THE OPERATOR THEORY BASIS OF

QUANTUM COMPUTING

by

CANLIN ZHANG

A research essay

presented to the University of Waterloo

in fulfilment of the

research essay requirement for the degree of

Master of Mathematics

in

Pure Mathematics

Waterloo, Ontario, Canada, 2014

c©CANLIN ZHANG 2014

AUTHOR’S DECLARATION

I hereby declare that I am the sole author of this research essay. This is

a true copy of the research essay, including any required final revisions, as

accepted by my examiners.

I understand that my research essay may be made electronically available to

the public.

CANLIN ZHANG

ii

Abstract

This paper will introduce some operator theory which has connec-

tions with quantum computation and quantum information. We first

introduce some basic ideas and notations in operator theory. Then

we will discuss quantum algorithms. We mainly focus on quantum

channels, which are also called completely positive trace preserving

maps. After that, we will outline the main theorems for quantum er-

ror detection and correction, which is also the most interesting part of

this paper. Finally, we conclude with a discussion of a special passive

method in quantum error detection.

iii

Acknowledgements

I am grateful so much to Professor Kenneth Davidson, my supervisor, for

his patient instruction and farsighted enlightenment. I would not complete

my paper without the help from him. I am very grateful to Professor David

Kribs. This paper is mainly based on [7], which is a paper of Kribs. I am

also very grateful to Professor Laurent Marcoux for his helpful suggestion

and guiding. Moreover, I am also grateful to Cameron Williams and Michael

Hartz for their help on Latex. Thanks also to members of Department of

Pure Mathematics at University of Waterloo for kind supporting during the

preparation of this paper.

iv

Contents

1 Introduction 1

2 Quantum Computing Basics 2

3 Quantum Algorithms 5

4 Quantum Channels 9

5 Quantum Error Correction 25

6 Noiseless Subsystems via The Noise Commutant 34

v

1 Introduction

In the past few decades, researchers have being devoting themselves to es-tablishing the theoretical basis for quantum computation and quantum in-formation. Although many important results have been found and proved,there are still more problems, theoretical and experimental, that need to beovercome. And for those theoretical ones, many are related to deep mathe-matical problems. This paper mainly tries to provide a basic idea of quantumcomputing for researchers interested in operator theory or operator algebras.But we note that in order to understand this paper, a reader only needs tohave knowledge of linear algebra and basic functional analysis.

Following standard practice in quantum computing, we use the physicalnotation for operators on a Hilbert space.

This paper is organized as follows:Section 1 is the background introduction.Section 2 mainly introduce the basic concepts, terms and notations in the

field of quantum computing;Section 3 will provide the basic ideas in quantum algorithms by describ-

ing Deutsch’s algorithm [1,2], which is an elementary example of quantumalgorithm, to show the power of quantum computation;

Section 4 primarily provides the mathematical basis required for the de-scription of evolution within a quantum system. This section would mostlyfocus on quantum channels, or namely the completely positive and trace pre-serving maps [8] in pure mathematics.

Section 5 would be the most interesting part of this paper. Although, thispaper only provides basic ideas, we will provide a quite detailed discussion ofquantum error correction methods in section 5. The fundamental theoremsfor quantum error detection and correction would be presented in the ‘stan-dard model’;

In section 6 (the last section), we would describe the ‘noiseless subsystemvia noise commutant’ [3], which is a very basic and simple method of quan-tum error prevention.

1

2 Quantum Computing Basics

The properties of operators on Hilbert space is a central issue in the studyof mathematical basis for quantum information and quantum computing.

Let H be a (complex) Hilbert space. For vectors and vector duals in H,we use the Dirac notation: A typical vector in H will be denoted as a ‘ket’|ψ〉, and the linear functional on H determined by this |ψ〉 will be denoted asa ‘bra’ 〈ψ|. (Here the “linear function determined by 〈ψ|” means the innerproduct produced by 〈ψ| and other vectors.)

Notice that a bra and a ket yield a inner product 〈ψ1||ψ2〉, while a ket anda bra yield a rank one operator |ψ2〉〈ψ1|. Particularly, for a unit vector |ψ〉,the rank one projection from H to the subspace {λ|ψ〉 : λ ∈ C} is written as|ψ〉〈ψ|. Let B(H) denote the set of bounded operators (or equivalently, theset of continuous operators) acting on H. We use the physics symbol U † torepresent the adjoint of the operator U .

Now, we are ready to show the postulate of quantum mechanics:(i) Typically, a quantum system can be regarded as a portion of the

physical universe chosen for analysis. And the portion outside our chosenquantum system is called the environment. A quantum system is calledclosed if it has no mass or energy exchange with the environment.

(ii) We use a Hilbert space H to represent a closed quantum system, anduse a mapping |ψ〉〈ψ| (where |ψ〉 is a unit vector) to represent a certain stateof the quantum system at a precise moment of time. Also, we usually onlywrite |ψ〉 rather than |ψ〉〈ψ| for simplicity. However, the state of a quantumsystem is usually not certain. In this case, we use a probability density func-tion to describe the state space of the quantum system. In quantum physics,this density function is a positive operator ρ on H with trace equals to one.It is called a density operator .

(iii) We use a unitary transformation to describe an evolution in a closedquantum system. This means: Let the initial state of the quantum systembe ρ. Then after an evolution, the state would become UρU †.

(iv) We use finitely many operators to describe a measurement of a quan-tum system. That is, a set of operators Mk, 1 ≤ k ≤ r, such that

r∑k=1

M †kMk = 1,

2

where 1 is the identity operator.We call a measurement projective if all the Mk are projections. Then the

Mk have mutually orthogonal ranges. (Mk is a projection implies M †k = Mk.

Then∑r

k=1M†kMk =

∑rk=1M

2k =

∑rk=1Mk. So,

∑rk=1Mk = 1. Then ob-

viously their ranges should be mutually orthogonal.) Moreover, we call apositive measurement classical when each of the Mk is rank one. In ex-periments, if the initial state of a quantum system is certain (for example|ψ〉), then the probability of the occurrence of the evolution Mk|ψ〉〈ψ|M †

k is

p(k) = 〈ψ|M †kMk|ψ〉 (See [5], page 13-16). Notice that:∑

k

p(k) =∑k

〈ψ|M †kMk|ψ〉 = 〈ψ|(

∑k

M †kMk)|ψ〉 = 〈ψ||ψ〉 = 1.

So p(k) indeed defines a probability distribution.(v) Let H1 , ... ,Hm be the Hilbert spaces associated with m quantum

systems. We call the quantum system associated with the Hilbert spaceH1 ⊗ ... ⊗Hm the composite quantum system.

If H1 and H2 have dimensions n1 and n2 respectively, the dimension ofH1⊗H2 will then be n1n2. Note that a typical vector in H1⊗H2 is a linearcombination of the form

∑i

hi ⊗ ki. If {e1, · · · , en1} is an orthogonalized

basis of H1 and {f1, · · · , fn2} is an orthogonalized basis of H2, then {ei⊗fj :1 ≤ i ≤ n1, 1 ≤ j ≤ n2} is an orthogonalized basis of H1 ⊗ H2. Also, thecomposite inner product is:

〈h1 ⊗ k1, h2 ⊗ k2〉 = 〈h1, h2〉〈k1, k2〉 ,

for any h1 ⊗ k1, h2 ⊗ k2 ∈ H1 ⊗H2.

So, given the (pure) state |ψi〉 of Hi, we obtain the (pure) statem∏i=1

⊗|ψi〉

of the composite system.

These concepts are basic in quantum mechanics. After introducing them,we are ready to show some basic concepts and conventions in quantum com-putation.

The most frequently discussed Hilbert spaces in quantum computationand quantum information are those with dimension N = dn for any positiveintegers n ≥ 1 and d ≥ 2. For simplicity, we always suppose d = 2 in this

3

paper. So, we write the Hilbert space as HN = H2n = C2⊗ ...⊗C2 = (C2)⊗n,which is a n-fold tensor product. We drop the N when convenient.

Now consider H2 = C2 : let {|0〉, |1〉} be an orthonormal basis for H2.This is a typical representation of the classical base in many two level quan-tum systems, such as the ground and excited states of an electron capturedby an atomic nucleus, the ‘spin-up’ and ‘spin-down’ of an electron, and thetwo polarizations of a photon of light. To describe the standard orthonormalbasis of H2n = (C2)⊗n ' C2n , we use the abbreviated form in quantum me-chanics. For example, the standard orthonormal basis of H4 shall be writtenas {|ij〉 : i, j ∈ Z2} or {|i〉|j〉 : i, j ∈ Z2}, where |ij〉 = |i〉|j〉 = |i〉 ⊗ |j〉 isthe tensor product of two vectors. In other words, the basis of H4 can bewritten as {|00〉, |01〉, |10〉, |11〉} or {|0〉|0〉, |0〉|1〉, |1〉|0〉, |1〉|1〉}.

In quantum information, we call a vector |ψ〉 in H2 a quantum bit, or a‘qubit’. Notice that any vector |ψ〉 in H2 can be written as |ψ〉 = a|0〉+ b|1〉.The vector |ψ〉 is called a superposition of the classical states |0〉 and |1〉, ifboth a and b are not zero. A ‘qudit’ is a unit vector in C d. In the compositespace HN , a vector (state) |ψ〉 is called entangled if it cannot be written asa tensor product of vectors from the component systems. For instance, the‘EPR pairs’ |ψ〉 = |00〉+|11〉√

2in H4 is an entangled vector.

In the following part, we do not distinguish a quantum system from itscorresponding Hilbert space. We would directly say “ the quantum systemHN”.

For an evolution on a quantum system HN , let U be the correspondingunitary operator. Since HN is finite dimensional, U is actually a unitarymatrix. Then, decoherence can be regarded as the process to vanish theoff-diagonal entries of U .

The following discussions are about several classical unitary matrices(evolutions) on a quantum system HN .

In the 1-qubit case (N = 21), the Pauli matrices are given by:

X =

(0 11 0

), Y =

(0 −ii 0

), Z =

(1 00 −1

).

Let 12 be the 2 × 2 identity matrix. These matrices can be regarded asoperators acting on H2 under the basis {|0〉, |1〉}. And in the n-qubit case(N = 2n), the set of ‘single qubit unitary gates’ generated by the Pauli

matrices is {Xk, Yk, Zk : 1 ≤ k ≤ n}, where Xk = 1⊗(k−1)2 ⊗X ⊗ 1

⊗(n−k)2 and

Yk, Zk are similar.

4

Let UCN be an operator on H4 such that UCN(|ij〉) = |i〉|(i+ j) mod 2 〉.Then it is easy to see that UCN is unitary. We call this UCN the ‘controlled-NOT gate’, or CNOT gate for short. The CNOT gate has natural extensions{U (k,l)

CN : 1 ≤ k 6= l ≤ n} to unitary operators (gates) on HN : If N = 2n,let the basis of HN be {|i1〉|i2〉...|in〉 : ik = 0 or 1 , k = 1, 2...n}. Then

we have: U(k,l)CN : |i1〉...|in〉 7−→ |i1〉...|ik〉...|(ik + il) mod 2 〉...|in〉. Note that

UCN = U(1,2)CN .

Note that all the N × N unitary matrices form a group under matrixmultiplication. Denote this group as U(N). Then, it is easy to see that

{Xk, Yk, Zk, U(k,l)CN : 1 ≤ k 6= l ≤ n} generates U(N).

At the end of this section, we introduce the Hadamard gate H and thespin-1

2Pauli matrices σk on H2:

H =1√2

(1 11 −1

),

and:

σk =1

2K , for k = x, y, z and K = X, Y, Z ,

where X, Y, Z are the Pauli matrices.The Hadamard gate was first used by J. Hadamard in the year 1893,

while the Pauli matrices and Pauli-12spin matrices were first studied by W.

Pauli in 1932. These operators were widely used in the research of quantuminformation.

3 Quantum Algorithms

Generally speaking, a quantum algorithm is a collection of initial states ρ to-gether with their evolutions UρU † under a unitary matrix U . Many quantumalgorithms, such as the factoring algorithm of Shor [9,10] and the search al-gorithm of Grover [4], have being taking into useage recently. In this section,we will first introduce the Deutsch algorithm (or Deutsch-Josza algorithm)[1,2], which is a simple but useful quantum algorithms. After that, wewill provide some examples to show how powerful the quantum computationcould be comparing with classical computations.

Before continuing, we shall use a simple example to show how the basicoperations (such as addition) can be performed by a quantum algorithm: Let

5

dimHN = 2n. Then we can see that there is a natural bijection between thebasis vectors |i1 ... in〉 and integers 0, 1, ..., 2n− 1 (the basis has 2n elements).Then define x ⊕ y = x + y mod N , and define the unitary operator U onHN ⊗HN by U |x〉|y〉 = |x〉|x ⊕ y〉. Then the corresponding quantum algo-rithm would be associated with an addition operation modulo N . (Note thatwhen N = 2, U is just the CNOT gate.)

Note that the tensor product of matrix is: Let H =

(a bc d

), then

H⊗n =

(aH⊗n−1 bH⊗n−1

cH⊗n−1 dH⊗n−1

).

Now we are able to discuss the Deutsch algorithm. Let Hn = H⊗n be then-fold tensor product of the Hadamard gate on HN . We have:

Hn|0〉⊗n =1√2n

2n−1∑x=0

|x〉 .

Fix positive integers k,m ≥ 1. Let Hm,k = H2m ⊗ H2k . Then Hm,k hasthe basis |x〉|y〉 = |x〉 ⊗ |y〉, where x ∈ Zm2 and y ∈ Zk2 . Let f : Zm2 7→ Zk2 beany function. Then, define Uf ∈ B(Hm,k) via:

Uf : |x〉|y〉 7−→ |x〉|y ⊕ f(x)〉 .

Then it is easy to see that Uf is an unitary operator. Also we can see thatU permutes |y〉 in the basis vectors {|x〉|y〉 : y ∈ Zk2} .

Note 3.1. Notice that for any x ∈ Zm2 , we have Uf (|x〉|0〉) = |x〉|f(x)〉. Thisis how Uf simulates the function f on a quantum computer. Therefore, anyclassical function can be simulated in this way on a quantum computer.

Then we have:

Uf ((Hm|0〉⊗m)⊗ |0〉⊗k) = Uf (1√2m

2m−1∑x=0

|x〉 ⊗ |0〉)

=1√2m

2m−1∑x=0

Uf (|x〉 ⊗ |0〉) =1√2m

2m−1∑x=0

|x〉 ⊗ |f(x)〉

This means Uf acting on Hm ⊗ 12k would yield a simultaneous compu-tation (parallel computation) of f on every possible value of x. This is the

6

so-called quantum parallelism. We put its diagram (which is called a ‘circuit-gate’) here:

|0〉⊗k

|0〉⊗m Hm

Uf1√2m

∑2m−1x=0 |x〉 ⊗ |f(x)〉

In the diagram, the states of every component systems ( |0〉⊗m and |0〉⊗k)are called ‘circuits’ and are drawn at the left hand side. The unitary opera-tors corresponding to the evolutions of every component systems are called‘gates’ and are drawn as the boxes in the middle. The result at the righthand side is the final state of the composite system.

Note 3.2. Comparing with a classical computer, one of the most importantadvantages of a quantum computer is the simultaneous computing ability,which is reflected perfectly by the quantum parallelism in the above exampleof the Deutsch algorithm. Hence, we can understand why the Deutsch algo-rithm can become very powerful on a quantum computer. Moreover, Deutschalgorithm is widely used in the research of quantum computation since it isvery simple and hence efficient.

We conclude this section by introducing the Deutsch-Josza generalization[2].

Let f : Zm2 → Z2 be any function. Then f is called constant if f(x) = f(y)for any x, y ∈ Zm2 , and f is called balanced if |f−1(0)| = |f−1(1)| = 2m−1.Suppose f is either constant or balanced, and we wish to know which situ-ation f is in. If we work on a classical computer, we have to test 2m−1 + 1values to know for sure the situation of f , which means 2m−1+1 steps of eval-uation need to be done. On the other hand, the Deutsch-Josza generalizationallowed us to get the result on a quantum computer with a single operation(algorithm), which contains only four steps of evaluation. This shows thata quantum computer can have tremendous advantage in calculating if thealgorithms are suitable.

Here is the diagram for this algorithm:

7

|1〉

|0〉⊗m Hm

Uf g

Hm

H

The initial state is |0〉⊗m ⊗ |1〉 on H2m ⊗ H2, where Hm is the tensor

products of Hadamard gate. Let |+〉 = |0〉+|1〉√2

and |−〉 = |0〉−|1〉√2

, and let

g = 〈0|⊗m ⊗ 〈−| be a linear functional on H2m ⊗H2.The first stage of the algorithm is:

(Hm ⊗H)(|0〉⊗m ⊗ |1〉) = (Hm|0〉⊗m)⊗ (H|1〉)

= (1√2m

2m−1∑x=0

|x〉)⊗ |0〉 − |1〉√2

= S ⊗ |−〉

Recall that Uf (|x〉|y〉) = |x〉|y ⊕ f(x)〉 for any |x〉 ∈ Zm2 and |y〉 ∈ Z2.Then the second stage is:

Uf (S ⊗ |−〉) = Uf (S ⊗|0〉 − |1〉√

2)

=1√2Uf (S ⊗ |0〉 − S ⊗ |1〉)

=1√

2m+1

[∑2m−1x=0 Uf (|x〉 ⊗ |0〉)−

∑2m−1x=0 Uf (|x〉 ⊗ |1〉)

]=

1√2m+1

[∑2m−1x=0 (|x〉 ⊗ |f(x)〉)−

∑2m−1x=0 (|x〉 ⊗ |1⊕ f(x)〉)

]= ± 1√

2m

[∑2m−1x=0 (−1)f(x)|x〉 ⊗ |−〉

]We can see that the first row of Hm is (1, 1...1), while each of the other

rows consists of 2m−1 “1” and 2m−1 “−1”.If f is constant, then we have:

Hm

(∑2m−1x=0 (−1)f(x)|x〉

)= ±Hm

(∑2m−1x=0 |x〉

)= ±|0〉⊗m .

8

If f is balanced, then the first row of Hm acting on∑2m−1

x=0 (−1)f(x)|x〉yields zero. Then we have:

Hm

(∑2m−1x=0 (−1)f(x)|x〉

)=∑2m−1

x=1 kx|x〉 ,

where kx is of the form 2m−2P2m

, for P in {0, 1...2m−1}.

Hence, after passing the second last gate (Hm ⊗ 12), the state becomes:

(Hm ⊗ 12)(∑2m−1

x=0 (−1)f(x)|x〉 ⊗ |−〉)

=

{±|0〉⊗m ⊗ |−〉 , if f is constant;

( k2|0...1〉⊗m + ...+ k2m |1...1〉⊗m )⊗ |−〉 , if f is balanced;

Then, after the final gate, we get the resulting state to be:{1 , if f is constant;

0 , if f is balanced;

Notice that the result contains no uncertainty. It would be a certainnumber rather than a probability density function: If we get 1 (0), then weknow the probability for f to be constant (balanced) is 1.

Note 3.3. Readers who want to acquire more knowledge about quantumgate may have a look at chapters six, nine and ten of [5].

4 Quantum Channels

A quantum system is called open if it has mass or energy exchange with theenvironment. Mathematically, an open quantum system is represented by asubset in a larger Hilbert space (or equivalently, a subset in a larger closedquantum system). And quantum channels are often applied to deal withopen quantum systems.

Before continuing, we shall introduce some basic theories in operator the-ory:

(i) Let H(k) = H ⊕H... ⊕H. Then there is a natural way to define the

9

norm and inner product on H(k) in order to make it a Hilbert space. Namely,∥∥∥∥∥∥∥h1...hk

∥∥∥∥∥∥∥2

= ‖h1‖2 + · · ·+ ‖hk‖2

and ⟨h1...hk

,

l1...lk

⟩H(k)

= 〈h1, l1〉H + · · ·+ 〈hk, lk〉H

where h1...hk

and

l1...lk

belong to H(k).

Let Mk(B(H)) denote the “tensor” of B(H), which is the set of k ×k matrices with entries from B(H). Let (Tij) denote a typical element ofMk(B(H)). Define:

(Tij)

h1...hk

=

∑k

j=1 T1jhj...∑k

j=1 Tkjhj

Then Tij becomes an operator on H(k). It is straightforward to verify that

Mk(B(H)) = B(H(k)).

(ii) If A is a C*-algebra, then Mk(A) is a C*-algebra as well. And sinceevery C*-algebra is isometrically ∗-isomorphic to a concrete C*-algebra, allthe theories of the “tensor” of B(H) work for a general C*-algebra A as well.

(iii) Let A,B be two C*-algebras and let E be a linear map from A to B.Then:

E is called positive if for any a ∈ A, a ≥ 0 implies E(a) ≥ 0.E is called k -positive if for the integer k (k ≥ 2), the ‘amplification

10

map’:E (k) : Mk(A)→Mk(B)

defined by E (k)((aij)) = (E(aij)) is a positive map.E is called completely positive if E (k) is a positive map for any integer

k.Completely positivity is a very strong condition. But W. Forrest Stine-

spring proved that every completely positive map can be regarded as a ‘com-pression’ of a ∗-homomorphism on a larger Hilbert space containing the for-mer space. We now give the description and proof of this result according to[8], page 43-45:

Theorem 4.2. (Stinespring’s dilation theorem). Let A be a unital C*-algebra, and let φ: A → B(H) be a completely positive map. Then thereexists a Hilbert space K, a unital ∗-homomorphism π: A → B(K), and abounded operator V : H → K with ‖φ(1)‖ = ‖V ‖2 such that

φ(a) = V †π(a)V .

Proof. Let A⊗H denote the algebraic tensor product of A and H. Then forany a⊗ x, b⊗ y in H, we define a symmetric product 〈, 〉 by

〈a⊗ x, b⊗ y〉 = 〈φ(b†a)x, y〉H

and make this product to be bilinear, where 〈, 〉H is the inner product on H.Define an inner product on the direct sum H(n) = H⊕ · · · ⊕ H by

⟨x1...xn

,

y1...yn

⟩H(n)

= 〈x1, y1〉H + · · ·+ 〈xn, yn〉H ,

and by this definition we can easily prove that H(n) is still a Hilbert space.Then, since φ is completely positive, we have that

⟨n∑j=1

aj ⊗ xj,n∑i=1

ai ⊗ xi

⟩=

⟨φn((a†iaj))

x1...xn

,

x1...xn

⟩H(n)

≥ 0 ,

which means 〈, 〉 is positive semidefinite on A⊗H.

11

Hence, because of the bilinearity and the positive semidefinite property,〈, 〉 satisfies the Cauchy-Schwarz inequality: |〈u, v〉|2 ≤ 〈u, u〉 · 〈v, v〉. So wehave that

{u ∈ A⊗H|〈u, u〉 = 0} = {u ∈ A⊗H|〈u, v〉 = 0 for any v ∈ A⊗H}

is a subspace of A ⊗ H, which we denote as N . Also, the induced bilinearform on the quotient space A⊗H/N defined by 〈u+N , v+N〉 = 〈u, v〉 willbe an inner product.

Define K to be the completion of the inner product space A⊗H/N . ThenK is a Hilbert space.

For any a ∈ A, we define a linear map π(a): A⊗H → A⊗H to be

π(a)(∑

ai ⊗ xi)

=∑

(aai)⊗ xi .

Also, notice that in Mn(A) we have:

(a†iaj) =

a†1...a†n

·a1...an

T

and

(a†ia†aaj) =

(aa1)†

...(aan)†

·aa1...aan

T

So it is not hard to get that (a†ia†aaj) ≤ ‖a†a‖ · (a†iaj) always holds true in

Mn(A)+. Therefore, we have⟨π(a)

(∑aj ⊗ xj

), π(a)

(∑ai ⊗ xi

)⟩=∑i,j

〈φ(a†ia†aaj)xj, xi〉H

≤ ‖a†a‖ ·∑i,j

〈φ(a†iaj)xj, xi〉H

= ‖a‖2 ·⟨∑

aj ⊗ xj,∑

ai ⊗ xi⟩

12

Thus, since π(a) leaves N invariant, it induces a quotient linear transfor-mation on A ⊗ H/N , which we still denote by π(a). The above inequalityalso indicates that ‖π(a)‖ ≤ ‖a‖. Thus, π(a) extends to a bounded linearoperator on K, which we still denote by π(a).

Also, it is easy to see that the map π: A → B(K) is a unital ∗-homomorphism.Now define the mapping V from H to K by

V (x) = 1⊗ x+N .

Then since

‖V x‖2 = 〈1⊗ x, 1⊗ x〉 = 〈φ(1)x, x〉H ≤ ‖φ(1)‖ · ‖x‖2 ,

we know that V is bounded.Also, it is easy to see that ‖V ‖2 = sup{〈φ(1)x, x〉H : ‖x‖ ≤ 1} = ‖φ(1)‖ .Finally, observe that

〈V †π(a)V x, y〉H = 〈π(a)1⊗ x, 1⊗ y〉K = 〈φ(a)x, y〉H

holds for any x and y in H. Therefore, we get V †π(a)V = φ(a).

We call the triple (π, V,K) a Stinespring representation. Then, a Stine-spring representation is called minimal if the closure of span{π(A)VH} isK.

Another result from [8] shows that any two minimal Stinespring rep-resentations of the same completely positive map φ are actually unitarilyequivalent. We now provide this result according to [8], page 46-47:

Proposition 4.3. Let A be a C*-algebra, let φ: A → B(H) be completelypositive, and let

(πi, Vi,Ki) , i = 1, 2,

be two minimal Stinespring representations for φ. Then there exists a unitaryU : K1 → K2 satisfying UV1 = V2 and Uπ1U

† = π2.

Proof. If U exists, then it has to satisfy

U

(∑i

π1(ai)V1hi

)=∑i

π2(ai)V2hi .

Also note that by the minimal condition, U will have dense range and

13

hence be onto. Hence, if we can show that the above formula yields a well-defined isometry from K1 to K2, we will complete the proof.

To this end, observe that∥∥∥∥∥∑i

π1(ai)V1hi

∥∥∥∥∥2

=∑i,j

〈V †1 π1(a†iaj)V1hj, hi〉

=∑i,j

〈φ(a†iaj)hj, hi〉 =

∥∥∥∥∥∑i

π2(ai)V2hi

∥∥∥∥∥2

.

So, U is isometric and therefore well defined, as desired.

Researchers in the fields of quantum physics and operator theory havebeen devoting themselves to the research of completely positive maps for thelast three decades. However, many important results were obtained indepen-dently in these two fields without knowing the works of the other. The proofof Stinespring’s dilation theorem is an example of this. The books of Kraus[5] and Paulsen [8] discuss the subject from the perspectives of physicistsand mathematicians respectively.

Now we are ready to introduce the main theories of this section:

Definition 4.4. A quantum channel is a map E from B(H) to B(H) whichis completely positive and trace preserving.

Let dimH = N , then we have B(H) ∼= MN , and we may use the nota-tion HN here. Then under a given basis of HN , the trace of an operator isjust the trace of the corresponding matrix. Mathematically, trace preservingmeans: for any ρ ∈ B(HN), Tr(E(ρ)) = Tr(ρ). In quantum information, itis equivalent to requiring that the probabilities remain the same as a stateevolve through the channel.

We require a quantum channel to be positive because density operatorsmust evolve to density operators, and we require a quantum channel to becompletely positive because the tensor product of the initial system and an-other quantum system should also have this ‘density operator preserving’property.

The following theorem was proved by Choi [7] and Kraus [5] indepen-dently. We provide Choi’s operator proof here, which is cited from [7].

14

Theorem 4.5. Let E: B(HN)→ B(HN) be a completely positive map. Thenthere are operators Ek ∈ B(HN), where k = 1, 2, ..., r and 1 ≤ r ≤ N2, suchthat

E(ρ) =r∑

k=1

EkρE†k for all ρ ∈ B(HN) . (1)

Proof. Define eij = |i〉〈j|, where |i〉, |j〉 are from the standard basis of HN .

It is straightforward to show that (eij) is a positive matrix in B(H(N)N ) =

MN(B(HN)). Then R = E (N)((eij)) is also a positive matrix by the N-positivity of E .

And since R is positive, it can be diagonalized. Therefore, there is adecomposition: R =

∑rk=1 |ak〉〈ak|, where the |ak〉’s are the normalized

eigenvectors of R and r ≤ N2. (It is straightforward to justify that thenormalization can always been done. And obviously |ak〉 are linearly in-dependent.) It is easy to see that |ak〉 ∈ CN2

. We decompose CN2into

CN2= CN ⊕ ... ⊕ CN . Let {Pi : 1 ≤ i ≤ N} be the set of projections onto

each CN . Then Pi have mutually orthogonal ranges to each other and satisfy

PiRPj =

0 · · · 0... E(eij)

...0 · · · 0

(zero matrix in every N ×N sub-block except an

E(eij) in the block of i’th “row” and j’th “column”). Also, |ak〉 =∑N

i=1 Pi|ak〉.Define operators Ek : CN → CN to be Ek|i〉 ≡ Pi|ak〉. Then

R =∑k

∑i,j

Pi|ak〉〈ak|Pj =∑i,j

Pi

(∑k

Ek|i〉〈j|E†k

)Pj .

Therefore,

E(ei,j) = E(|i〉〈j|) = PiRPj =r∑

k=1

Ek|i〉〈j|E†k .

Finally, we get equation (1) by the linearity of E .

In quantum information, equation (1) is called the operator -sum representationof E . The operators Ek are referred to as error or the noise operators of thechannel. Moreover, since we only work on finite dimensional Hilbert space,

15

we can regard operators as matrices. Then for any operators A,B on HN ,we have Tr(AB) = Tr(BA).

Then, for any ρ ∈ B(HN), we have

Tr(E(ρ)) = Tr(∑k

EkρE†k) =

∑k

Tr((Ekρ)E†k) =∑k

Tr(E†k(Ekρ))

= Tr(∑k

E†k(Ekρ)) = Tr((∑k

E†kEk)ρ)

Then, Tr(E(ρ)) = Tr(ρ) if and only if Tr((∑

k E†kEk)ρ) = Tr(ρ). Hence

the trace preservation of E is equivalent to requiring its noise operators Ekto satisfy ∑

k

E†kEk = 1 ,

where 1 is the identity operator.

Remark 4.6. Sometimes we want to fully recover an unknown quantumchannel from only a small portion of experimental data. Now, with Choi’swork, we only need to recover the noise operators of a quantum channel inorder to achieve this. Hence, Choi’s work provides us a manner to recover theunknown, complicated quantum channel from known, simple experimentaldata.

The following theorem indicates the connections between different sets ofnoise operators for the same channel. This theorem is frequently used in thefollowing part of this paper.

Theorem 4.7. On a Hilbert space HN , let {E1, ... , Er} and {F1, ... , Fs} betwo sets of linearly independent noise operators for channels E and E ′ re-spectively. Then E = E ′ if and only if r = s and there exists an r × r scalarunitary matrix U = (uij), such that

Ei =s∑j=1

uijFj for 1 ≤ i ≤ r (2)

Proof. We can write (1) as

E(ρ) =r∑

k=1

E†kρEk , (3)

16

by replacing Ek with E†k. The form (3) is more convenient for us to use theStinespring’s dilation theorem, we replace (1) by (3) in the following proof.But we still go back to (1) later on, since the form (1) is the convention ofquantum physics.

Suppose E = E ′. Then we have

E(ρ) =r∑i=1

E†i ρEi =s∑j=1

F †j ρFj

=(E†1, ..., E

†r

)ρ . . .

ρ

E1

...Er

=(F †1 , ..., F

†s

)ρ . . .

ρ

F1

...Fs

Let K1 =∏

r times

⊗HN , K2 =∏

s times

⊗HN . We can see that K1 ' CNr and

K2 ' CNs. Moreover:

Define

V1 =

E1...Er

and V2 =

F1...Fs

.

So, V1: HN → K1 and V2: HN → K2.

Define

π1(ρ) =

ρ . . .

ρ

= ρ⊗ Ir and π2(ρ) =

ρ . . .

ρ

= ρ⊗ Is .

Then it is straightforward to prove π1: MN = B(HN) → B(K1) and π2:MN = B(HN)→ B(K2) are two ∗-homomorphism between C*-algebras.

Therefore, E(ρ) = V †1 π1(ρ)V1 = V †2 π2(ρ)V2. Then we get that (π1, V1,K1)and (π2, V2,K2) are two Stinespring representations.

Now look at span{π1(MN)V1HN}:

17

Let hk = (0, 0, ..., 1, ..., 0)T be the element in the standard basis of HN ,where the 1 is at the k’th position and 0 at all the others. Let eij =0 · · · 0

... 1...

0 · · · 0

be the element in the standard basis of MN , where the 1 is

in the i’th row and j’th column and 0 at all the other places.Let fijk = π1(eij)V1(hk), then

fijk =

eij . . .

eij

E1

...Er

(hk) =

eij . . .

eij

E1(hk)

...Er(hk)

=

eijE1(hk)...

eijEr(hk)

=

E(1)jk hi...

E(r)jk hi

where E

(τ)jk is the entry in the j’th row and k’th column of the matrix Eτ . It

is a scalar. (i.e. fijk is the column vector with zero in all positions except

E(τ)jk in the (τ + i)’th position for τ = 1, 2, · · · , r.)

Fix i, and define

Ai = (fi11 · · · fi1N · · · fiN1 · · · fiNN) =

E(1)11 hi · · · E

(1)NNhi

... · · · ...

E(r)11 hi · · · E

(r)NNhi

.

Then, Ai is a r ×N2 matrix.Now, we note the τ ’th row of Ai to be gτ . So gτ is a N2-dimensional

vector. Then for any scalars x1, x2, ..., xr, consider the linear combination:

x1g1 + x2g2 + · · ·+ xrgr = 0 ,

which is equivalent toE

(1)11 x1 + · · ·+ E

(r)11 xr = 0

...

E(1)NNx1 + · · ·+ E

(r)NNxr = 0

,

18

which is equivalent tox1E1 + · · ·+ xrEr = 0 .

Then x1 = x2 = · · · = xr = 0 because of the linearly independence ofEk’s.

So rankAi = rank{fi11, fi12 , ..., fiNN} = rank{g1, g2, ..., gr} = r. Sorank{A1, A2, ..., AN} = Nr. (fijk and fi′j′k′ are linearly independent if i 6= i′,

since the E(i)jk and E

(i′)j′k′ are not in the same positions.)

Then we have:

span{fijk} = span{π1(eij)V1(hk)} = span{π1(MN)V1HN} = CNr = K1 ,

which means (π1, V1,K1) is minimal. Similarly, (π2, V2,K2) is minimal aswell.

Hence, by proposition 4.3, there exists an unitary map U : K1 → K2, suchthat V2 = UV1 and π2 = Uπ1U

†. Thus we get r = s from the unitary. Soactually, π1 = π2.

Then for any ρ ∈ B(HN) and any h ∈ HN , we have{π2(ρ)V2h = π1(ρ)V2h = π1(ρ)UV1h

π2(ρ)V2h = Uπ1(ρ)U †V2h = Uπ1(ρ)U †UV1h = Uπ1(ρ)V1h

So Uπ1(ρ)V1h = π1(ρ)UV1h for any ρ ∈ B(HN), any h ∈ HN . So we haveUπ1(ρ) = π1(ρ)U for any ρ ∈ B(HN).

Let U =

U11 · · · U1r...

. . ....

Ur1 · · · Urr

. Then we have

U11 · · · U1r...

. . ....

Ur1 · · · Urr

ρ . . .

ρ

=

ρ . . .

ρ

U11 · · · U1r

.... . .

...Ur1 · · · Urr

.

Or equivalently,U11ρ · · · U1rρ...

. . ....

Ur1ρ · · · Urrρ

=

ρU11 · · · ρU1r...

. . ....

ρUr1 · · · ρUrr

.

19

So, Uijρ = ρUij for i, j ∈ {1, 2, ..., r}. Hence, we get Uij = uijI, where

U0 =

u11 · · · u1r...

. . ....

ur1 · · · urr

is unitary because U is unitary.

Finally, we have

V2 =

F1...Fr

= UV1 =

u11I · · · u1rI...

. . ....

ur1I · · · urrI

E1

...Er

= U0 ⊗ I

E1...Er

.

Hence, E1...Er

= U †

F1...Fr

= U †0 ⊗ I

F1...Fr

,

which is just the form of (2).

Conversily, suppose r = s and there exists a unitary matrix U = (uij)such that the form (2) holds true. Then for any ρ ∈ B(HN), we have:

E(ρ) =r∑i=1

EiρE†i =

r∑i=1

( r∑j=1

uijFj

)ρ

(r∑j=1

uijFj

)†=

r∑i=1

[(r∑j=1

uijFj

)ρ

(r∑j=1

uijF†j

)]=

r∑i=1

[(r∑j=1

uijFj

)ρ

(r∑j=1

u†jiF†j

)]

=r∑i=1

(F1 · · · Fr)ui1...

uir

ρ(u†1i · · · u†ri

)F†1...F †r

=r∑i=1

(F1 · · · Fr) (uiju

†ki

)ρ . . .

ρ

F

†1...F †r

20

=(F1 · · · Fr

) (∑ri=1 u

†kiuij

)ρ . . .

ρ

F

†1...F †r

=(F1 · · · Fr

)U †U

ρ . . .

ρ

F

†1...F †r

=(F1 · · · Fr

)ρ . . .

ρ

F

†1...F †r

=r∑j=1

FrρF†r = E ′(ρ)

Thus, since ρ is arbitrary, we have E = E ′.

The following corollary show that any set of noise operators can be ex-pressed by a set of linearly independent noise operators of the same channel.

Corollary 4.8. Let E be a quantum channel with noise operators {E1, ... , Er}on the Hilbert space HN , where Rank{Ei} = s. Then there exists a set oflinearly independent noise operators {F1, ... , Fs} of E, such that

Ei =s∑j=1

λijFj for 1 ≤ i ≤ r

Proof. Without loss of generality, suppose {E1, ... , Es} is a basis of {E1, ... , Er}.

Then, let Ei =∑s

j=1 aijEj for s+1 ≤ i ≤ r. And let A =

as+1,1 · · · ar,1...

......

as+1,s · · · ar,s

be the scalar matrix consisting of aij. Then we have(

E1 · · · Er)

=(E1 · · · Es

)·(Is A

),

where each matrix Ei is multiplied by a scalar at the right hand side of theequation.

Let(Is A

)·(

IsA†

)= K, then K ≥ Is. So K is positive and invertible

in B(HN), which means that it is diagonalizable under an orthogonal basis.

21

In other words, K ∼=

µ1

. . .

µs

under an orthogonal basis {u1, ..., us}.

Let ξj =√µjuj, then we have K =

∑sj=1 ξjξ

†j .

Then, let Fj =(E1 · · · Es

)yj1...yjs

=∑s

l=1 yjlEl for 1 ≤ j ≤ s, where

yj1...yjs

= ξj. It is not hard to show that {F1, ..., Fs} is linearly independent.

And for any ρ ∈ B(H), we have

E(ρ) =r∑i=1

EiρE†i =

(E1, ..., Er

)ρ . . .

ρ

E

†1

...E†r

=(E1, ..., Es

) (Is A

)ρ . . .

ρ

( IsA†

)E†1

...E†s

=(E1, ..., Es

)(

s∑j=1

ξjρξ†j )

E†1

...E†s

=s∑j=1

FjρF†j ,

which means that {F1, ..., Fs} is a set of linearly independent noise operatorsof E . And obviously, there exists a s× s scalar matrix (λij) such that Ei =∑s

j=1 λijFj for 1 ≤ i ≤ r. Therefore, we complete the proof.

Then, we can immediately get:

Corollary 4.9. For a fixed quantum channel E, the rank of its noise opera-tors remains invariant under transformations of noise operators.

Hence, we can define the rank of a quantum channel to be the rank of itsnoise operators.

There is a simple but widely used C*-algebra in the study of quantumcomputing:

22

Consider the algebra generated by a pair of adjoint operators: A =Alg{Ei, E†i }. Then A is the set of polynomials of Ei and E†i . We call A theinteraction algebra in quantum computation, and obviously A is †-closed.Since A is a subspace of B(HN), it is finite dimensional.

The interaction algebra A is widely used in the study of quantum channel.

Note 4.10. The study of quantum channel capacities is highly active in theresearch of quantum channel. A series of deep mathematical problems havebeen realized to have connections with computing the capacity of a quantumchannel to carry information.

Now let us consider more specific examples.

Examples 4.11. (i) Let the basis of H2 be {|0〉, |1〉}. For any p ∈ (0, 1),define two operators on H2 by

E1 = (√

1− p)12 , E2 = (√p)X ,

where X is one of the Pauli matrices described in section 2. Notice that E†1 =E1 and E†2 = E2. Then the bit flip channel is defined as E = E1ρE

†1+E2ρE

†2.

It flips the vector (state) |0〉 to |1〉 with probability p and vice versa. Forinstance, for an vector v ∈ H2, let v = a|0〉+ b|1〉, we have

E(|0〉〈0|)(v) = E1|0〉〈0|E1(v) + E2|0〉〈0|E2(v)

= E1|0〉〈0|(√

1− p)(a|0〉+ b|1〉) + E2|0〉〈0|(√p)(b|0〉+ a|1〉)

= (√

1− p)E1(a|0〉) + (√p)E2(b|0〉)

= (1− p)a|0〉+ p · b|1〉= ((1− p)|0〉〈0|+ p|1〉〈1|) · (a|0〉+ b|1〉)= ((1− p)|0〉〈0|+ p|1〉〈1|) (v)

Hence, since v is arbitrary, we get

E(|0〉〈0|) = (1− p)|0〉〈0|+ p|1〉〈1|

(ii) Now, let E1 = 1212 (again, E†1 = E1) and define the channel E to be

E(ρ) = E1ρE1 + σxρσx + σyρσy + σzρσz ,

where σk = 12K for k = x, y, z and K = X, Y, Z are the spin -1

2Pauli matrices

described in section 2.

23

Also, an operator ρ on a separable Hilbert space H is called density,if ρ =

∑i pi|ψi〉〈ψi|, where 0 < pi < 1,

∑i pi = 1 and 〈ψi||ψi〉 = 1. Let

ρ =

(a bc d

)be a density operator on H2. Then for any v = r|0〉 + s|1〉, we

have

E(ρ)(v) =(E1ρE1 + σxρσx + σyρσy + σzρσz)(v)

=E1ρE1(v) + σxρσx(v) + σyρσy(v) + σzρσz(v)

=1

4[(ar + bs)|0〉+ (cr + ds)|1〉] +

1

4[(cs+ dr)|0〉+ (as+ br)|1〉]

+1

4[(−cs+ dr)|0〉+ (as− br)|1〉] +

1

4[(ar − bs)|0〉+ (−cr + ds)|1〉]

=1

2[(a+ d)r|0〉+ (a+ d)s|1〉]

=1

2tr(ρ) · [r|0〉+ s|1〉]

Since ρ is density, we have

tr(ρ) =∑τ=0,1

∑i

〈τ ||ψi〉Pi〈ψi||τ〉 =∑i

〈ψi||ψi〉Pi = 1

So, E(ρ)(v) = 12[r|0〉 + s|1〉] = 1

212(v). Hence by the arbitrary of v, we

have E(ρ) = 1212, which means E turns every density operator (matrix) into

the same density operator (matrix) 1212.

(iii) Let 0 < r < 1 and define operators on H2 by

E1 =

(1 00√

1− r

)E2 =

(0√r

0 0

)Then E(ρ) = E1ρE

†1 +E2ρE

†2 is called the amplitude damping channel. This

channel is related to the energy dissipation in a quantum system.(iv) Choose positive real numbers r1, · · · , rd such that

∑i ri = 1, and let

U1, · · · , Ud be unitaries acting on a common Hilbert space H. Then defineE(ρ) =

∑di=1 riUiρU

†i . In section 6 we will work on the ‘unital’ channel and

this one is a prototypical example of the unital channel. Moreover, it is easyto see that every unital channel on H2 can be written as a (convex) summa-tion of unitaries. But this is not true to a non-unital channel.

(v) An entanglement breaking channel is the channel which can be writ-

24

ten asE(ρ) =

∑k

|ψk〉〈ψk|〈φk|ρ|φk〉 ,

for some vectors |ψk〉 and |φk〉. Then since tr(ρ|φk〉〈φk|) = 〈φk|ρ|φk〉, thetrace preserving of E is equivalent to

∑k |φk〉〈φk| = 1. This channel is called

entanglement breaking because E (d)(Γ) will never be entangled for any d ≥ 1and any initial operator Γ.

5 Quantum Error Correction

There is no computing system, including the quantum computing system,that can fully eliminate errors. Moreover, errors occurred in a quantumcomputing system can be more complicated than those occurred in a clas-sical computing system, since the only kind of errors occurred in classicalcomputing is bit flips. But fortunately, many precise and delicate methodshave been developed for quantum error detection and correction.

Because of the lack of knowledge, we only discuss the quantum error de-tection and correction with respect to a quantum channel. We shall firstpresent the basic theories in quantum error detection and correction. Thenwe introduce some simple methods in quantum error correction.

I. Quantum Error Detection. Simply speaking, quantum error de-tection with respect to a quantum channel means to figure out all the noiseoperators of that channel. The theory of quantum error detection with re-spect to a quantum channel is fundamental and crucial in the entire quantumerror detection theory. This is because the operating of all kinds of quantumsystems is deeply involved with quantum channels. Moreover, we will seethat the manner used here is simple and hence useful.

Before continuing, we present the following settings:For a given closed quantum system, let H be the corresponding Hilbert

space. Then a quantum code on H can be any subspace C of H. Also, let PCbe the projection of H onto C. Then the projections PC and P⊥C are appliedto determine whether a given state |ψ〉 ∈ H belongs to the given quantumcode. So we can regard {PC, P⊥C } as a measurement (See section 2).

Correctly using the Projection PC is important to the quantum error de-tection with respect to a quantum channel. We need the following definition

25

for later discussion:

Definition 5.1. Let C be a quantum code on some Hilbert space H, and letE be an error (noise operator) corresponding to a given quantum channel onH. Suppose that C is fixed. Then we say C can detect the error E, if thereexists a scalar λE which only depends on E such that

PCE|ψ〉 = λE|ψ〉 for any |ψ〉 ∈ C .

We shall note that all the operators that can be detected by a fixedquantum code C form a subspace in B(H).

The following theorem will provide several equivalent conditions for errordetecting.

Theorem 5.2. Let C be a quantum code on some Hilbert space H, and letE be an error (noise operator) corresponding to a given quantum channel onH. Then the following conditions are equivalent:

(i) E can be detected by C, with scaling factor λE.(ii) PCEPC = λEPC, where PC is the projection from H to C.(iii) 〈ψ1|E|ψ2〉 = λE〈ψ1||ψ2〉 for any |ψi〉 ∈ C, i = 1, 2.(iv) For any pair of orthogonal vectors {|ψ1〉, |ψ2〉} in C, the vectors E|ψ1〉

and |ψ2〉 are also orthogonal.

Proof. The most difficult part is (iv) ⇒ (iii). We start from it.(iv) ⇒ (iii). Obviously we can assume that dim C ≥ 2, otherwise there

will be no orthogonal vectors in C. Define Ψ = {|ψ1〉, |ψ2〉, · · · } to be anorthonormal basis for C. Then for any |ψi〉, |ψj〉 ∈ Ψ and i 6= j, let |+〉 =|ψi〉 + |ψj〉 and |−〉 = |ψi〉 − |ψj〉. Then |+〉 and |−〉 are orthogonal to eachother in C. So by (iv) we have

〈ψi|E|ψi〉 − 〈ψj|E|ψj〉 = 〈ψi|E|ψi〉 − 〈ψi|E|ψj〉+ 〈ψj|E|ψi〉 − 〈ψj|E|ψj〉= 〈ψi|(E|ψi〉 − E|ψj〉) + 〈ψj|(E|ψi〉 − E|ψj〉)= (〈ψi|+ 〈ψj|)E(|ψi〉 − |ψj〉)= 〈+|E|−〉 = 0

So 〈ψi|E|ψi〉 = 〈ψj|E|ψj〉 holds for any |ψi〉, |ψj〉 ∈ Ψ and i 6= j.Therefore, define λE = 〈ψi|E|ψi〉. So λE is independent of i.Now, for any two vectors |ψ〉 and |φ〉 in C, let |ψ〉 = α1|ψ1〉+α2|ψ2〉+ · · ·

26

and |φ〉 = β1|ψ1〉+ β2|ψ2〉+ · · · . Then we have that

〈ψ|E|φ〉 =∑i,j

αiβj〈ψi|E|ψj〉 =∑i

αiβi〈ψi|E|ψi〉

= λE

(∑i

αiβi

)= λE

(∑i

αiβi〈ψi||ψi〉

)

= λE

(∑i,j

αiβj〈ψi||ψj〉

)

= λE

(∑i

αi〈ψi|

)(∑j

βj|φj〉

)= λE〈ψ||φ〉 ,

which is the result we want.(iii) ⇒ (ii). For any |x〉, |y〉 ∈ H, let |x〉 = |ψx〉 + |ψx〉⊥ and |y〉 =

|ψy〉+ |ψy〉⊥, where |ψx〉 and |ψy〉 belong to C while |ψx〉⊥ and |ψy〉⊥ belongto C⊥. Then we can get

〈y|PCEPC|x〉 = 〈y|PCE|ψx〉 = 〈ψy|E|ψx〉= λE〈ψy||ψx〉 = λE〈ψy|PC|x〉= λE〈y|PC|x〉 = 〈y|λEPC|x〉 .

Hence, since |x〉 and |y〉 are arbitrary, we get PCEPC = λEPC.(ii) ⇒ (i). Trivial.(i) ⇒ (iv). If |ψ1〉 and |ψ2〉 are orthogonal, then

〈ψ2|E|ψ1〉 = 〈ψ2|PCE|ψ1〉 = 〈ψ2|λE|ψ1〉 = λE〈ψ2||ψ1〉 = 0

So |ψ2〉 and E|ψ1〉 are orthogonal.

Remark 5.3. Since the Hilbert space H we work on is always finite dimen-sional, we can take a further discussion.

Let C be a fixed quantum code in H. Then we can write PC as

(1C 00 0

)if we appropriately choose the basis of H, and it is a finite matrix since B(H)

27

is finite dimensional. Also let E =

(A BC D

)be an operator in B(H) which

can be detected by C. Then the condition (ii) of Theorem 5.2 shows that(1C 00 0

)(A BC D

)(1C 00 0

)= λE

(1C 00 0

)=

(λE1C 0

0 0

).

But since(1C 00 0

)(A BC D

)(1C 00 0

)=

(1C 00 0

)(A 0C 0

)=

(A 00 0

),

we can get that an operator E can be detected by C if and only if

E =

(λE1C ∗∗ ∗

).

Or in other words,

E ∈{(

λ1C ∗∗ ∗

): λ ∈ C

}.

Now we provide a simple example of undetectable errors.

Example 5.4. Define the quantum code C in H8 to be

C = span{|000〉, |111〉} .

Then consider the operator E = Z1 = Z ⊗ 12 ⊗ 12, where Z is the Paulimatrix. Observe that E|000〉 = |000〉 and E|111〉 = −|111〉. Assume that Ecan be detected by C. Then we have

λE = 〈000|E|000〉 = 1 and λE = 〈111|E|111〉 = −1 ,

which is a contradiction. Hence, the operator E cannot be detected by thequantum code C.

We shall mention that the above operator E = Z1 is a noise operatorof the so-called 3-qubit depolarizing channel. And the n-qubit depolarizingchannel is E = {12n , Z1 , ... , Zn}, where

Zi = 12 ⊗ · · · ⊗ 12 ⊗ Z ⊗ 12 ⊗ · · · ⊗ 12

28

with Z in the i’th position.

II. Quantum Error Correction. On a finite dimensional Hilbert spaceH, let E be the given quantum channel and let C be the given quantum code.The basic idea of quantum error detection corresponding to this setting istrying to find another quantum channelR such that the compounded channelR◦E can “work well” on C, which means that all operators that are supportedon C will be invariant under R ◦ E . We present the following definition as amore clear description:

Definition 5.5. Let E be the given quantum channel and let C be the givenquantum code with projection PC. Then C is correctable for E if there existsa quantum channel R such that

R ◦ E(ρ) = ρ

for any ρ supported on C, that is, any ρ with ρ = PCρPC.

There are several equivalent conditions of correctable quantum codes,which will be discussed in the following theorem. Note that in the proof of(iii) ⇒ (i) below, the error correction channel R is explicitly constructed.This provides us a way to construct R.

Theorem 5.6. Let E be the given quantum channel with noise operators{Ei : 1 ≤ i ≤ p} and let C be the given quantum code with projection PC.Then the following conditions are equivalent:

(i) C is correctable for E.(ii) All the operators in the set {E†iEj : 1 ≤ i, j ≤ p} can be detected by

C.(iii) There exist a scalar matrix Λ = (λij) such that

PCE†iEjPC = λijPC for all 1 ≤ i, j ≤ p .

Proof. For (i) ⇒ (iii), let R be the corresponding error correction channelwith noise operators {Rs : 1 ≤ s ≤ q}. Define a new channel EC via EC(ρ) ≡E(PCρPC). Then by condition (i), we have

R(EC(ρ)) = R(E(PCρPC)) =∑s,i

RsEiPCρPCE†iR†s = PCρPC (4)

29

holds for any ρ ∈ B(H), since PCρPC is always supported on C.Then by (4), we can see that {RsEiPC : 1 ≤ s ≤ q, 1 ≤ i ≤ p} and {PC}

are two sets of noise operators for the channel R◦EC. Then by Corollary 4.8,there exists a set of linearly independent noise operators {Fk : 1 ≤ k ≤ r}of R ◦ EC such that RsEiPC =

∑rk=1 µsikFk. But by Theorem 4.7, we have

that {Fk : 1 ≤ k ≤ r} and {PC} must have the same cardinality. Theonly way this can happen is that Fk = βkPC for some scalars βk, where∑r

k=1 βk = 1. Hence, we can see that RsEiPC = αsiPC for some scalars αsi,where

∑qs=1

∑pi=1 αsi = 1. Hence,

PCE†iR†sRsEjPC = αsiαsjPC for all i, j, s. (5)

Then since R is trace preserving, we have that∑q

s=1R†sRs = 1. So sum

over (5) with respect to s, we have

PCE†iEjPC = λijPC for all i, j,

where λi,j =∑q

s=1 αsiαsj.

For (iii) ⇒ (i), let A = [E1PC E2PC · · · ]. Then the formula in condition(iii) can be written as A†A = (λijPC). Since A†A is positive, it is straight-forward to show that Λ = (λij) is positive, and hence diagonalizable. So wecan find a unitary scalar matrix U such that U †ΛU = diag(dkk) = D. Thus,∑

k

dkk = Tr(D) = Tr(U †ΛU) = Tr(ΛUU †) = Tr(Λ) =∑i

λii .

But since E is trace preserving, we have∑

iE†iEi = 1. Hence,(∑

i

λii

)PC =

∑i

λiiPC =∑i

PCE†iEiPC = PC

(∑i

E†iEi

)PC = PC .

So we have that Tr(D) =∑

k dkk = 1.Let Fk =

∑i uikEi. Then {Fk} is another set of noise operators of E

according to Theorem 4.7. And a simple computation can show that

PCF†kFlPC = dklPC for all k, l.

30

The polar decomposition of FkPC is

FkPC = Uk

√PCF

†kFkPC =

√dkkUkPC ,

where we may assume Uk to be unitary rather than partial isometry as H isfinite dimensional.

Define projections Pk ≡ UkPCU†k . Recall that U †ΛU = diag(dkk) = D. So

dlk = 0 if l 6= k. Then

PlPk = UlPCU†l UkPCU

†k = Ul(UlPC)

†(UkPC)U†k

=Ul(FlPC)

†(FkPC)U†k√

dlldkk=UlPCF

†l FkPCU

†k√

dlldkk

=dlk√dlldkk

UlPCU†k = 0 , if k 6= l .

So the Pk have mutually orthogonal ranges.Without loss of generality, we can assume that

∑k Pk = 1 (Otherwise

we can add the projection which is onto the orthogonal complement.) Thendefine a new channel via

R(ρ) =∑k

U †kPkρPkUk .

Then for any operator ρ that satisfies ρ = PCρPC, we have

R(E(ρ)) =∑k,l

U †kPkFlρF†l PkUk =

∑k,l

U †kUkPCU†kFlρF

†l UkPCU

†kUk

=∑k,l

PCU†kFlρF

†l UkPC =

∑k,l

(UkPC)†FlρF

†l (UkPC)

=∑k,l

(FkPC√dkk

)†FlρF

†l

(FkPC√dkk

)=∑k,l

1

dkkPCF

†kFlPCρPCF

†l FkPC

=∑k,l

dkldlkdkk

PCρPC =∑k

dkkρ = ρ .

31

And hence C is correctable for E .

Condition (ii) and (iii) are equivalent by Theorem 5.2. So finally wecomplete the proof.

Now we provide an example to show how to determine whether a quantumcode is correctable for a given channel by using Theorem 5.6.

Example 5.7. Define two orthogonal vectors in H29 via

|0L〉 =(|000〉+ |111〉)⊗ (|000〉+ |111〉)⊗ (|000〉+ |111〉)

2√

2

|1L〉 =(|000〉 − |111〉)⊗ (|000〉 − |111〉)⊗ (|000〉 − |111〉)

2√

2

Then Shor’s 9-qubit code is defined as CS = span{|0L〉, |1L〉}.For a fixed k in {1, 2, ... , 9}, let ES be the quantum channel with noise

operators {Xk, Yk, Zk}, where

Zk = 12 ⊗ · · · ⊗ 12 ⊗ Z ⊗ 12 ⊗ · · · ⊗ 12

and Z is in the k’th position. Xk, Yk are similarly defined.Let B = {|0...0〉, ... , |1...1〉} be a basis of H29 , where the basis ele-

ments can be regarded as a list of the first 2n binary numbers. Then both|0L〉 and |1L〉 are the linear combination of eight basis elements, which are|000〉|000〉|000〉, ... , |111〉|111〉|111〉, which are the τ1’th basis element, τ2’thbasis element, ... and τ8’th basis element in B. Note that all the coefficientsof the basis elements in |0L〉 and |1L〉 have the same absolute value 1

2√2. Then

the value of each τi and the corresponding signals for |0L〉 and |1L〉 are:

signalτi’s value

1 23 26−23 +1 26

|0L〉 + + + +|1L〉 + − − +

signalτi’s value

29−26 +1 29−26+23 29−23 +1 29

|0L〉 + + + +|1L〉 − + + −

32

Observe that we can separate the eight basis elements into four groups.In each group the difference of the two τ is 7.

And under the basis defined above, we have

Xk =

[0 12k−1

12k−1 0

][

0 12k−1

12k−1 0

]. . .

,

Yk =

[0 i12k−1

−i12k−1 0

][

0 i12k−1

−i12k−1 0

]. . .

,

and

Zk =

[12k−1 0

0 −12k−1

][12k−1 0

0 −12k−1

]. . .

.

So we can see that Xk = X†k, Yk = Y †k and Zk = Z†k.Now we consider the position change of a basis element under Xk, Yk and

Zk (Or in other words, the τ ’s change under these operators): Suppose thatτ = p · 2k−1 + q, where p, q ∈ N and q < 2k−1. Then we have

Zk : τ 7→ τ ,

and

Xk, Yk : τ 7→

{(p− 1)2k−1 + q , if p is even

(p+ 1)2k−1 + q , if p is odd.

Then we have XkYk : τ 7→ τ , and so is YkXk, XkXk, and YkYk. Hence,

33

we have {PCX

†kXkPC = PC , PCX

†kYkPC = iPC

PCY†kXkPC = iPC , PCY

†k YkPC = −i · iPC = PC

(6)

Finally, since τ is invariant under Zk but variant under Xk and Yk, wehave

PCX†kZkPC = PCY

†k ZkPC = PCZ

†kXkPC = PCZ

†kYkPC = 0 . (7)

Observe that (6) and (7) establish the condition (iii) in Theorem 5.6.Hence, CS is correctable for ES.

6 Noiseless Subsystems via The Noise Com-

mutant

In section 5, we mainly focused on two problems: whether a given quantumcode is correctable for a given quantum channel, and what kind of quantumchannel a given quantum code is correctable for. But in experiments, weoften need to consider another problem, that is, what kind of quantum codeis correctable for a given quantum channel. In section 6, we will provide abasic but effective method to show how to solve this problem when the givenchannel is unital. We call this method the noiseless subsystem method, orthe noiseless subsystem via noise commutant method.

Let E : B(H)→ B(H) be a unital channel with noise operators {E1, ..., Er}.Recall that E is unital if and only if E(1) =

∑ri=1EiE

†i = 1. Define

A = Alg{E1, ..., Er} and A† = Alg{E†1, ..., E†r}. Define the noise commutantto be

A′ = {ρ ∈ B(H) : ρA = Aρ for any A ∈ A}= {ρ ∈ B(H) : ρEi = Eiρ for i = 1, ..., r} .

And defineFix(E) = {ρ ∈ B(H) : E(ρ) = ρ} .

Also, let C be a quantum code which is correctable for E and let R be thecorresponding correction channel for C as in Definition 5.5. Then we say thatC is a noiseless subsystem if R is the identity channel, that is, R(ρ) = ρ for

34

any ρ ∈ C.The main result of this section is showing that A′ is a noiseless subsystem.We shall note that Fix(E) is †-closed. Indeed, if ρ ∈ Fix(E), then we have

E(ρ†) =r∑i=1

Eiρ†E†i =

r∑i=1

(EiρE†i )† = (

r∑i=1

EiρE†i )† = (E(ρ))† = ρ† .

Thus Fix(E) is †-closed. Also, it is straightforward to show that Fix(E) is asubspace of B(H).

Before continuing, we shall provide the supportive theories of this section,which are Lemma 6.1 to Lemma 6.5. We shall mention that all the proofs inthis section are provided in [6].

Lemma 6.1. Let H be a general Hilbert space and let ρ be a positive andcontractive operator in B(H). Then for any vector |ψ〉 in H, we have that(ρ|ψ〉, |ψ〉) ≤ (|ψ〉, |ψ〉). Moreover, the equality holds if and only if ρ|ψ〉 =|ψ〉.

Proof. Since ρ is positive, we have ρ = δδ† for some δ ∈ B(H). Since ρis contractive and B(H) is a C*-algebra, we have ‖ρ‖ = ‖δδ†‖ = ‖δ‖2 =‖δ†‖2 ≤ 1. So δ and δ† are contractive as well. Then

(ρ|ψ〉, |ψ〉) = (δδ†|ψ〉, |ψ〉) = (δ†|ψ〉, δ†|ψ〉) = ‖δ†|ψ〉‖2 ≤ ‖|ψ〉‖2 = (|ψ〉, |ψ〉) ,

which proves the inequality.If (ρ|ψ〉, |ψ〉) = (|ψ〉, |ψ〉), we assume that ρ|ψ〉 = |ψ〉+ |φ〉. Then we have

(ρ|ψ〉, |ψ〉) = (|ψ〉+ |φ〉, |ψ〉) = (|ψ〉, |ψ〉) + (|φ〉, |ψ〉) = (|ψ〉, |ψ〉) ,

which shows that (|φ〉, |ψ〉) = 0. Then, since ρ is contractive, we have

‖ρ|ψ〉‖2 = ‖|ψ〉+ |φ〉‖2 = (|ψ〉+ |φ〉, |ψ〉+ |φ〉) = ‖|ψ〉‖2 + ‖|φ〉‖2 ≤ ‖|ψ〉‖2 .

Thus obviously |φ〉 = 0, which means ρ|ψ〉 = |ψ〉. And the other direction istrivial. Thus we complete the proof.

Lemma 6.2. Suppose E : B(H)→ B(H) is a unital quantum channel withnoise operators {E1, ... , Er}, where H is a finite dimensional Hilbert space.Let ρ be an operator in B(H) with 0 ≤ ρ ≤ E(ρ). Then the subspaceker(ρ− ‖ρ‖1) is E†i -invariant for all 1 ≤ i ≤ r.

35

Proof. Without loss of generality, suppose ‖ρ‖ = 1. Then ρ is contractive.Let M = ker(ρ − 1). Then according to Lemma 6.1, for any |ψ〉 ∈ M, wehave

‖|ψ〉‖2 = (ρ|ψ〉, |ψ〉) ≤r∑i=1

(EiρE†i |ψ〉, |ψ〉)

=r∑i=1

(ρE†i |ψ〉, E†i |ψ〉) ≤

r∑i=1

(E†i |ψ〉, E†i |ψ〉)

= (r∑i=1

EiE†i |ψ〉, |ψ〉) = (|ψ〉, |ψ〉) = ‖|ψ〉‖2

Thus, all the inequalities above are actually equalities. In particular, wehave

∑ri=1(ρE

†i |ψ〉, E

†i |ψ〉) =

∑ri=1(E

†i |ψ〉, E

†i |ψ〉). Again, by Lemma 6.1,

we get ρE†i |ψ〉 = E†i |ψ〉 for all 1 ≤ i ≤ r. So (ρ − 1)(E†i |ψ〉) = 0 for all1 ≤ i ≤ r. Then, since |ψ〉 is arbitrary in M, we get M is E†i -invariant forall 1 ≤ i ≤ r.

Now, we are able to provide the following theorem, which is necessary forthe proof of the main theorem.

Theorem 6.3. Suppose E : B(H)→ B(H) is a unital quantum channel withnoise operators {E1, ... , Er}, where H is a finite dimensional Hilbert space.If P is an orthogonal projection in B(H), then the following results holdstrue:

(i) E(P ) ≥ P if and only if Ran(P ) is E†i -invariant for all 1 ≤ i ≤ r.(ii) E(P ) ≤ P if and only if Ran(P ) is Ei-invariant for all 1 ≤ i ≤ r.(iii) E(P ) = P if and only if Ran(P ) is Ei-reducing for all 1 ≤ i ≤ r.

Proof. Note that since E is unital, E(P ) ≥ P if and only if E(1−P ) ≤ 1−P .In addition, Ran(P ) is E†i -invariant if and only if Ran(1− P ) = Ran(P )⊥ isEi-invariant. Thus, we only need to prove (i).

Suppose E(P ) ≥ P holds true. Since P is a projection, we have that0 ≤ P ≤ E(P ) and ker(P − ‖P‖1) = Ran(P ) (actually ‖P‖ = 1). Then byLemma 6.2, Ran(P ) is E†i -invariant for all 1 ≤ i ≤ r.

To see the converse, consider the decomposition H = PH⊕ P⊥H. Then

36

we can write Ei in the form

Ei =

[Bi 0Ci Di

]for 1 ≤ i ≤ r .

Then since E(1) =∑r

i=1EiE†i = 1, we have

r∑i=1

BiB†i = 1PH ,

r∑i=1

BiC†i = 0 and

r∑i=1

(CiC†i +DiD

†i ) = 1P⊥H .

Then, writing P =

[1 00 0

]according to the above decomposition, we will get

E(P ) =r∑i=1

EiPE†i =

[1 0

0∑r

i=1CiC†i

]≥[1 00 0

]= P .

This completes the proof of (i).Then, since E(P ) = P means E(P ) ≥ P and E(P ) ≤ P hold true simul-

taneously, we can easily get (iii).

Now we are close to the main result of this section. But before going on,we still need the following two lemmas.

Lemma 6.4. Suppose E : B(H)→ B(H) is a unital quantum channel withnoise operators {E1, ... , Er}, where H is a finite dimensional Hilbert space.Then A = B(H) will imply Fix(E) = C1. In other words, Fix(E) consists ofscalars.

Proof. Since A = B(H), we have A† = B(H). Now, assume that there existsan operator ρ ∈ Fix(E) which is non-scalar. Since Fix(E) is †-closed and E isunital, without loss of generality we can suppose ρ to be positive. Then byLemma 6.2, M = ker(ρ− ‖ρ‖1) are E†i -invariant for all 1 ≤ i ≤ r.

Since B(H) is finite dimensional, the spectrum σ(ρ) of ρ is a finite set.But since ρ is positive, we know that ‖ρ‖ is an extreme point of σ(ρ). Sowe can see that ‖ρ‖ ∈ σ(ρ), which means M 6= ∅. Also, we have M 6=B(H) (otherwise, we would get ρ = ‖ρ‖1). Then, M is a proper invariantsubspace of A† = B(H), which is absurd. Hence we complete the proof bycontradiction.

37

Lemma 6.5. Suppose the same conditions in Lemma 6.4 hold true. Then forany projection P , E(P ) = P holds true when either E(P ) ≤ P or E(P ) ≥ Pholds true. Moreover, every subspace that is invariant for E† = {E†i : 1 ≤i ≤ r} is also reducing for E†. And the same result holds true for E = {Ei :1 ≤ i ≤ r}.

Proof. Assume that there is a projection P such that E(P ) ≤ P and E(P ) 6=P . Then P−E(P ) is positive. So tr(P−E(P )) =

∑i λi > 0, where λi are the

eigenvalues of P − E(P ). But tr(P − E(P )) = 0 since E is trace preserving,contradiction. Hence, E(P ) ≤ P will imply E(P ) = P . The same discussionworks for the case E(P ) ≥ P .

Then, the rest of this lemma can be obtained directly form Theorem6.3.

Finally we can provide the main result of this section. The proof is from[6]. We should say that some simple results in [6] are directly used here sincethey can be easily verified according to some basic knowledge in C*-algebra.

Theorem 6.6. Let E be a unital quantum channel. Then A = A† is aC*-algebra and Fix(E) coincides with A′.

Proof. We first prove that A = A† is a C*-algebra. Define {Pj} to bea maximal family of pairwise orthogonal projections, where each PjH is aminimal reducing subspace for the family {E1, ..., Er}. For simplicity, wewill just say that Pj is minimal reducing for {Ei}. And let Hj = PjH foreach j. It is easy to see that EiPj = PjEi for all i, j, and the maximalityof {Pj} implies 1 =

∑j Pj. Hence, we have that A =

∑j PjAPj and A is

block diagonal with respect to {Pj}. Moreover, for a fixed j, we have thatAPj is an algebra and APj = PjA = PjAPj. But the only subspaces of Hthat are invariant for APj are the trivial subspaces. (Indeed, if a subspacePH is Ei-invariant for all 1 ≤ i ≤ r, then PH is Ei-reducing according toLemma 6.5. So by the minimality of Pj, we get either P = 0 or P = Pj.) Soapplying Burnside’s classical theorem, we can get that APj = B(Hj), whichshows that A = A† is a finite dimensional C*-algebra.

Now we prove that Fix(E) = A′.For any ρ ∈ A′, we have Eiρ = ρEi for i = 1, ..., r. Hence we have

E(ρ) =r∑i=1

EiρE†i = ρ

r∑i=1

EiE†i = ρ · 1 = ρ .

38

So ρ belongs to Fix(E). Thus A′ ⊂ Fix(E).To see the converse inclusion, let Ei,j = EiPj for all j and i = 1, ..., r.

Let Fj = (E1,j, ..., Er,j) ∈ B(H(r)j ,H). Given any ρ ∈ B(H), build the block

decomposition ρ = (ρjk) according to {Pj}. Then it is not hard to showthat ρjk = PjρPk. If E(ρ) = ρ holds true, then a simple deduction canshow that Ej(ρjj) = ρjj, where Ej : B(Hj) → B(Hj) is defined as Ej(ρ) =∑r

i=1Ei,jρE†i,j, and hence all the Ej are unital and completely positive. Then

since APj = B(Hj), according to Lemma 6.4 we have that ρjj = λjjPj forsome scalar λjj.

For j 6= k (j and k are fixed here), we will show that either ρjk = ρkj = 0for all ρ ∈ Fix(E) with ρ = ρ†, or there exists a unitary Wjk : Hk → Hj such

that Ei,j = WjkEi,kW†jk for i = 1, ..., r. Suppose that there is a ρ ∈ Fix(E)

with ρ = ρ† such that ρjk 6= 0. Then without loss of generality, we can let‖ρjk‖ = 1. Define M = {|ψ〉 ∈ Hk : ‖ρjk|ψ〉‖ = ‖|ψ〉‖} and let N = ρjkM(Since j, k are fixed, we do not need to use the notationMk and Nk). Thenfor any |ψ〉 ∈ M, we have

ρjk|ψ〉 = (PjE(ρ)Pk)|ψ〉 = E(ρjk)|ψ〉 = (Fjρ(r)jk F

†k )|ψ〉 , (8)

where ρ(r)jk =

ρjk . . .

ρjk

. The form (8) indicates that M is invariant

under each E†i,k. (Indeed, it is easy to show that ‖Fj‖, ‖F †k‖ ≤ 1. Also,

‖ρjk|ψ〉‖ = ‖|ψ〉‖. Then, we can see that F †k |ψ〉 ∈ M(r), where M(r) = {ξ ∈H(r)k : ‖ρ(r)jk ξ‖ = ‖ξ‖}. So it is easy to see that M is invariant under each

E†i,k.) Then by Lemma 6.5, we can see that M is reducing of each Ei. Also,M is a non-zero subspace contained in Hk. Hence, by the minimality of Pk,we get M = Hk. And we get N = Hj since ρkj = ρ†jk. Also, ρjk and ρ†jkare partial isometries and hence the operator Wjk = ρjk|Hk

: Hk → Hj is aunitary operator, which is the one we desire.

Hence, it is not hard to show that Wjk = FjW(r)jk F

†k . Then for any

|ψ〉 ∈ Hk, we have

‖|ψ〉‖ = ‖Wjk|ψ〉‖ = ‖FjW (r)jk F

†k |ψ〉‖ ≤ ‖W

(r)jk F

†k |ψ〉‖ ≤ ‖|ψ〉‖ .

Thus Fj is isometric from RanW(r)jk F

†k to RanWjk = Hj. (Recall that the

39

domain of Fj is H(r)j . Here we only guarantee that Fj is isometric when ξ ∈

RanW(r)jk F

†k .) Also, since Fj is a contraction (‖Fj‖ ≤ 1), Fj(ξ) can only be

zero if ξ ∈ (RanW(r)jk F

†k )⊥. Hence, it is not hard to show that F †j is isometric

from Hj to RanW(r)jk F

†k (Again, here we guarantee the isometry only when

|ψ〉 ∈ Hj). Therefore, we can get F †jWjk = W(r)jk F

†k . Then after simplification

it becomes E†i,j = WjkE†i,kW

†jk for i = 1, ..., r. Hence Ei,j = WjkEi,kW

†jk, as

desired.Now suppose E(ρ) = ρ = ρ† = (ρjk) for an operator ρ ∈ B(H), where the

decomposition (ρjk) is based on {Pj}. Then fix one arbitrary pair (j, k) withj 6= k, we have

WjkEk(W †jkρjk) =

r∑i=1

WjkEi,kW†jkρjkE

†i,k

=r∑i=1

Ei,jρjkE†i,k

= PjE(ρ)Pk = ρjk .

Thus by Lemma 6.4, we get that W †jkρjk = λ′kkPk for some scalar λ′kk. Hence

we have ρjk = µjkWjk for some scalar µjk, and also ρkj = ρ†jk = µjkW†jk. All

the other off-diagonal entries of ρ are either zero or in the same form.Then decomposing each Ei according to {Pj} as well, we get Ei =∑j EiPj =

∑j Ei,j. So, it is easy to see that ρEi = Eiρ for all i = 1, ..., r

when ρ ∈ Fix(E) and ρ = ρ†.Hence, all the self-adjoint elements in Fix(E) is contained in A′. Then, let

{ρ1, ..., ρn} be a basis of Fix(E). Then {(a1ρ1+a1ρ†1), ..., (anρn+anρ

†n)}, which

consists of self-adjoint elements, can also be a basis if the scalars a1, ..., an aresuitable. Hence, Fix(E) is spanned by its self-adjoint elements. Then, sinceA′ is a subspace and all the self-adjoint elements of Fix(E) are contained inA′, we get Fix(E) ⊂ A′.

Therefore, Fix(E) = A′.

Remark 6.7. Note that Fix(E)′ = (A)′′ = A. But A = A since A is asubspace in the finite dimensional B(H). Hence we have Fix(E)′ = (A)′′ = A.

Moreover, since Fix(E) is finite dimensional and †-closed, we can see thatFix(E) = A′ is a C*-subalgebra of B(H), where H is a finite dimensionalHilbert space. This means that A′ is a C*-algebra of compact operators.

40

Hence, A′ is unitarily equivalent to a unique direct sum of amplified matrixalgebras, which means

A′ '∑k

⊕k(1mk⊗Mnk

) =∑k

⊕kM(mk)nk

=∑k

⊕kB(Hnk)(mk) ,

where Mnkis the space of nk × nk matrices, and Hnk

is the nk dimensionalHilbert space. And therefore we can decompose ρ accordingly.

Theorem 6.6 shows that A′ is a noiseless subsystem. This is why Theorem6.6 is valuable.

However, when E is not guaranteed to be unital, Theorem 6.6 is nothelpful any more, and we cannot use A′ for error correction. We show thisby the following proposition.

Proposition 6.8. Let E be a quantum channel with noise operator {E1, ..., Er}such that AE = E(1) is not invertible. Let PE be the projection onto the sub-space HE = Ran(AE). If Ei = PEEiPE for any Ei, then H⊥E is non-zero.Moreover, for any operator ρ in B(H⊥E ), we have ρ ∈ A′ and E(ρ) = 0.

Proof. Since AE is not invertible, we can see that H⊥E is non-zero. Let ρ ∈B(H⊥E ) and ρ ≥ 0, then ρ = P⊥E ρP

⊥E . Then ρ ∈ A′ since ρEi = 0 = Eiρ for

any i. At last, we have E(ρ) = E(P⊥E ρ) = 0.

Remark 6.9. A more general definition of unital channel is a channel thatmake the identity operator evolve to a multiple of a projection. That is,E(1H) = mP for some projection P . Suppose dimH = N . Then since E istrace preserving, we have N = tr(1) = tr(E(1)) = tr(mP ) = mtr(P ). So mdivides the dimension of H. Then, the noise commutant A′ works for thiskind of ‘general unital channel’. A typical example is the channel E withnoise operators Ai = |0〉〈i| for 1 ≤ i ≤ dimH ≡ d. Then we have

E(1d) =d∑i=1

AiA†i =

d∑i=1

(|0〉〈i|)(|i〉〈0|) = d|0〉〈0| .

At the end of this paper, we provide some special examples of unitalchannels.

41

Examples 6.10. (i) For any 0 < p < 1, let E1, E2 be operators defined onthe standard basis of H2 by

E1 = (√

1− p)12 and E2 = (√p)Z .

Then the phase flip channel is the quantum channel E : B(H2) → B(H2)with noise operators {E1, E2}. Since E1 = E†1 and E2 = E†2, we have

E(12) = E112E†1 + E212E

†2 = E1E1 + E2E2 = (1− p)12 + p12 = 12 ,

and hence E is unital.Notice that the only difference between the phase flip channel and the

bit flip channel is that E2 = (√p)Z in the former one, while E2 = (

√p)X in

the later one. Moreover, define |+〉 = |0〉+|1〉√2

and |−〉 = |0〉−|1〉√2

as in section 3.Then by applying the exactly same method in the discussion of the bit flipchannel, we can prove that

E(|+〉〈+|) = (1− p)|+〉〈+|+ p|−〉〈−| .

Hence, E flips the phases of |+〉〈+| and |−〉〈−| with probability p. (This isalso why E is called the phase flip channel.)

For any operator ρ ∈ B(H2), write ρ as

(a bc d

). Also, letA = Alg{E1, E2}

and let A′ be the noise commutant of E . Then we have

A′ = {ρ ∈ B(H2) : ρA = Aρ for any A ∈ A}= {ρ ∈ B(H2) : ρEi = Eiρ for i = 1, 2}= {ρ ∈ B(H2) : ρE2 = E2ρ}

=

{(a bc d

):

(a bc d

)(1 00 −1

)=

(1 00 −1

)(a bc d

)}=

{(a bc d

):

(a −bc −d

)=

(a b−c −d

)}=

{(a 00 d

): a, b ∈ C

}' C1⊕ C1 .

Hence, any noiseless subsystem of the phase flip channel is non-trivial. Be-cause of this nice property, the phase flip channel is widely used in quantumerror correction.

42

(ii) For any 0 < p < 1, let E1, E2 be operators defined on the standardbasis of H4 by

E1 =√

1− p(

12 00 12

)and E2 =

√p

(Z 00 −Z

).

Again, E1 = E†1 and E2 = E†2.Similarly as in (i), write ρ in the form of (aij), where 1 ≤ i, j ≤ 4. Since

M4 = B(H4), we have

A′ = {ρ ∈ B(H4) : ρEi = Eiρ for i = 1, 2}= {ρ ∈ B(H4) : ρE2 = E2ρ}

=

{(aij) ∈M4 : (aij)

(Z 00 −Z

)=

(Z 00 −Z

)(aij)

}

=

(aij) ∈M4 :

a11 −a12 −a13 a14a21 −a22 −a23 a24a31 −a32 −a33 a34a41 −a42 −a43 a44

=

a11 a12 a13 a14−a21 −a22 −a23 −a24−a31 −a32 −a33 −a34a41 a42 a43 a44

=

a11 0 0 a140 a22 a23 00 a32 a33 0a41 0 0 a44

: aij ∈ C

'M2 ⊕M2

(iii) Suppose E1 = λ1U1 , ... , Er = λrUr, where Ui are unitaries ona n-dimensional Hilbert space H and λi are scalars with

∑ri=1 |λi|2 = 1.

Let E be the unital channel with noise operators {E1, ..., Er}. Define A =

Alg{E1, ..., Er} as usual, and define A = Alg{E1, E†1 , ... , Er, E

†r}.

Let f(λ) = |λI − U1| be the characteristic polynomial of U1. ThenU−11 = U †1 since U1 is unital. Then by Cayley-Hamilton theorem, we have

f(U1) = Un1 + cn−1U

n−11 + cn−2U

n−21 + · · ·+ c1U1 + c0I = 0 . (9)

Now, let U1 = TJ1T−1, where J1 is the Jordan form of U1. Then

|λI − U1| = |λI − J1| =s∏j=1

(λ− λ(1)j )kj ,

43

where λ(1)j are the eigenvalues of U1 and J1 (Jordan transformation does

not change the characteristic polynomial and hence does not change theeigenvalues.) Hence, we have

f(U1) =s∏j=1

(U1 − λ(1)j I)kj

= Un1 + cn−1U

n−11 + cn−2U

n−21 + · · ·+ c1U1 + (−1)n

(s∏j=1

(λ(1)j )kj

)I .

But

(s∏j=1

(λ(1)j )kj

)= |J1| = |U1|, and |U1| = 1 since U1 is unitary. Hence

c0 = (−1)n 6= 0. Taking the value of c0 back into (9), we have

(−1)n+1I = U1(Un−11 + cn−1U

n−21 + cn−2U

n−21 + · · ·+ c1I) .

In other words,

I = (−1)n+1U1(Un−11 + cn−1U

n−21 + cn−2U

n−21 + · · ·+ c1I) .

Multiplying U †1 (which is equal to U−11 ) from left at both side, we get

U †1 = (−1)n+1(Un−11 + cn−1U

n−21 + cn−2U

n−21 + · · ·+ c1I) .

Thus, U †1 can be expressed by a polynomial of U1, which is also true forU †2 , ... , U

†r . Hence, we have {U †1 , ... , U †r} ⊂ Alg{E1, ... , Er} = A, which indi-

cates that A ⊂ A. And obviously we have A ⊂ A. So finally, we get A = A.(iv) An important case of the above unital channel is the class of ‘collec-

tive rotation channels’.Recall that σk = 1/2K for k = x, y, z and K = X, Y, Z, where X, Y, Z

are the Pauli matrices. Then a collective rotation channel defined on B(H2n)

is the unital channel E with noise operators Jk =∑n

m=1 J(m)k for k = x, y, z,

where J(1)k = σk ⊗ (12)

⊗(n−1), J(2)k = 12 ⊗ σk ⊗ (12)

⊗(n−2), etc. The collectiverotation channel is widely used in the research of quantum information.

44

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