2005 q 0031 density matrices 2
TRANSCRIPT
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Richard Cleve
Lectures 10 ,11 and 12
DENSITY
MATRICES, traces,
Operators andMeasurements
Michael A. Nielsen
Michele Mosca
Sources:
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Review: Density matrices
of pure statesWe have represented quantum states as vectors(e.g. ,and all such states are called pu re states)
An alternative way of representing quantum states is in terms
of densi ty m atr ices(a.k.a. densi ty operators)
The density matrix of a pure state is the matrix =
Example:the density matrix of 0+1is
2
2
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Example:Notation of Density
Matrices and traces
Notice that 0=0|, and 1=1|.
So the probability of getting 0 when measuring |is:220 0)0( p
0000
0000
0000
TrTr
Tr
where = || is called
the density matrixfor thestate |
10 10
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Review:Mixture of pure states
A state described by a state vector |is called apure state.
What if we have a qubit which is known to be in thepure state |1with probabilityp1, and in |2withprobabilityp2?
More generally, consider probabilistic mixturesofpure states (called mixed states):
...,,,, 2211 pp
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Density matrices of mixed states
A probability distribution on pure states is called a mixed state:( (1, p1), (2, p2), , (d, pd))
The densi ty matr ixassociated with such a mixed state is:
d
kkkkp
1
Example: the density matrix for ((0, ), (1, ))is:
10
01
2
1
10
00
2
1
00
01
2
1
Question:what is the density matrix of
((0+1,
),(0 1,
)) ?
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Density matrix of a mixed
state (use of trace)then the probability of measuring 0 is given byconditional probability:
i
iipp statepuregiven0measuringofprob.)0(
00
00
00
Tr
pTr
Trp
i
iii
i
iii
where i
iiip is the density matrixfor the mixedstate
Density matrices contain all the useful information about anarbitrary quantum state.
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Recap: operationally
indistinguishable states
Since these are expressible in
terms of density matrices alone(independent of any specific
probabilistic mixtures), states with
identical density matrices areoperat ional ly ind ist inguishable
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Applying Unitary Operator to a
Density Matrix of a pure state
If we apply the unitary operation U tothe resulting state is
with density matrix
U
tt UUUU
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Density Matrix
If we apply the unitary operation U to
the resulting state is
with density matrix
kkq , kk Uq ,
t
t
t
UU
UqU
UUq
k
k
kk
k
k
kk
Applying Unitary Operator to a
Density Matrix of a mixedstate
How do quantum operations work for these mixedstates?
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Operators on Density matrices of
mixed states.
Effect of a unitary operation on a density matrix:
applying U to st i l lyields UU
Effect of a measurement on a density matrix:
measuring statewith respect to the basis1, 2,..., d,st i l lyields the kthoutcome with probability kk
Why?
Thus this
is true
always
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Effect of a measurement on a density matrix:
measuring state with respect to the basis1,2,..., d, yields the kthoutcome with probabilitykk
How do quantum operations
work using density matrices?
(this is because kk=kk=k2)
and thestate collapses tok
k
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More examples of density matrices
The densi ty matr ixof the mixed state
((1, p1), (2,p2), ,(d,pd)) is:
d
k
kkk p1
1. & 2. 0+1and 01both have
3. 0with prob. 1with prob.
4. 0+1 with prob.
01 with prob.
6. 0 with prob. 1 with prob. 0+1 with prob.
01 with prob.
Examples (from previous lecture):
11
11
2
1
1001
21
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5. 0 with prob. 0+1 with prob.
7. The first qubit of 0110
Examples (continued):
4/12/1
2/14/3
2/12/1
2/12/1
2
1
00
01
2
1has:
...?(later)
More examples of density matrices
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To Remember:Three Properties of Density
Matrices
Three properties of :
Tr=1(TrM =M11+M22 + ... +Mdd )
=(i.e.is Hermitian)
0, for all states
d
kkkk p
1
Moreover, for anymatrix satisfying the above properties,
there exists a probabilistic mixturewhose density matrix is
Exercise:show this
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Use of Density Matrix and Trace to
Calculate the probability of obtaining
state in measurement
If we perform a Von Neumann measurementof the state wrt a basiscontaining , the probability ofobtaining is
Tr2 This is for a pure
state.
How it would be for a
mixed state?
U f D it M t i d T t C l l t th
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Density Matrix
If we perform a Von Neumann measurementof the state
wrt a basis containing the probabilityof obtaining is
kkq ,
Tr
qTr
Trqq
k
kkk
k
kkk
k
kk
2
Use of Density Matrix and Trace to Calculate the
probability of obtaining state in measurement (now
for measuring a mixed state)
The same
state
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Conclusion: Density Matrix Has
Complete Information
In other words, the density matrix containsall the information necessary to computethe probability of any outcome in anyfuture measurement.
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Spectral decomposition can be used to
represent a useful form of density matrix
Often it is convenient to rewritethedensity matrix as a mixture of itseigenvectors
Recall that eigenvectors with distinct
eigenvalues are orthogonal;for the subspace of eigenvectors with a
common eigenvalue(degeneracies), wecan select an orthonormal basis
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Continue - Spectral decomposition used
to diagonalize the density matrix
In other words, we can alwaysdiagonalizea density matrix so that it
is written ask
k
kkp
where is an eigenvectorwitheigenvalue and forms anorthonormal basis
kkp k
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Taxonomy ofvarious normal
matrices
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Normal matrices
Definition:A matrixMis normalifM
M =MM
Theorem:Mis normal iff there exists a unitary U such that
M =UDU, whereD is diagonal (i.e. unitarily diagonalizable)
Examples of abnormal matrices:
10
11 is not even
diagonalizable
20
11 is diagonalizable,but not unitarily
eigenvectors:
d
D
00
0000
2
1
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Unitary and Hermitian matrices
d
M
00
0000
2
1 with respect to someorthonormal basis
Normal:
Unitary:MM =I which implies |k |2=1, for all k
Hermitian:M =M which implies kR, for all k
Question:which matrices areboth unitary andHermitian?
Answer:reflections (k{+1,1}, for all k)
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Positive semidefinite matrices
Positive semidefinite:Hermitian and k0, for all k
Theorem:Mis positive semidefinite iffMis Hermitian and,
for all , M 0
(Positive defini te:k>0, for all k)
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Projectors and density matrices
Projector:Hermitian andM2 =M, which implies thatMispositive semidefinite and k{0,1}, for all k
Density matrix:positive semidefinite and TrM=1, so 11
d
k
k
Question:which matrices areboth projectors anddensity
matrices?
Answer:rank-one projectors(k=1if k = k0and k=0 ifk k0)
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Taxonomy of normal matrices
normal
unitary Hermitian
reflectionpositive
semidefinite
projector densitymatrix
rank one
projector
If Hermitian then
normal
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Review:Bloch sphere for qubits
Consider the set of all 2x2 density matrices
Note that the coefficient ofI is , sinceX,Y,Zhave trace zero
They have a nice representation in terms of the Paul i m atr ices:
01
10 Xx
0
0
i
iYy
10
01 Zz
Note that these matricescombined withIform a basisforthe vector space of all 2x2 matrices
We will express density matrices in this basis
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Bloch sphere for qubits: polar
coordinates
2
ZcYcXcI
zyx We will express
First consider the case of pure states , where, withoutloss of generality, =cos()0+e2isin()1 (, R)
2cos12sin
2sin2cos1
2
1
sinsincos
sincoscos
2
2
22
22
i
i
i
i
e
e
e
e
Therefore cz= cos(2),cx= cos(2)sin(2),cy= sin(2)sin(2)
These are po lar co ordinatesof a unit vector (cx ,cy ,cz)R3
Bloch sphere for qubits: location of
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Bloch sphere for qubits: location of
pure and mixed states
+
0
1
+i
i
+i=0+i1
i=0i1
=01+=0+1
Pure statesare on the surface, and mixed statesare inside
(being weighted averages of pure states)
Note that or thogonalcorresponds to ant ipodalhere
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General
quantum
operationsDecoherence, partial traces,
measurements.
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General quantum operations (I)
Example 1(unitary op): applying U to yields UU
General quantum operationsare also calledcompletely positive trace preserving maps,or
admissible operations
IAA jm
j
j 1
t
Then the mapping m
jjj AA1
t
is a general quantumoperator
LetA1,A2, ,Ambe matrices satisfying
condition
General quantum operations:
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General quantum operations:
Decoherence Operations
Example 2(decoherence):letA0 =00andA1 =11This quantum op maps to 0000+1111
Corresponds to measuring without looking at the outcome
2
2
2
2
0
0
For =0+1,
After looking at the outcome, becomes 00 with prob. ||211 with prob. ||2
General quantum operations: measurement
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General quantum operations: measurementoperations
Example 3(trine statemeasurement):
Let 0=0, 1=1/20+ 3/21, 2=1/203/21
Then IAAAAAA 221100ttt
The probability that state kresults in outcome stateAk is 2/3.This can be adapted to actually yield the value of
kwith this success
probability
00
01
3
2DefineA0 =2/300
A1=
2/311 A2=
2/322
62
232
4
1
62
232
4
1
We apply the general quantum mapping operator
m
j
jj AA1
t
Condition satisfied
General quantum operations: Partial trace discards
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General quantum operations: Partial tracediscards
the second of two qubits
Example 4(discarding the second of two qubits):
LetA0=I0 andA1=I1
0100
0001
1000
0010
State becomes
State becomes110011002
1
2
1
2
1
2
1
Note 1:its the same density matrix as for ((0, ), (1, ))
10
01
2
1
Note 2:the operation is the partial traceTr2
We apply the general quantum mapping operator
m
j
jj AA1
t
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Distinguishing
mixed statesSeveral mixed states can have the same
density matrixwe cannot distinguish
between them.
How to distinguish by two different density
matrices?
Try to find an orthonormal basis 0,1in which both
density matrices are diagonal:
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Distinguishing mixed states (I)
10
01212
0 with prob. 0+1 with prob.
0 with prob. 1 with prob.
412121431
////
0 with prob. cos2(/8)1 with prob. sin2(/8)
0
+
0
1
0 with prob. 1 with prob.
Whats the best distinguishing strategybetween these two
mixed states?
1also arises from this
orthogonal mixture: as does 2from:
/8=180/8=22.5
Di ti i hi i d t t (II)
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Distinguishing mixed states (II)
8sin0
08cos2
2
2/
/
0
+
0
1
10
01
2
11
Weve effectively found an orthonormal basis 0,1inwhich both density matrices are diagonal:
Rotating 0,1to 0, 1the scenario can nowbe examined using classical probability theory:
Question:what do we do if we arent so lucky to get two
density matrices that are simultaneously diagonalizable?
Distinguish between two classicalcoins, whose probabilities
of headsare cos2(/8)and respectively (details: exercise)
1
Density matrices 1and 2are simultaneously diagonalizable
B i ti f
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Reminder:Basic properties of
the trace
d
k
k,kMM1
Tr
NMNM TrTrTr
NMNM TrTrTr
MNNM TrTr
adcbdcba Tr
d
k
kMUUM1
1 TrTr
The t raceof a square matrix is defined as
It is easy to check that
The second property implies
and
Calculation maneuvers worth remembering are:
aMMa bb Tr and
Also, keep in mind that, in general,
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Partial Trace How can we compute probabilities fora partial system?
E.g.
yx
p
p
yx
yx
y x y
xy
y
y xxy
yxxy
,
Partial
measurement
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Partial Trace
If the 2ndsystem is taken away and neveragain (directly or indirectly) interacts withthe 1stsystem, then we can treat the firstsystem as the following mixture
E.g.
22,2 Trx
p
p
yx
p
p
x y
xy
y
Trace
y x y
xy
y
From previous
slide
Partial Trace: we derived an important
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Partial Trace:we derived an important
formula to use partial trace
22,2 Trx
pp
yxp
p
x y
xy
y
Trace
y x y
xy
y
yy
y
ypTr 2 x y
xy
y xp
Derived in
previous
slide
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Why?
the probability of measuring e.g. inthe first register depends only on
2
2
2
TrwwTr
pwwTr
wwTrp
pp
yy
y
y
yy
y
y
y y y
wy
ywy
w2Tr
Partial Trace can be calculated in
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Partial Trace can be calculated in
arbitrary basis
Notice that it doesnt matter in whichorthonormal basis we trace outthe2ndsystem, e.g.
11001100 222 Tr
In a different basis
12
1
02
1
102
1
1100
1
2
10
2
110
2
1
(cont) Partial Trace can be calculated in
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Partial Trace
1100
10102
1
10102
1
22
**
**2
Tr
1
2
10
2
110
2
1
12
1
02
1
102
1
(cont) Partial Trace can be calculated in
arbitrary basis
Which is the same as in
previous slide for otherbase
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Methods to calculate the Partial Trace
Partial Traceis a linear mapthat takesbipartite statesto single system states.
We can also trace out the first system
We can compute the partial trace directlyfrom the density matrixdescription
kijljlki
ljTrkiljkiTr
2
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Partial Trace using matrices
Tracing out the 2ndsystem
33223120
13021100
3332
2322
3130
2120
1312
0302
1110
0100
33323130
23222120
13121110
03020100
2
aaaa
aaaa
aa
aaTr
aa
aaTr
aa
aaTr
aa
aaTr
aaaa
aaaaaaaa
aaaa
Tr
Tr 2
E l P ti l t (I)
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Examples: Partial trace (I)
In such circumstances, if the second register (say) is discardedthen the
state of the first register remains
Two quantum registers (e.g. two qubits) in states and (respectively) are independentif then the combined system
is in state =
In general, the state of a two-register system may not be of the
form (it may contain entanglementor correlat ions)
We can define the part ial trace, Tr2,as the unique linearoperator satisfying the identity Tr2()=
For example, it turns out that
110011002
1
2
1
2
1
2
1
10
01
2
1Tr2( )=
index means
2ndsystem
traced out
E l P ti l t (II)
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Examples:Partial trace (II)
Weve already seen this defined in the case of 2-qubit systems:discarding the second of two qubits
LetA0=I0 andA1=I1
0100
0001
1000
0010
For the resulting quantum operation, state becomes
For d-dimensional registers, the operators areAk =Ik,
where0, 1, , d1are an orthonormal basisAs we see in last slide, partial trace is a
matrix.
How to calculate this matrix of partial trace?
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Unitary transformations dont
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Unitarytransformations don t
change the local density matrix
A unitary transformation on the systemthat is traced outdoes not affect theresult of the partial trace
I.e.
22,2
Trp
UIyUp
yy
Trace
y
yy
Distant transformations dont
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Distant transformationsdon t
change the local density matrix
In fact, any legal quantum transformationon the traced out system, includingmeasurement (without communicatingback the answer) does not affect thepartial trace
I.e.
22,2 ,
Trp
yp
yy
Trace
yy
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Why??
Operations on the 2ndsystem should notaffect the statistics of any outcomes ofmeasurements on the first system
Otherwise a party in control of the 2ndsystem could instantaneouslycommunicate information to a partycontrolling the 1stsystem.
Principle of implicit
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Principle of implicit
measurement
If some qubits in a computationarenever used again, you can assume (if
you like) that they have beenmeasured (and the result ignored)
The reduced density matrix of theremaining qubits is the same
POVMs (I)
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POVMs (I)Posit ive operator valued measurement(POVM):
LetA1,A2 , ,Ambe matrices satisfying IAA jm
j
j 1
t
Then the corresponding POVM is a stochastic operation on
that, with probability produces the outcome:
j (classicalinformation) t
jj AATr
t
t
jj
jj
AA
AA
Tr
(the collapsed quantum state)
Examp le 1:Aj = jj(orthogonal projectors)
This reduces to our previously definedmeasurements
POVMs (II): calculating the measurement
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POVMs (II):calculating the measurement
outcome and the collapsed quantum state
Moreover,
tjj AATr
jj
j
jjjj
jj
jj
AA
AA
2Tr t
t
WhenAj=
j
jare orthogonal projectors and= ,
= Trjjjj
= jjjj
= j2
(the collapsed quantum state)
probability
of the
outcome:
The measurement postulate
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The measurement postulateformulated
in terms of observablesA measurement is described by a complete set of
projectors onto orthogonal subspaces. Outcome occurs
with probability Pr( ) .
The corresponding post-measure
Our
ment state is
form
:
j
j
P j
j P
P
.j
jP
This is aprojector
matrix
The measurement postulate formulatedf l
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pin terms of observables
A measurement is described by a complete set ofprojectors onto orthogonal subspaces. Outcome occurs
with probability Pr( ) .
The corresponding post-measure
Our
ment state is
form
:
j
j
P j
j P
P
.j
jP
A measurement is described by an ,
a Hermitian operator , with spectral decomposiOld form: o
tion
bservable
.j jj
MM P
The possible measurement outcomes correspond to theeigenvalues , and the outcome occurs with probability Pr( ) .
j j
j jP
The corresponding post-measurement state is
.j
j
P
P
The same
An example of observables in action
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An example of observables in action
Suppose we "measE uxample: re ".Z
has spectral decomposition 0 0 - 1 1 , so
this is just like measuring in the computational basis,and calling the outcomes "1" and "-1", respectively, for0 and 1.
Z Z
Find the spectral decomposition of .
Show that measuring corresponds to measuringthe parity of two qubits, with the result +1 correspondingto even parity, and the result
Exercis
-1 correspon
:
i
e
d
Z Z
Z Z
ng to oddparity.
00 00 11Hint: 11 10 10 01 01Z Z
An example of observables in action
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An example of observables in action
Suppose we measure the observable for astate which is an eigenstate of that observable. Showthat, with certainty, the outcome of the measurement isthe corresponding eigenvalue
Exerci
of the ob
se: M
servable.
What can be measured in
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What can be measured inquantum mechanics?
Computer science can inspire fundamental questions aboutphysics.
We may take aninformatic approach to physics.(Compare thephysical approach to information.)
Problem:What measurements can be performed in
quantum mechanics?
What can be measured in quantum mechanics?
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Traditional approach to quantum measurements:A quantum measurement is described by an observable MM is a Hermitian operator acting on the state spaceof the system.
Measuring a system prepared in an eigenstate of
Mgives the corresponding eigenvalue of Mas themeasurement outcome.
The question now presents itself Can every observable
be measured? The answer theoretically is yes. In practiceit may be very awkward, or perhaps even beyond the ingenuityof the experimenter, to devise an apparatus which couldmeasure some particular observable, but the theory alwaysallows one to imagine that the measurement could be made.
- Paul A. M. Dirac
What can be measured in quantum mechanics?
V N t
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Von Neumann measurement
in the computational basis
Suppose we have a universal set of quantumgates, and the ability to measure each qubitin the basis
If we measure we getwith probability
}1,0{
2
bb
)10( 10
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In section 2.2.5, this is described as follows
00P0 11P1
We have the projection operatorsand satisfying
We consider the projection operatororobservable
Note that 0 and 1 are the eigenvalues
When we measure this observable M, theprobability of getting the eigenvalue isand we are in
that case left with the state
IPP 10
110 PP1P0M
b2
)Pr( bbPb bb
)b(p
P
b
bb
What is an Expected value
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What is an Expected valueof an observable
b If we associate with outcome theeigenvalue then the expected outcome is
)Pr(
MTrbPTr
bPPb
bb
b
b
b
b
b
b
b
b
Von Neumann measurement in
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Von Neumann measurement in
the computational basis
Suppose we have a universal set of quantumgates, and the ability to measure each qubitin the basis
Say we have the state If we measure all n qubits, then we obtain
with probability
Notice that this means that probability ofmeasuring a in the first qubit equals
}1,0{x
n}1,0{xx
x2
x
0
1n}1,0{0x
2
x
Partial measurements
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Partial measurements
(This is similar to Bayes Theorem)
x
p1n}1,0{0x 0
x
0
1n}1,0{0x
2
x0p
If we only measure the first qubit and leavethe rest alone, then we still get with
probability The remaining n-1 qubits are then in the
renormalized state
M t l t
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Most general measurement
kk
000 U
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In section 2.2.5
This partial measurementcorresponds tomeasuring the observable
1n1n I111I000M
V N M t
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Von Neumann Measurements
A Von Neumann measurement is a type ofprojective measurement. Given anorthonormal basis , if we perform a
Von Neumann measurement with respect to of the state thenwe measure with probability
}{ k
kk}{ kk
kkkk
kk2
k2
k
TrTr
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Von Neumann Measurements
E.x. Consider Von Neumann measurement ofthe state with respect tothe orthonormal basis
Note that
2
10,
2
10
2
10
22
10
2
)10(
We therefore get with probability
2
10
2
2
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Von Neumann Measurements
Note that
22
10
22 10
**
22
10
2
10Tr
2
10
2
10
2
How do we implement
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How do we implement
Von Neumann measurements?
If we have access to a universal set ofgates and bit-wise measurements in the
computational basis, we can implement VonNeumann measurements with respect to anarbitrary orthonormal basis asfollows.
}{ k
How do we implement
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How do we implement
Von Neumann measurements?
Construct a quantum network thatimplements the unitary transformation
kU k Then conjugate the measurement
operation with the operation U
kk U k2
kprob
1U k
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Example: Bell basis change
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Example:Bell basis change
100101
Consider the orthonormal basisconsistingof the Bell states
110000
110010 100111 Note that
xyx
y
H
We discussed Bell basis in
lecture about superdense
coding and teleportation.
Bell measurements:destructiveand
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non-destruct ive
We can destructivelymeasure
Or non-destructivelyproject
xy
y,x
y,x x
y
H2
xyprob
xyy,x
y,x xyy,xH
2
xyprob 00
H
Most general measurement
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Most general measurement
000 U
0000002 Tr
Simulations among operations:
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Simulationsamong operations:general quantumoperations
Fact 1:any general quantum operat ioncan be simulatedby applying a unitary operation on a larger quantum system:
U000
Example:decoherence
0
0+1
2
2
0
0
output
discard
input
zeros discard
Simulations among operations:
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g p
simulations of POVM
Fact 2:any POVMcan also be simulated by applying aunitary operation on a larger quantum system and then
measuring:
U0
00
quantum outputinput
classical outputj
Separable states
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Separable states
m
j
jjjp1
product stateif =
separablestateif
A bipartite (i.e. two register) stateis a:
Question: which of the following states are separable?
110011001100110021
21
2
(i.e. a probabilistic mixture
of product states)
(p1 ,,pm 0)
1100110021
1
Continuous-time evolution
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Continuous-time evolution
Although weve expressed quantum operations in discreteterms, in real physical systems, the evolution is continuous
0
1LetHbe any Hermitianmatrix and tR
Then eiHtis uni tarywhy?
H =UDU, where
d
D
1
Therefore eiHt=UeiDtU= Ue
e
Uti
ti
d
1
t (unitary)
P ti ll d i 2007
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Partially covered in 2007:
Density matrices and indistinguishable states Taxonomy of normal operators
General Quantum Operations
Distinguishing states Partial trace
POVM
Simulations of operators
Separable states
Continuous time evolution