(e1⊗e2⊗e3+e2⊗e3⊗e1+e3⊗e1⊗e2+e1⊗e3⊗e2+e3⊗e2⊗e1+e2⊗e1⊗e3)

Entanglement 125

1√6

(e1⊗e2⊗e3+e2⊗e3⊗e1+e3⊗e1⊗e2−e1⊗e3⊗e2−e3⊗e2⊗e1−e2⊗e1⊗e3)

1√6

((e1⊗e2⊗e3+e2⊗e1⊗e3)+ε(e2⊗e3⊗e1+e1⊗e3⊗e2)+ε∗(e3⊗e1⊗e2+e3⊗e2⊗e1).

Problem 40. Find the entanglement (three tangle) as a function of θ ofthe normalized state in C8

|ψ〉 = cos(θ)e1 ⊗ e1 ⊗ e1 − i sin(θ)e2 ⊗ e2 ⊗ e2

where

e1 =(

10

), e2 =

(01

)and 0 < θ < π/4.

Problem 41. Given the eigenvalue equations Ax = λx, Ay = λy andx∗y = 0. Then A(x + y) = λ(x + y). Thus x + y is also an eigenvectorwith eigenvalue λ. Consider the 4× 4 matrix

σ1 ⊗ σ1 =

0 0 0 10 0 1 00 1 0 01 0 0 0

.

The eigenvalues are +1 (twice) and −1 (twice). The normalized eigenvec-tors for +1 are

1√2

(11

)⊗ 1√

2

(11

)=

12

1111

,1√2

(1−1

)⊗ 1√

2

(1−1

)=

12

1−1−11

.

These two states are orthonormal to each other and obviously not entangled.The normalized eigenvectors for the eigenvalue −1 are

1√2

(11

)⊗ 1√

2

(1−1

)=

12

1−11−1

,1√2

(1−1

)⊗ 1√

2

(11

)=

12

11−1−1

.

These two states are orthonormal to each other and obviously not entangled.All four vectors form an orthonomal basis in C4. Find linear combinationsof the two cases so that the eigenvectors are entangled and still form anorthonormal basis in C4.


Problem 42. Are the states in C4

|±〉 =1√2

(10

)⊗(

10

)± 1

2

(01

)⊗(

10

)± 1

2

(10

)⊗(

01

)entangled?

Problem 43. (i) Consider the two states in C4

|ψ〉 =1√2

1001

, |φ〉 =(

cosαsinα

)⊗(

cosβsinβ

)≡

cos(α) cos(β)cos(α) sin(β)sin(α)cos(β)sin(α) sin(β)

.

One definesG(|ψ〉) = max

α,β|〈φ|ψ〉|

as the maximum overlap between |ψ〉 and the product state |φ〉. FindG(|ψ〉).(ii) Given the state

|χ〉 =12

1111

.

Find G(|χ〉) with the product state given at (i). Discuss.

Problem 44. Consider a bipartite system and the product Hilbert spaceH = H1 ⊗H2. Let |ψ〉 ∈ H and normalized. Then a density matrix (purestate)

ρ12 := |ψ〉〈ψ|

is entangled when the density matrices

ρj = trk(ρ12), j, k = 1, 2, j 6= k

provided by partial tracing as non-zero von Neumann entropy, i.e.

S(ρj) = −tr(ρj log(ρj)) 6= 0, j = 1, 2.

There is no entanglement if S(ρj) = 0. Consider the Hilbert spaces H1 =H2 = C3 and H = C9. Is the normalized state in C9

|ψ〉 =1√3

( 1 0 0 0 1 0 0 0 1 )T

entangled?

Entanglement 127

Problem 45. An entanglement measure is the relative entropy of entan-glement. It is defined for a density matrix σ as

ER(σ) := minρ∈D

S(σ‖ρ)

where D is the set of density matrices with positive partial transpose (PPTstates) and

S(σ‖ρ) := tr(σ log2(σ)− σ log2(ρ)).

Find S(σ‖ρ) for the density matrix (one of the Werner states)

σ =16

2 0 0 00 1 1 00 1 1 00 0 0 2

.

Problem 46. Let

|1〉 =(

10

), |0〉 =

(01

).

Show that the normalized state

12

(|0〉⊗|0〉⊗|0〉⊗|0〉+|1〉⊗|0〉⊗|0〉⊗|1〉+|0〉⊗|0〉⊗|1〉⊗|0〉+|1〉⊗|0〉⊗|1〉⊗|1〉)

in the Hilbert space C16 is three-separable and thus biseparable.

Problem 47. Are the vectors

v1 =

1

cos(π/4)cos(π/2)cos(3π/4)

, v2 =

0

sin(π/4)sin(π/2)sin(3π/4)

entangled?

Problem 48. Can the normalized vector in C16

12

(|0〉⊗|0〉⊗|0〉⊗|0〉+|0〉⊗|1〉⊗|0〉⊗|1〉+|1〉⊗|0〉⊗|1〉⊗|0〉+|1〉⊗|1〉⊗|1〉⊗|1〉)

be written as Kronecker product of lower dimensional vectors?

Problem 49. Let H be the finite dimensional Hilbert space Cd. Let Idbe the d × d identity matrix and A an arbitrary d × d matrix over C. We


call a vector |Ψ〉 ∈ H ⊗ H maximally entangled, if it normalized, and itsreduced density matrix is maximally mixed, i.e., a multiple of Id

〈Ψ(A⊗ Id)|Ψ〉 =1d

tr(A).

(i) Let d = 2. Consider the normalized state

|Ψ〉 =1√2

1001

.

Calculate 〈Ψ|(A⊗ I2)|Ψ〉 and 1d tr(A).

(ii) Let d = 2. Consider the normalized state

|Ψ〉 =12

1111

.

Calculate 〈Ψ|(A⊗ I2)|Ψ〉 and 1d tr(A).

Problem 50. (i) Is the state in C4

|ψ〉 =1√2

1010

entangled?(ii) Is the state in C4

|ψ〉 =1√2

0−i0 i

entangled?

Problem 51. We set

| ↑〉 =(

10

), | ↓〉 =

(01

).

Can the state in C16

12

(| ↑↑↑↑〉+ | ↑↓↓↑〉+ | ↓↑↑↓〉+ | ↓↓↓↓〉)

Entanglement 129

be written as the Kronecker product of 2×8, 8×2, 4×4 normalized vectors?

Problem 52. Consider the three spin-1 matrices

S1 =1√2

0 1 01 0 10 1 0

, S2 =1√2

0 −i 0i 0 −i0 i 0

, S3 =

1 0 00 0 00 0 −1

.

Apply the vec-operator to these matrices and then normalize them. Canthe vectors be written as Kronecker products of vectors in C3?

Chapter 11

Bell Inequality

Problem 1. Consider four observers: Alice (A), Bob (B), Charlie (C)and Dora (D) each having one of the qubits. Every observer is allowedto choose between two dichotomic observables. Denote the outcome ofobserver X’s measurement by Xi (X = A,B,C,D) with i = 1, 2. Underthe assumption of local realism, each outcome can either take the value+1 or −1. The correlations between the measurement outcomes of all fourobservers can be represented by the product AiBjCkDl, where i, j, k, l =1, 2. In a local realistic theory, the correlation function of the measurementperformed by all four observers is the average of AiBjCkDl over many runsof the experiment

Q(AiBjCkDl) := 〈ψ|AiBjCkDl|ψ〉

The Mermin-Ardehali-Belinskii-Klyshko inequality is given by

Q(A1B1C1D1)−Q(A1B1C1D2)−Q(A1B1C2D1)−Q(A1B2C1D1)−Q(A2B1C1D1)−Q(A1B1C2D2)−Q(A1B2C1D2)−Q(A2B1C1D2)−Q(A1B2C2D1)−Q(A2B1C2D1)−Q(A2B2C1D1) +Q(A2B2C2D2)+Q(A2B2C2D1) +Q(A2B2C1D2) +Q(A2B1C2D2) +Q(A1B2C2D2) ≤ 4 .

Each observer X measures the spin of each qubit by projecting it eitheralong nX1 or nX2 . Every observer can independently choose between twoarbitrary directions. For a four qubit state |ψ〉, the correlation functionsare thus given by

Q(AiBjCkDl) = 〈ψ|(nAi · σ)⊗ (nBj · σ)⊗ (nCk · σ)⊗ (nDl · σ)|ψ〉 .

130

Bell Inequality 131

where · denotes the scalar product, i.e. nXj · σ := nXj1σ1 + nXj2σ2 + nXj3σ3.Let

nA1 =

100

, nA2 =

001

, nB1 =

010

, nB2 =

001

nC1 =

010

, nC2 =

001

, nD1 =1√2

−101

, nD2 =1√2

101

.

Show that the Mermin-Ardehali-Belinskii-Klyshko inequality is violated forthe state

|ψ〉 =1

2√

2(|0000〉−|0011〉−|0101〉+|0110〉+|1001〉+|1010〉+|1100〉+|1111〉)

where

|0〉 =(

10

), |1〉 =

(01

).

and |0000〉 ≡ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 etc..

Chapter 12

Quantum Channels

We consider the Hilbert space H of n × n matrices over C with the scalarproduct (Frobenius inner product)

〈A,B〉 := tr(AB∗)

with A,B ∈ H. A state is described using n × n density matrices ρ, i.e.tr(ρ) = 1 and ρ ≥ 0 (positive semidefinite). The space of trace-class oper-ators acting in this Hilbert space is denoted by S(H). A quantum channelfrom a Hilbert space HA to a Hilbert space HB is represented by a com-pletely positive trace-preserving map Φ : S(HA) → S(HB). Such a posi-tive trace-preserving map can be represented in Stinespring representation,Kraus operator representation and Choi-Jamiolkowski representation.

Let Hn denote the vector space of n × n Hermitian matrices over the realnumbers. We say that ρ ∈ Hn is positive semi-definite (or ρ ≥ 0) if x∗ρx ≥0 for all x ∈ Cn, or equivalently: all of the eigenvalues of ρ are non-negative.A linear map ψ : Hn → Hp is TPCP (trace-preserving completely positive)if

1. TP (trace-preserving): ∀ρ ∈ Hn, trρ = trψ(ρ)

2. CP (completely positive): ∀m ∈ N, ρ ∈ Hmn,

ρ ≥ 0 ⇒ (ψ ⊗ Im×m)(ρ) ≥ 0

where Im×m is the identity operator on m×m matrices.

132

Quantum Channels 133

Problem 1. Let Hn be the vector space of n × n hermitian matrices.The adjoint (conjugate transpose) of a matrix A ∈ Cn×n is denoted by A∗,Consider a family V1, V2, . . . , Vm of n × n matrices over C. We associatewith this family the completely positive map ψ : Hn → Hn defined by

ψ(X) =m∑j=1

VjXV∗j .

The map ψ is said to be a Kraus map if ψ(In) = In, i.e.

m∑j=1

VjV∗j = In

and the matrices V1, V2, . . . , Vm are called Kraus operators.

Let m = n = 2 and

V1 =(

0 10 0

), V2 =

(0 01 0

).

Show that V1 and V2 are Kraus operators and find the associated Krausmap.

Problem 2. Let ψ : Hn → Hn be a Kraus map. Thus ψ is linear. Showthat there exists Ψ ∈ Cn×n such that for all X ∈ Hn

vec(ψ(X)) = Ψ vec(X)

where 1 is an eigenvalue of Ψ. What is a corresponding eigenvector?

Problem 3. Find all Kraus maps ψ : H2 → H2, associated with familiesof 2 Kraus operators (V1 and V2), which provide the transformation

ψ

(1 00 0

)=(

0 00 1

).

Calculate

ψ

(0 00 1

).

Is there a Kraus map associated with a single Kraus operator which alsoprovides this transformation?

Problem 4. Let p ∈ [0, 1] and σ1, σ2, σ3, σ0 = I2 be the Pauli spinmatrices.


(i) Show that the four 2× 2 matrices

K0 =√

1 + 3p2

σ0, K1 =√

1− p2

σ1, K2 =√

1− p2

σ2, K3 =√

1− p2

σ3

are Kraus operators.(ii) Show that the sixteen 4× 4 matrices

Kj ⊗K`, j, ` = 0, 1, 2, 3

are Kraus operators, where ⊗ denotes the Kronecker product.(iii) Show that the sixteen 4× 4 matrices

Kj ? K`, j, ` = 0, 1, 2, 3

are Kraus operators, where ? denotes the star product.

Problem 5. Let Kj (j = 1, . . . ,m) be n× n matrices over C with

m∑j=1

KjK∗j = In.

Show that

m∑j=1

m∑`=1

(Kj ⊗K`)(K∗j ⊗K∗` ) = In ⊗ In ≡ In2 .

Problem 6. Let Kj (j = 1, . . . ,m) be 2× 2 matrices over C with

m∑j=1

KjK∗j = I2.

Show thatm∑j=1

m∑`=1

(Kj ? K`)(K∗j ? K∗` ) = I2 ⊗ I2 = I4.

Problem 7. (i) Let A be an n×n matrix over C. Let G be a finite groupgiven by n× n matrices over C and g ∈ G. Consider the linear map

A 7→ A =1|G|

∑g∈G

gAg−1


where |G| denotes the number of elements in the finite group G. Show thattr(A) = tr(A).(ii) Is the determinant preserved under the linear map?(iii) Let A be positive semi-definite. Is A positive semi-definite?(iv) Apply it to the case of 4× 4 matrices with

A = ρ =12

1 0 0 10 0 0 00 0 0 01 0 0 1

and the group is given by the 4×4 permutation matrices with |G| = 4! = 24.

Problem 8. Let ρ1, ρ2 ∈ Hn be positive semi-definite matrices.

(i) Is ρ3 = ρ1 + ρ2 positive semi-definite?

(ii) Is ρ4 = kρ1 (k ∈ C) positive semi-definite?

(iii) Let ρ5 ∈ Hn such that ρ1 + ρ5 is positive semi-definite. Is ρ5 positivesemi-definite?

Problem 9. Show that a linear map ψ : Hn → Hp is a TP map ifand only if ψ∗(In) = In where ∗ denotes the adjoint with respect to theFrobenius inner product and In is the n× n identity matrix.


ρ0 :=n∑

i,j=1

Eij ⊗ Eij ∈ Hn2

is positive semi-definite, where Eij is the n × n matrix with a 1 in row iand column j and 0 elsewhere.

Problem 11. An orthonormal basis, with respect to the Frobenius innerproduct, for Hn (n ≥ 2) is given by B = B1 ∪B2 where

B1 =

1√2

(Ejk + Ekj) : j, k = 1, . . . , n, j ≤ k

B2 =

i√2

(Ejk − Ekj) : j, k = 1, . . . , n, j < k

.

Express

ρ0 :=n∑

i,j=1

Eij ⊗ Eij ∈ Hn2


in terms of this basis.

Problem 12. Show that a linear map ψ : Hn → Hp is a CP map if andonly if (ψ ⊗ In×n)(ρ0) is positive semi-definite where

ρ0 :=n∑

i,j=1

Eij ⊗ Eij ∈ Hn2 .

Problem 13. Is the map ψ : Hn → Hp given by ψ(ρ) = ρT completelypositive?

Problem 14. Let ψ : Hn → Hp given by

ψ(ρ) :=n2∑k=1

VkρV∗k

be a CP map. Find the condition on V1, . . . , Vn2 such that ψ is TP (andhence TPCP).

Problem 15. Let ψ : Hn → Hp given by

ψ(ρ) :=n2∑k=1

VkρV∗k

be a CP map. Show that there exists a matrix V such that

ψ(ρ) = trmV (ρ⊗ In2)V ∗.

Problem 16. A minimal Stinespring representation of a CP map ψ :Hn → Hp is a represenatation

ψ(ρ) = trmV (ρ⊗ Im)V ∗

where m is minimal. This corresponds to minimizing the number of non-zero Kraus operators Vk in a Kraus representation

ψ(ρ) :=m∑k=1

VkρV∗k .

Given a Kraus representation

ψ(ρ) :=n2∑k=1

VkρV∗k .


Consider Let

A =n2∑k=1

(vecVk)(vecVk)∗.

The matrix A is positive definite and thus has a spectral decomposition

A =m∑k=1

λkvkvk∗

where m is the rank of A and λ1, . . . , λm > 0. We find the Kraus operatorsVk for the minimal representation from vecVk =

√λkvk. Find a minimal

representation for the completely map on H2 given by

V1 =(

0 11 0

), V2 =

(0 10 0

), V3 =

(0 01 0

), V4 =

(0 00 0

).

Problem 17. Consider the Kraus operators K1 and K2

K1 =(

0 10 0

)⇒ K∗1 =

(0 01 0

), K2 =

(0 01 0

)⇒ K∗2 =

(0 10 0

)and an arbitrary 2× 2 matrix A = (ajk). Then

K1AK∗1 +K2AK

∗2 =

(a22 00 a11

).

So the trace is preserved under this transformation. Let c†1, c†2, c1, c2be Fermi creation and annihilatin operators, respectively. Consider theoperators

K1 =(c†1 c†2

)( 0 10 0

)= c†1c2 K†1 = c†2c1

K2 =(c†1 c†2

)( 0 01 0

)= c†2c1 K†2 = c†1c2

and

A =(c†1 c†2

)A

(c1c2

)= a11c

†1 + a12c

†1c2 + a21c

†2c†2c1 + a22c

†2c2.

Find the operatorK1AK

†1 + K2AK

†2 .

Chapter 13

Miscellaneous

Problem 1. Let H0 and V be n × n hermitian matrices and ε ∈ R.Consider the hermitian matrix H = H0 + εV . Let

U(β) = e−β(H0+εV )

with β ≥ 0. Then

dU(β)dβ

= −(H0 + εV )e−β(H0+εV ) = −(H0 + εV )U(β)

where U(β = 0) = In. Let

U(β) = e−βH0W (β).

(i) Show that W (β) is given by

W (β) =∞∑k=0

(−1)kεk∫ β

0

∫ β1

0

· · ·∫ βk−1

0

dβ1dβ2 · · · dβkV (β1)V (β2) · · · V (βk).

where V (β) := eβH0V e−βH0 .(ii) Apply (i) to

H0 = ~ω(

1 00 1

), V = ∆

(0 11 0

).

Problem 2. Let H, A, B be hermitian matrices. Let

A(t) := eiHtAe−iHt, B(s) := eiHsBe−iHs

138

Miscellaneous 139

where s, t ∈ R. Show that

tr(A(t)B(s)e−βH) = tr(eiH(t−s)Ae−iH(t−s)Be−βH).

Problem 3. Let H(t) be a given time-dependent hermitian Hamiltonoperator given as an n×n matrix. We assume that H(t) depends smoothlyon t. Find the solution of the initial value problem of the matrix differentialequation

dU(t)dt

= − i~H(t)U(t), U(0) = In

where In is the n×n identity matrix. Apply the ansatz (Magnus expansion)

U(t) = exp(Ω(t))

and Ω(t) =∑∞k=1 Ωk(t). Find the first two terms in the expansion, i.e. find

Ω1(t) and Ω2(t).


H = ~ωσ1 · σ2 ≡ ~ω(σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3).

Find the eigenvalues and normalized eigenvectors of H.

Problem 5. Let A, H be n × n hermitian matrices, where H plays therole of the Hamilton operator. The Heisenberg equations of motion is givenby

dA(t)dt

=i

~[H,A(t)].

with A = A(t = 0) = A(0). Let Ej (j = 1, 2, . . . , n2) be an orthonormalbasis in the Hilbert space H of the n× n matrices with scalar product

〈X,Y 〉 := tr(XY ∗), X, Y ∈ H.

Now A(t) can be expanded using this orthonormal basis as

A(t) =n2∑j=1

cj(t)Ej

and H can be expanded as

H =n2∑j=1

hjEj .


Find the time evolution for the coefficients cj(t), i.e. dcj/dt, where j =1, 2, . . . , n2.

Problem 6. Consider the standard basis in the Hilbert space C9

|00〉, |01〉, |02〉, |10〉, |11〉, |11〉, |12〉, |20〉, |21〉, |22〉

where |00〉 ≡ |0〉 ⊗ |0〉, and |0〉, |1〉, |2〉 is the standard basis in C3. Showthat the normalized states

|ψ〉nm =1√3

2∑j=0

e2πijn/3|j〉 ⊗ |(j +m) mod 3〉

i.e.

|ψ〉00 =1√3

(|00〉+ |11〉+ |22〉)

|ψ〉10 =1√3

(|00〉+ e2πi/3|11〉+ e4πi/3|22〉)

|ψ〉20 =1√3

(|00〉+ e4πi/3|11〉+ e2πi/3|22〉)

|ψ〉01 =1√3

(|01〉+ |12〉+ |20〉)

|ψ〉11 =1√3

(|01〉+ e2πi/3|12〉+ e4πi/3|20〉)

|ψ〉21 =1√3

(|01〉+ e4πi/3|12〉+ e2πi/3|20〉)

|ψ〉02 =1√3

(|02〉+ |10〉+ |21〉)

|ψ〉12 =1√3

(|02〉+ e2πi/3|10〉+ e4πi/3|21〉)

|ψ〉22 =1√3

(|02〉+ e4πi/3|10〉+ e2πi/3|21〉)

form an orthonormal basis in the Hilbert space C9.


|ψ〉 = −E1E2(ei(k1·r+k2·r′)−i(ω1t+ω2t

′) + ei(k2·r+k1·r′)−i(ω2t+ω1t′))|0〉 ⊗ |0〉.

Findw ∝ 〈ψ|ψ〉.

Miscellaneous 141

Problem 8. Let A be a nonzero bounded linear operator in a Hilbertspace H. Let |n〉, |m〉 be normalized states in the Hilbert space H. Wedefine

S(|m〉, |n〉, A) :=〈m|AA†|n〉√

〈m|AA†|m〉√〈n|AA†|n〉

.

Consider the Hilbert space C2. Calculate S for A = σ1 and the normalizedstates

|u〉 =(

10

), |v〉 =

1√2

(1−1

).

Problem 9. Consider the state |ψ〉

|ψ〉 =1∑

j0,j1,...,jn=0

cj0,j1,...,jN−1 |j0〉 ⊗ |j1〉 ⊗ · · · ⊗ |jN−1〉

in the Hilbert space C2n

. The bitstring j0j1 . . . jN−1 can be mapped one-to-one into a non-negative integer j

j =N−1∑k=0

jk2k

where jk ∈ 0, 1. Thus we can write the state as

|ψ〉 =2N−1∑j=0

cj |j〉.

We can associate a polynomial with the state |ψ〉 via

p(|ψ〉, x) =2N−1∑j=0

cjxj .

(i) Consider the Bell state (N = 2)

|ψ〉 =1√2

(|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉).

Find the polynomial of |ψ〉 and calculate the roots.(ii) Consider the state (N = 2)

|φ〉 =12

(|0〉 ⊗ |0〉+ |1〉 ⊗ |0〉 − |0〉 ⊗ |1〉 − |1〉 ⊗ |1〉).


Find the polynomial of |φ〉 and calculate the roots.

Problem 10. Let A, B be observable, i.e. hermitian matrices. Then theuncertainty relation is given by

∆2A ·∆2B ≥ 14|〈[A,B]〉|2 + cov(A,B)

where [ , ] denotes the commutator,

cov(A,B) :=12

(〈AB〉+ 〈BA〉)− 〈A〉〈B〉

and∆2A := cov(A,A).

This inequality can generalized to 2n observable A1, A2, . . . , A2n. We have

det(Σ) ≥ det(C)

whereΣk` = cov(Ak, A`), Ck` = − i

2〈[Ak, A`]〉.

Let

A = σ1, B = σ2, |ψ〉 =1√2

(1−1

).

Find the left-hand side and right-hand side of the inequality.

Problem 11. The most general real three-qubit state can be written as

|ψ〉=−c3 cos2 θ|0〉 ⊗ |0〉 ⊗ |1〉 − c2|0〉 ⊗ |1〉 ⊗ |0〉+ c3 sin(θ) cos(θ)|0〉 ⊗ |1〉 ⊗ |1〉−c1|1〉 ⊗ |0〉 ⊗ |0〉 − c3 sin(θ) cos(θ)|1〉 ⊗ |0〉 ⊗ |1〉+ (c0 + c3 sin2(θ))|1〉 ⊗ |1〉 ⊗ |1〉

where c0, c1, c2, c3, θ are real parameters. Classify the state with respect toentanglement.

Problem 12. Let A, B be n×n matrices acting in the Hilbert space Cn.Then A, B can be considered as observable. The two overvable A and Bare called complementary if their eigenvalues are non-degenerate and anytwo normalized eigenvectors aj of A and bj of B satisfy

|a∗jbk| =1√n

where ∗ means transpose and conjugate complex. Give an example for suchhermitian matrices in C2.

Miscellaneous 143

Problem 13. Two orthonormal bases

uj : j = 1, 2, . . . , n , vk : k = 1, 2, . . . , n

in the Hilbert space Cn are called mutually unbiased if

u∗jvk =1√n

for all j, k ∈ 1, 2, . . . , n .

(i) Give an example in C2.(ii) Give an example in C4.

Problem 14. Solve the initial value problem of the optical Bloch equa-tions.

dρ11

dt= −dρ22

dt= i

b

2(e−i(ω−α)tρ12 − ei(ω−α)tρ21)

dρ12

dt=dρ∗21dt

= ib

2ei(ω−α)t(ρ11 − ρ22)

with the initial conditions

ρ11(0) = ρ12(0) = ρ21(0) = 0, ρ22(0) = 1.

Problem 15. Consider a finite dimensional Hilbert space. Let H be ahermitian Hamilton operator. Let |ψ〉 be the normalized ground state ofthe system, i.e. H|ψ〉 = E0|ψ〉. Let |φ〉 be another normalized state. LetF be a positive semidefinite operator and 〈φ|F |φ〉 > 0. Then we have theinequality

〈ψ|F |ψ〉 ≥ (〈φ|ψ〉〈φ|F |φ〉 − (∆F )(1− 〈φ|ψ〉2)1/2)2

〈φ|F |φ〉

where(∆F )2 := 〈φ|F 2|φ〉 − 〈φ|F |φ〉2.

The inequality follows from the non-negativity of the Gramian determinantof the vectors |ψ〉, |φ〉, and F |φ〉. Consider the Hilbert space C2 and theHamilton operator

H = ~ω(

0 11 0

)the positive semidefinite operator

F =(

1 11 1

)


and the normalized states

|ψ〉 =1√2

(1−1

), |φ〉 =

(10

)with H|ψ〉 = E0|ψ〉 and E0 = −~ω. Apply the inequality to these operatorsand states, i.e. calculate the left and right-hand side of the inequality.

Problem 16. Consider the normalized states in the Hilbert space C3

|ψ1〉 =1√2

101

, |ψ2〉 =

010

, |ψ3〉 =1√2

10−1

.

Find the unitary matrices U12, U23, U31 such that

|ψ2〉 = U12|ψ1〉, |ψ3〉 = U23|ψ2〉, |ψ1〉 = U31|ψ3〉.


UH =1√2

(1 11 −1

).

Is UH ∈ SU(2)? Is iUH ∈ SU(2)?

Problem 18. The available uncertainty relations in finite-dimensionalHilbert spaces are those of Robertson and Schrodinger. Let ρ be the stateof the quantum system (density matrix), i.e. a positive semi-definite, self-adjoint linear operator with tr(ρ) = 1. The mean value functional is

〈 · 〉 := tr(ρ·).

Then for two self-adjoint operators, A and B, the variance is defined by

(∆A)2 := 〈A2〉 − 〈A〉2, (∆B)2 := 〈B2〉 − 〈B〉2.

We have the inequalities

(∆A)(∆B) ≥ |〈AB〉 − 〈A〉〈B〉| ≥ 12|〈[A,B]〉|.

Note that we have the identity

|〈AB〉|2 =14|〈[A,B]+〉+ 〈[A,B]〉|2 =

14|〈[A,B]+〉|2 +

14|〈[A,B]〉|2.

Miscellaneous 145

(i) Let

A = σ1, B = σ2, ρ =(

1/2 00 1/2

).

Calculate the left-hand side of the inequalty and the right-hand sides of theinequality. Discuss.(ii) Let

A = σ1 ⊗ σ1, B = σ2 ⊗ σ2, ρ =

1/4 0 0 00 1/4 0 00 0 1/4 00 0 0 1/4

.

Calculate the left-hand side of the inequalty and the right-hand sides of theinequality. Discuss.

Problem 19. A spin- 12 system in a time-dependent magnetic fields S(t)

is described by the Hamilton operator

H(t) =12

~ωσ3 +12

~γS(t)σ1

where σ1 and σ3 are the Pauli matrices

σ1 =(

0 11 0

), σ3 =

(1 00 −1

).

Then the Schrodinger equation

i~∂ψ

∂t= H(t)ψ

for the spinor ψ = (ψ1, ψ2)T takes the form

idψ1

dt= −1

2ωψ1 +

12γS(t)ψ2, i

dψ2

dt=

12ωψ2 +

12γS(t)ψ1.

Rewrite this system in terms of the observable Bloch variables

A(t) := |ψ2|2 − |ψ1|2, B(t) := i(ψ2ψ∗1 − ψ1ψ

∗2), C(t) := ψ2ψ

∗1 + ψ1ψ

∗2 .

Problem 20. Consider the operators

H1 = 1, H2 = x, H3 =∂2

∂x2, H4 = i

∂

∂x.

(i) Show that we have a nilpotent Lie algebra under the commutator.


(ii) Let

α1(t) = cf(t), α2(t) = c, α3(t) = −12, α4(t) =

df

dt.

Consider the Hamilton operator

K =4∑j=1

αj(t)Hj

and the Schrodinger equation

i∂ψ

∂t= Kψ.

We write the solution of the Schrodinger equation in the form

ψ(x, t) = U(t, 0)ψ(x, 0)

where the unitary time evolution operator is give by

U(t, 0) = exp(β1(t)H1) exp(β2(t)H2) exp(β3(t)H3) exp(β4(t)H4).

Find the system of ordinary differential equations for βj(t) (j = 1, 2, 3, 4)and solve them.

Problem 21. Consider the spin Hamilton operator

H = ∆1σ1 ⊗ σ1 + ∆2σ2 ⊗ σ2 + ∆3σ3 ⊗ σ3.

Let K = βH. The partition function is

Z(β) = tr exp(−K).

The logarithm of the partition function is given by the cumulant expansion

ln(Z(β)) = ln tr(I)− 〈K〉+12!

(〈K2〉 − 〈K〉2)

− 13!

(〈K3〉 − 3〈K2〉〈K〉+ 2〈K〉3)

+14!

(〈K4〉 − 4〈K3〉〈K〉 − 3〈K2〉2 + 12〈K2〉〈K〉2 − 6〈K〉4)

− · · ·

Here I is the identity operator given by In ⊗ In with n = 2 and

〈· · ·〉 :=tr(· · ·)

trI.

Miscellaneous 147

Calculate the function ln(Z(β)) up to this order.

Problem 22. Let |ψ〉, |φ〉 be normalized states in the Hilbert space Cn.Let K be a positive semi-definite matrix in Cn. Show that

G := det

1 〈φ|ψ〉〈φ|K|ψ〉〈φ|ψ〉 1 〈φ|K|φ〉〈φ|K|ψ〉〈φ|K|φ〉〈φ|K2|φ〉

≥ 0.

G is called the Gramian. Apply it to the Hilbert space C2 and

|ψ〉 =(

10

), |φ〉 =

1√2

(11

), K =

(1 11 1

).

Problem 23. Let C = (cjk) (j, k = 1, . . . , n) be an n × n matrix withreal entries. Then C is called a quantum correlation matrix if there areself-adjoint operators Aj , Bk (j, k = 1, . . . , n) on a Hilbert space H with‖Aj‖ ≤ 1, ‖Bk‖ ≤ 1 and u in the unit sphere of H⊗H such that

cjk = 〈(Aj ⊗Bk)u,u〉

where 〈 , 〉 denotes the scalar product. If the self-adjoint operators Aj ,Bk (j, k = 1, . . . , n) commute the matrix C is called a classical correlationmatrix. Consider the case with n = 3, the Hilbert space C2, A1 = B1 = σ1,A2 = B2 = σ2, A3 = B3 = σ3 and the Bell state

uT =1√2

( 1 0 0 1 ) .

Find the correlation matrix C. What is the significance of this matrix?

Problem 24. Let

T (X,P ) = exp(i(Px−Xp)/~)

be the phase space translation operator. Show that

T (X1, P1)T (X2, P2) = exp(i(X2P1 −X1P2)/(2~))T (X1 +X2, P1 + P2).

Problem 25. The group generator of the compact Lie group SU(2) canbe written as

J1 =12

(z1

∂

∂z2+ z2

∂

∂z1

), J2 =

i

2

(z2

∂

∂z1− z1

∂

∂z2

), J3 =

12

(z1

∂

∂z1− z2

∂

∂z2

).


(i) FindJ+ = J1 + iJ2, J− = J1 − iJ2.

(ii) Let j = 0, 1, 2, . . . and m = −j,−j + 1, . . . , 0, . . . , j. We define

ejm(z1, z2) =1√

(j +m)!(j −m)!zj+m1 zj−m2 .

FindJ+e

jm(z1, z2), J−e

jm(z1, z2), J3e

jm(z1, z2)

(iii) Let

J2 = J21 + J2

2 + J23 ≡

12

(J+J− + J−J+) + J23 .

FindJ2ejm(z1, z2).

Problem 26. Show that the operators

L+ = zz, L− = − ∂

∂z

∂

∂z

L3 = −12

(z∂

∂z+ z

∂

∂z+ 1), L0 = −1

2

(z∂

∂z− z ∂

∂z+ 1).

form a basis for the Lie algebra su(1, 1) under the commutator.

Miscellaneous 149

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Index

W -state, 116π-meson, 59

Aharonov state, 115

Bell matrix, 43, 50, 53, 54, 96, 101Bell states, 69Bloch variables, 145Braid-like relation, 103

Cayley transform, 74Clifford group, 98Complementary, 142

Dicke states, 18

Entropic uncertainty relation, 108Exceptional points, 6

Feynman gate, 89Fredkin gate, 90–92

Generalized Fredkin gate, 91Generalized Toffoli gate, 91GHZ-state, 13, 116Gramian, 147Gramian determinant, 143

Hadamard basis, 21Half-adder, 92Hyperdeterminat, 116

Klein inequality, 85Kraus map, 133Kraus operators, 133

Magnus expansion, 139Moller wave operators, 57Mutually unbiased, 27Mutually unbiased bases, 8

Optical Bloch equations, 143

Partition function, 14, 146Pauli group, 97, 123Pauli spin matrices, 52Phase operator, 120

Quantum correlation function, 42Quantum correlation matrix, 147

Remoteness, 124

Schrodinger equation, 96Shannon information entropy, 108Spin coherent state, 35Spin-1, 44, 51Stinespring representation, 136

Three tangle, 116Three-tangle, 120Toffoli gate, 56, 90, 91Translation operator, 147

Uncertainty relation, 142Uncertainty relations, 144

Variance, 3, 124

W-state, 13Werner state, 77

155

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