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Photons and Quantum Information. Stephen M. Barnett. 1. A bit about photons 2. Optical polarisation 3. Generalised measurements 4. State discrimination Minimum error Unambiguous Maximum confidence. 1. A bit about photons. Photoelectric effect - Einstein 1905. BUT … - PowerPoint PPT Presentation
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Photons and Quantum InformationStephen M. Barnett
1. A bit about photons
2. Optical polarisation
3. Generalised measurements
4. State discriminationMinimum errorUnambiguousMaximum confidence
1. A bit about photons
Photoelectric effect - Einstein 1905
WheV
BUT …
Modern interpretation: resonance with the atomic transition frequency.
We can describe the phenomenon quantitatively by a model in which the matter is described quantum mechanically but the light is described classically.
Photons?
Single-photon (?) interference - G. I. Taylor 1909
Light source Smoked glass screens Screen with two slits
Photographic plate
Longest exposure - three months
“According to Sir J. J. Thompson, this sets a limit on the size of the indivisible units.”
Single photons (?) Hanbury-Brown and Twiss
1
2 1)0(
)2()1(
)2,1(
2
2)2(
2
I
Ig
ITPIRP
IRTTIRIP
Blackbody light g(2)(0) = 2
Laser light g(2)(0) = 1
Single photon g(2)(0) = 0 !!!
violation of Cauchy-Swartz inequality
Single photon source - Aspect 1986
Detection of the first photon acts as a herald for the second.
Second photon available for Hanbury-Brown and Twiss measurement or interference measurement.
Found g(2)(0) ~ 0 (single photons) andfringe visibility = 98%
Two-photon interference - Hong, Ou and Mandel 1987
“Each photon then interferes only with itself. Interference between different photons never occurs” Dirac
Two photons in overlapping modes
50/50 beam splitter R=T=1/2
P = R T 2 = 1/2
Boson “clumping”. If one photon is present then it is easier to add a second.
Two-photon interference - Hong, Ou and Mandel 1987
“Each photon then interferes only with itself. Interference between different photons never occurs” Dirac
Two photons in overlapping modes
50/50 beam splitter R=T=1/2
P = R T 2 = 1/2
Boson “clumping”. If one photon is present then it is easier to add a second.
Two-photon interference - Hong, Ou and Mandel 1987
“Each photon then interferes only with itself. Interference between different photons never occurs” Dirac
Two photons in overlapping modes
50/50 beam splitter R=T=1/2
P = 0 !!!
Destructive quantum interference between the amplitudes for two reflections and two transmissions.
1. A bit about photons
2. Optical polarisation
3. Generalised measurements
4. State discriminationMinimum errorUnambiguousMaximum confidence
Maxwell’s equations in an isotropic dielectric medium take the form:
tc
t
EB
BE
BE
2
00
E, B and k are mutually orthogonal
For plane waves (and lab. beams that are not too tightly focussed) this means that the E and B fields are constrained to lie in the plane perpendicular to the direction of propagation.
k
E
B
BES 10
Consider a plane EM wave of the form
)(exp
)(exp
0
0
tkzitkzi
BBEE
If E0 and B0 are constant and real then the wave is said to be linearly polarised.
E
B
E
B
Polarisation is defined by an axis rather than by a direction:
If the electric field for the plane wave can be written in the form
)(exp)(0 tkziiE jiE
Then the wave is said to be circularly polarised.
B
E
For right-circular polarisation, an observer would see the fields rotating clockwise as the light approached.
The Jones representation
We can write the x and y components of the complex electric field amplitude in the form of a column vector:
y
x
iy
ix
y
x
eE
eEEE
0
0
0
0
The size of the total field tells us nothing about the polarisation so we can conveniently normalise the vector:
Horizontal polarisation
01
Vertical polarisation
10
Left circular polarisation
i1
21
Right circular polarisation
i1
21
One advantage of this method is that it allows us to describe the effects of optical elements by matrix multiplication:
Linear polariser(oriented to horizontal):
451111
,901000
,00001
21
Quarter-wave plate(fast axis to horizontal):
451
1,90
001
,00
012
1
i
iii
Half-wave plate(fast axis horizontal or vertical):
1001
The effect of a sequence of n such elements is:
BA
dcba
dcba
BA
nn
nn
11
11
We refer to two polarisations as orthogonal if
01*2 EE
This has a simple and suggestive form when expressed in terms of the Jones vectors:
0
0
0
iftoorthogonalis
1
1†
2
2
1
1*2
*2
1*21
*2
2
2
1
1
BA
BA
BABA
BBAA
BA
BA
There is a clear and simple mathematical analogy between the Jones vectors and our description of a qubit.
Poincaré Sphere
Optical polarization
Bloch Sphere
Electron spin
Spin and polarisation QubitsPoincaré and Bloch Spheres
Two state quantum system
We can realise a qubit as the state of single-photon polarisation
Horizontal
Vertical
Diagonal up
Diagonal down
Left circular
Right circular
0
102
1
102
1
102
1 i
102
1 i
1
1. A bit about photons
2. Optical polarisation
3. Generalised measurements
4. State discriminationMinimum errorUnambiguousMaximum confidence
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,,21
4)(exp2, 2
24/12
mmmmmm
mm
mm
pxpxdpdx
xxipxxdxpx
This leads us to the POM elements:
mmmmmm pxpxpx ,,21),(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.
1. A bit about photons
2. Optical polarisation
3. Generalised measurements
4. State discriminationMinimum errorUnambiguousMaximum confidence
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
0ˆˆ 22122
221212 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
23
2
23
1
Minimum error probabilityis 1/3.
This corresponds to a POMwith elements
ˆ j 23 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
2/32/1
2/32/1
01
2/3
02/3
000
10
1/ 20
1/ 2
0
6/112/1
6/112/1
3/23/1
06/1
03/2
06/1
3/2
06/1
06/1
0
6/1
06/1
03/2
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?
1121
2
2221
1
ˆˆIˆ
11ˆ
11ˆ
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.
Maximum confidence measurements seek to maximise theconditional probabilities
for each state. For unambiguous discrimination these are all 1.
Bayes’ theorem tells us that
so the largest values of those give us maximum confidence.
)|( iiP
k kikk
iiii
i
iiiii p
pP
PpP
ˆˆ
)()|()|(
The solution we find is
where
11 ˆˆˆˆ ji
iii
jjj
p
ˆˆ
ˆ
Croke et al Phys. Rev. Lett. 96, 070401 (2006)
Example
• 3 states in a 2-dimensional space
• Maximum Confidence Measurement:
• Inconclusive outcome needed
Optimum probabilities• Probability of correctly
determining state maximised for minimum error measurement
• Probability that result obtained is correct maximised by maximum confidence measurement:
Results:
Minimum error
Maximum confidence
• Photons have played a central role in the development of quantum theory and the quantum theory of light continues to provide surprises.
• True single photons are hard to make but are, perhaps, the ideal carriers of quantum information.
• It is now possible to demonstrate a variety of measurement strategies which realise optimised POMs
• The subject of quantum optics also embraces atoms, ions molecules and solids ...
Conclusions