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Quantum Information
Stephen M. BarnettUniversity of Strathclyde
The Wolfson Foundation
1. Probability and Information
2. Elements of Quantum Theory
3. Quantum Cryptography
4. Generalized Measurements
5. Entanglement
6. Quantum Information Processing
7. Quantum Computation
8. Quantum Information Theory
4.1 Ideal von Neumann measurements4.2 Non-ideal measurements4.3 Probability operator measures4.4 Optimised measurements4.5 Operations
Measurement model
What are the probabilities for the measurement outcomes?How does the measurement change the quantum state?
‘Black box’
4.1 Ideal von Neumann measurements
Red
von Neumann measurements
Mathematical Foundations of Quantum Mechanics
(i) An observable A is represented by an Hermitian operator:
(ii) A measurement of A will give one of its eigenvalues as a result. The probabilities are
or
(iii) Immediately following the measurement, the system is left in the associated eigenstate:
nnnA ˆ
nnnnP 2
)(
ˆTrˆ)( nnnnnP
nn ˆ
More generally for observables with degenerate spectra:
(i’) Let be the projector onto eigenstates with eigenvalue :
(ii’) The probability that the measurement will give the result is
(iii’) The state following the measurement is
nP n
n
nnn PPA ˆˆˆ
ˆˆTr)( nn PP
)ˆˆ(Tr
ˆˆˆˆ
n
nn
P
PP
Detector
a b
DetectdetectNo
Detect
ˆ-Iˆ
ˆ
PP
dxxxPb
a
Properties of projectors
I. They are Hemitian Observable
II. They are positive Probabilities
III. They are complete Probabilities
IV. They are orthonormal ??
nn PP ˆˆ †
0nP
n
nP Iˆ
iijji PPP ˆˆˆ
4.2 Non-ideal measurements
Real measurements are ‘noisy’ and this leads to errors
‘Black box’
‘Black box’
0
1
pP 1)0(
pP )1(
pP )0(
pP 1)1(
For a more general state:
0ˆ01ˆ1)1()1(
1ˆ10ˆ0)1()0(
ppP
ppP
We can write these in the form
)ˆˆ(Tr)1(
)ˆˆ(Tr)0(
1
0
P
P
0011)1(ˆ
1100)1(ˆ
1
0
pp
pp
where we have introduced the probability operators
These probability operators are
I. Hermitian
II. positive
III. complete
But
IV. they are not orthonormal
1,0†
1,0 ˆˆ
001)1(ˆ
010)1(ˆ
22
1
22
0
pp
pp
I1100ˆˆ 10
0I)1(ˆˆ 10 pp
We seem to need a generalised description of measurements
4.3 Probability operator measures
Our generalised formula for measurement probabilities is
The set probability operators describing a measurement is called a probability operator measure (POM) or a positiveoperator-valued measure (POVM).
The probability operators can be defined by the properties that they satisfy:
ˆˆTr)( iiP
Properties of probability operators
I. They are Hermitian Observable
II. They are positive Probabilities
III. They are complete Probabilities
IV. Orthonormal ??
nn ˆˆ †
0ˆn
n
n I
iijji ˆˆˆ
Generalised measurements as comparisons
S A
S A
S + A
Prepare an ancillary system ina known state:
Perform a selected unitary transformation to couple the systemand ancilla:
Perform a von Neumann measurementon both the system and ancilla:
AS
AU S ˆ
iiS Ai
The probability for outcome i is
The probability operators act only on the system state-space.
AUiiUA
AUiiUA
ˆˆ
ˆˆ
†
††
0ˆˆ2
AUi Si
A,Si
ii I
POM rules: I. Hermiticity:
II. Positivity:
III. Completeness follows from:
SS
SS
AUiiUA
iUAAUiiP
ˆˆ
ˆˆ)(
†
†
i
i
Generalised measurements as comparisons
We can rewrite the detection probability as
is a projector onto correlated (entangled) states of the system and ancilla. The generalised measurement is a von Neumann measurement in which the system and ancilla are compared.
APAiP SiS ˆ)(
UiiUPiˆˆˆ †
APA iiˆˆ
0ˆˆˆˆ APAAPA mnmn
Simultaneous measurement of position and momentum
The simultaneous perfect measurement of x and p would violatecomplementarity.
x
p Position measurement gives nomomentum information and depends on the position probabilitydistribution.
Simultaneous measurement of position and momentum
The simultaneous perfect measurement of x and p would violatecomplementarity.
x
p
Momentum measurement gives noposition information and depends on the momentum probabilitydistribution.
Simultaneous measurement of position and momentum
The simultaneous perfect measurement of x and p would violatecomplementarity.
x
p Joint position and measurement gives partial information on both the position and the momentum.
Position-momentum minimumuncertainty state.
POM description of joint measurements
Probability density:
),(ˆˆ),( mmmm pxTrpx
Minimum uncertainty states:
I,,2
1
4
)(exp2,
2
24/12
mmmmmm
mmmm
pxpxdpdx
xxp
ixx
dxpx
This leads us to the POM elements:
mmmmmm pxpxpx ,,2
1),(ˆ
The associated position probability distribution is
2
22
22
2
2
4)(Var&
)(Var
2
)(expˆ)(
pp
xx
xxxxdxx
m
m
mm
Increased uncertainty is the price we pay for measuring x and p.
The communications problem
‘Alice’ prepares a quantum system in one of a set of N possiblesignal states and sends it to ‘Bob’
Bob is more interested in
ijijP ˆˆTr)|(
)ˆˆ(Tr
ˆˆTr)|(
j
iij pjiP
Preparationdevice
i selected.
prob. ip
i Measurement
result j
Measurementdevice
In general, signal states will be non-orthogonal. No measurement can distinguish perfectly between such states.
Were it possible then there would exist a POM with
Completeness, positivity and
What is the best we can do? Depends on what we mean by ‘best’.
121212
222111
ˆ0ˆ
ˆ1ˆ
1ˆ 111
0ˆandpositiveˆˆˆ 1111 AAA
0ˆˆ2
21222
21212 A
Minimum-error discrimination
We can associate each measurement operator with a signalstate . This leads to an error probability
Any POM that satisfies the conditions
will minimise the probability of error.
ii
jjj
N
je TrpP ˆˆ1
1
jpp
kjpp
jj
N
kkkk
kkkjjj
0ˆˆˆ
,0ˆ)ˆˆ(ˆ
1
For just two states, we require a von Neumann measurement withprojectors onto the eigenstates of with positive (1) and negative (2) eigenvalues:
Consider for example the two pure qubit-states
2211 ˆˆ pp
221121min ˆˆTr1 ppPe
1sin0cos
1sin0cos
2
1
)2cos(21
1
1
2
2
The minimum error is achieved by measuring in the orthonormal basis spanned by the states and .
We associate with and with :
The minimum error is the Helstrom bound
12
1 1 2 2
2
122
2
211 ppPe
2/1221212
1min 411 ppPe
+
P = ||2
P = ||2
A single photon only gives one “click”
But this is all we need to discriminate between our two states with minimum error.
A more challenging example is the ‘trine ensemble’ of threeequiprobable states:
It is straightforward to confirm that the minimum-error conditionsare satisfied by the three probability operators
31
33
31
221
2
31
121
1
0
130
130
p
p
p
iii 32ˆ
Simple example - the trine states
Three symmetric states of photon polarisation
3
2
32
2
31
Minimum error probabilityis 1/3.
This corresponds to a POMwith elements
ˆ j 2
3 j j
How can we do a polarisationmeasurement with these threepossible results?
Polarisation interferometer - Sasaki et al, Clarke et al.
PBS
PBS
PBS
2
2
2/3
2/1
2/3
2/1
0
1
2/3
0
2/3
0
0
0
1
0
1/ 2
0
1/ 2
0
6/1
12/1
6/1
12/1
3/2
3/1
0
6/1
0
3/2
0
6/1
3/2
0
6/1
0
6/1
0
6/1
0
6/1
0
3/2
0
Unambiguous discrimination
The existence of a minimum error does not mean that error-freeor unambiguous state discrimination is impossible. A von Neumannmeasurement with
will give unambiguous identification of :
result error-free
result inconclusive
2
111111ˆˆ PP
21
?1
There is a more symmetrical approach with
21?
11
21
2
22
21
1
ˆˆIˆ
1
1ˆ
1
1ˆ
Result 1 Result 2 Result ?
State 0
State 0
211
2
1
211
21
21
How can we understand the IDP measurement?
Consider an extension into a 3D state-space
a
ba
b
Unambiguous state discrimination - Huttner et al, Clarke et al.
a
b
?
a
b
A similar device allows minimum error discriminationfor the trine states.
Measurement model
What are the probabilities for the measurement outcomes?
How does the measurement change the quantum state?
‘Black box’
Red
How does the state of the system change after a measurement is performed?
Problems with von Neumann’s description:
1) Most measurements are more destructive than von Neumann’s ideal.
2) How should we describe the state of the system after a generalised measurement?
4.5 Operations
Physical and mathematical constraints
What is the most general way in which we can change a density operator?
Quantum theory is linear so …
or more generally
BUT this must also be a density operator.
BA ˆˆˆˆ
iii
BA ˆˆˆˆ
Properties of density operators
I. They are Hermitian
II. They are positive
III. They have unit trace
ˆˆ †
0ˆ
1ˆTr
The first of these tells us that and this ensures that the second is satisfied.
The final condition tells us that
†ˆˆii AB
Iˆˆ † iii
AA
The operator is positive and this leads us to associate
(Knowing does not give us .)If the measurement result is i then the density operator changesas
This replaces the von Neumann transformation
ii AA ˆˆ †
iii AA ˆˆˆ †
)ˆˆ(Tr
ˆˆˆ
)ˆˆˆ(Tr
ˆˆˆˆ
†
†
†
i
ii
ii
ii AA
AA
AA
)ˆˆ(Tr
ˆˆˆˆ
n
nn
P
PP
iAi
If the measurement result is not known then the transformed density operator is the probability-weighted sum
which has the form required by linearity.
We refer to the operators and as Krauss operatorsor an an ‘effect’.
†
†
†† ˆˆˆ
)ˆˆˆ(Tr
ˆˆˆ)ˆˆˆ(Trˆ ii
iii
ii
iii AA
AA
AAAA
iA †ˆiA
Repeated measurements
Suppose we perform a first measurement with results i and effectoperators and then a second with outcomes j and effects .
The probability that the second result is j given that the first was i is
iA jB
)ˆˆˆˆˆ(Tr)()|(),(
)ˆˆˆ(Tr
)ˆˆˆˆˆ(Tr)|(
††
†
††
ijji
ii
iijj
ABBAiPijPjiP
AA
AABBijP
Hence the combined probability operator for the two measurementsis
If the results i and j are recorded then
If they are not known then
ijjiij ABBA ˆˆˆˆˆ ††
),(
ˆˆˆˆˆˆ
††
jiP
BAAB jiij
††
,
ˆˆˆˆˆˆ jiijji
BAAB
Unitary and non-unitary evolution
The effects formalism is not restricted to describing measurements,e.g. Schroedinger evolution
which has a single Krauss operator .
We can also use it to describe dissipative dynamics:
/ˆexp)0(ˆ/ˆexp)(ˆ)0(ˆ tHitHit
/ˆexpˆ tHiA
e
g
Spontaneous decay rate 2
We can write the evolved density operator in terms of two effects:
Measurement interpretation?
Has the atom decayed?
egetA
ggeeetA
AtAAtAt
tY
tN
YYNN
2
††
1)(ˆ
)(ˆ
ˆ)0(ˆ)(ˆˆ)0(ˆ)(ˆ)(ˆ
eeeAA
ggeeeAA
tYYY
tNNN
)1(ˆˆˆ
ˆˆˆ
2†
2†
e
g
e
g
No detection
122
)(ˆ
gg
tee
ggt
ge
teg
tee
ggge
egeee
e
ee
e
g
Detection
gg
Optimal operations: an example
What processes are allowed? Those that can be described by effects.
State separation
Suppose that we have a system known to have been preparedin one of two non-orthgonal states and . Our taskis to separate the states, i.e. to transform them into and with
so that the states are more orthogonal and hence more distinguishable. This process cannot be guaranteed but can succeed with some probability . How large can this be?
1
1
2
2
1122
SP
We introduce an effect associated with successful state separation
We can bound the success probability by considering the action of on a superposition of the states and noting that theresult must be an allowed state:
221 ||ˆ SS PA
1SA
22
11
2
1
1||
SP
There is a natural interpretation of this in terms of unambiguousstate discrimination:
Because this is the maximum allowed, state separation cannot better it so
)2(1)2(1)2(1?
)2(1Conc 11 PP
22
11
1Conc
2Conc
1
1
S
S
P
PPP
ssAs
A
A
clone
AAA
s
No cloning theorem - Wootters & Zurek, Dieks
Can we copy an unknown state ?Suppose it is possible:
But the superposition principle then gives:
Exact cloning? - Duan & Guo, Chefles
Ab
Aa bb
aa
?
Pclone
1 Pclone
Error-free discrimination probability
For the cloned states
baP 1IDP
ba
bbaaP
1
1clonedIDP
Cloning cannot increase the discrimination probability
baP
PPP
1
1clone
clonedIDPcloneIDP
separation bound
Can we clone exactly a quantum system known to be in the state |a or |b?
Summary
• The projective von Neumann measurements do not provide the most general description of a measurement.
• The use of probability operators (POMs) allows us to seek optimal measurements for any given detection problem.
• The language of operations and effects allows us to describe, in great generality, the post-measurement state.
Naimark’s theorem
All POMs correspond to measurements. Consider a POM for aqubit with N probability operators:
POM conditions:
Can we treat this as a von Neumann measurement in a enlarged state-space?
10,ˆ 10 jjjjjj
N
jjj
N
jjj
N
jj
N
jj
11
*0
10
*1
1
2
11
2
0
0
1
Our task is to represent the vectors as the components in the qubit space of a set of orthonormal states in an enlarged space.
Enlarged space spanned by N orthornormal states .
The extra states can be other states of the system or by introducing an ancilla:
An alternative orthonormal basis is
jj
j
jUjN
jjii
ˆ1
*
mn
N
jjnjmnm
1
*
22100 0,,0,0,1,0 NAAAAA
Because is unitary, we can also form the orthonormal basis:
These are the required orthonormal states for our von Neumannmeasurement:
All measurements can be described by a POM and all POMs describe possible measurements.
U
ijU ji
N
ij
1
0
†ˆ
ˆˆTr
ˆ
ˆ)(
j
jj
jjjP
Summary
The probabilities for the possible outcomes of any given measurement can be written in the form:
iTriP ˆˆ)(
The probability operators satisfy the three conditions
Iˆ,0ˆ,ˆˆ † ii
iii
All measurements can be described in this way and all sets of operators with these properties represent possible measurements.
Poincaré Sphere
Optical polarization
Bloch Sphere
Electron spin
Spin and polarisation QubitsPoincaré and Bloch Spheres
Two state quantum system
States of photon polarisation
Horizontal
Vertical
Diagonal up
Diagonal down
Left circular
Right circular
0
1
102
1
102
1
102
1 i
102
1 i
Example from quantum optics:
How can we best discriminate (without error) between the coherent states and ?
Coherent states:
Interfere like classical amplitudes.
Symmetric beamsplitter
r t
r t
nn
n
n !0
Unambiguous discrimination between coherent states - Hutner et al
i
0or2)(2 2/12/1 iii
2/12/1 2or0)11(2
This interference experiment puts all the light into a single output.Inconclusive results occur when no photons are detected. This happens with probability
)||2exp(02 22
2/1?P
