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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
eV  h  W
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
P (1,2)  RI  TI  RT I 2
1
P (1)  R I
2
Blackbody light
g(2)(0)
Laser light
g(2)(0) = 1
=2
g ( 2 ) ( 0) 
P ( 2)  T I
I2
I
Single photon
2
1
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) and
fringe visibility = 98%
Two-photon interference - Hong, Ou and Mandel 1987
“Each photon then interferes only with itself. Interference between different
photons never occurs” Dirac
50/50 beam splitter R=T=1/2
P = R T 2 = 1/2
Two photons in
overlapping
modes
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
50/50 beam splitter R=T=1/2
P = R T 2 = 1/2
Two photons in
overlapping
modes
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
50/50 beam splitter R=T=1/2
P = 0 !!!
Two photons in
overlapping
modes
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 discrimination
Minimum error
Unambiguous
Maximum confidence
Maxwell’s equations in an isotropic dielectric medium take the form:
E  0
E, B and k are mutually orthogonal
B  0
B
E  
t
 E
B  2
c t
E
S
k
B
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.
1
0 E  B
Consider a plane EM wave of the form
E  E 0 exp i (kz  t )
B  B 0 exp i (kz  t )
If E0 and B0 are constant and real then the wave is said to be linearly polarised.
B
Polarisation is defined by an axis
rather than by a direction:
B
E
E
If the electric field for the plane wave can be written in the form
E  E0 (i  ij) exp i(kz  t )
Then the wave is said to be circularly polarised.
For right-circular polarisation, an observer
would see the fields rotating clockwise as
the light approached.
B
E
The Jones representation
We can write the x and y components of the complex electric field amplitude
in the form of a column vector:
i
 E0 x   E0 x e x 
E   
i y 
 0 y   E0 y e 
The size of the total field tells us nothing about the polarisation so we can
conveniently normalise the vector:
Horizontal polarisation
Vertical polarisation
1 
0 
 
0 
1 
 
Left circular polarisation
1
2
1
i 

Right circular polarisation
1
2
1
 i 
 
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):
Quarter-wave plate
(fast axis to horizontal):
1 0  0 0  1  1  1

0
,
90
,

45
0 0 
0 1 

2

1
1






1 0  1 0  
0 i  0 , 0  i  90 ,




 1  i


45
 i 1 


1 0 
0 1


Half-wave plate
(fast axis horizontal or vertical):
The effect of a sequence
of n such elements is:
1
2
 A an
 B  c
   n
bn  a1


d n   c1
b1   A



d1   B 
We refer to two polarisations as orthogonal if
E*2 E1  0
This has a simple and suggestive form when expressed in terms of the Jones
vectors:
 A1 
 A2 
 B  is orthogonal to  B  if
 1
 2
A2* A1  B2* B1  0

A

*
2
 A2 
 
 B2 

B2*  A1 
B   0
 1
†
 A1 
B   0
 1
There is a clear and simple
mathematical analogy between
the Jones vectors and our
description of a qubit.
Spin and polarisation Qubits
Poincaré and Bloch Spheres
Two state quantum system
Bloch Sphere
Electron spin
Poincaré Sphere
Optical polarization
We can realise a qubit as the state of single-photon polarisation
Horizontal
0
Vertical
1
Diagonal up
1
2
Diagonal down
1
2
Left circular
1
2
Right circular
1
2
0
1
0
1
0
i 1 
0
i 1 
1. A bit about photons
2. Optical polarisation
3. Generalised measurements
4. State discrimination
Minimum error
Unambiguous
Maximum confidence
Probability operator measures
Our generalised formula for measurement probabilities is
P(i)  Tr ˆi ˆ 
The set probability operators describing a measurement is
called a probability operator measure (POM) or a positive
operator-valued measure (POVM).
The probability operators can be defined by the properties
that they satisfy:
Properties of probability operators
I. They are Hermitian
II. They are positive
III. They are complete
ˆ  ˆ n
†
n
 ˆ n   0  
 ˆ
n
 Î
Observable
Probabilities
Probabilities
n
IV. Orthonormal
ˆ iˆ j   ijˆ i
??
Generalised measurements as comparisons
Prepare an ancillary system in
a known state:
A
S
Perform a selected unitary
transformation to couple the system
and ancilla:
S+A
S
S  A
Uˆ  S  A
A
Perform a von Neumann measurement
on both the system and ancilla:
i   Si  Ai
The probability for outcome i is
P(i)  i Uˆ A  S  S A Uˆ † i
 S

A Uˆ † i

i Uˆ A  S
ˆi
The probability operators ˆ i
act only on the system
state-space.
POM rules:
I. Hermiticity:

A Uˆ † i
i Uˆ A
†
ˆ
 AU i

†
i Uˆ A
II. Positivity:
 ˆi   i Uˆ  S A
2
III. Completeness follows
from:

i
i
i  Î A,S
0
Generalised measurements as comparisons
We can rewrite the detection probability as
P(i)  A   S Pˆi  S  A
†
ˆ
ˆ
Pi  U i
i Uˆ
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.
ˆi  A Pˆi A
ˆ nˆ m  A Pˆn A A Pˆm A  0
Simultaneous measurement of position and momentum
The simultaneous perfect measurement of x and p would violate
complementarity.
p
Position measurement gives no
momentum information and
depends on the position probability
distribution.
x
Simultaneous measurement of position and momentum
The simultaneous perfect measurement of x and p would violate
complementarity.
Momentum measurement gives no
position information and
p
depends on the momentum probability
distribution.
x
Simultaneous measurement of position and momentum
The simultaneous perfect measurement of x and p would violate
complementarity.
p
Joint position and measurement
gives partial information on both
the position and the momentum.
x
Position-momentum minimum
uncertainty state.
POM description of joint measurements
Probability density:
( xm , pm )  Trˆˆ ( xm , pm )
Minimum uncertainty states:

xm , pm  2

2 1 / 4
 ( x  xm ) 2

 dx exp  4 2  ipm x x
1
dxm dpm xm , pm xm , pm  Î

2
This leads us to the POM elements:
1
ˆ ( xm , pm ) 
xm , p m xm , p m
2
The associated position probability distribution is
 ( x  xm ) 2 
( xm )   dx x ˆ x exp 

2
2



 Var ( xm )  x 2   2
2

& Var ( pm )  p 2 
4 2
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 discrimination
Minimum error
Unambiguous
Maximum confidence
The communications problem
‘Alice’ prepares a quantum system in one of a set of N possible
signal states and sends it to ‘Bob’
i selected.
prob. pi
Preparation
device
̂ i
Measurement
device
P( j | i)  Tr ˆ j ˆ i 
Bob is more interested in
Tr ˆ j ˆ i  pi
P (i | j ) 
Tr (ˆ j ˆ )
Measurement
result j
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
 1 ˆ1  1  1   2 ˆ 2  2
 2 ˆ1  2  0   1 ˆ 2  1
Completeness, positivity and 1 ˆ1 1  1
 ˆ1   1  1  Aˆ
  2 ˆ1  2   1  2
Aˆ positive and Aˆ  1  0
2
  2 Aˆ  2   1  2
2
0
What is the best we can do? Depends on what we mean by ‘best’.
Minimum-error discrimination
We can associate each measurement operator ˆ i with a signal
state ̂ i . This leads to an error probability
Pe  1   p jTr ˆ j ˆ j 
N
j 1
Any POM that satisfies the conditions
ˆ j ( p j ˆ j  pk ˆ k )ˆ k  0
N
 p ˆ ˆ
k 1
k
k
k
 p j ˆ j  0
will minimise the probability of error.
j , k
j
For just two states, we require a von Neumann measurement with
projectors onto the eigenstates of p1 ˆ1  p2 ˆ 2 with positive (1)
and negative (2) eigenvalues:
Pemin  12 1  Tr p1 ˆ1  p2 ˆ 2 
Consider for example the two pure qubit-states
 1  cos  0  sin  1
1  2  cos(2 )
 2  cos  0  sin  1
The minimum error is achieved by measuring in
the orthonormal basis spanned by the states 1
and 2 .
1
1
2
2
We associate 1 with  1 and 2 with  2 :
Pe  p1  1 2
2
 p2  2 1
2
The minimum error is the Helstrom bound
Pemin
 
1
 2 1  1  4 p1 p2  1  2
 
1/ 2 
2




A single photon only gives one “click”
P = |a|2
a
+b
P = | b| 2
But this is all we need to discriminate between our two states
with minimum error.
A more challenging example is the ‘trine ensemble’ of three
equiprobable states:

0 

31
 1   12 0  3 1
p1 
 2   12
p2 
3  0
1
3
p3 
1
3
1
3
It is straightforward to confirm that the minimum-error conditions
are satisfied by the three probability operators
ˆi  23  i  i
Simple example - the trine states
Three symmetric states of photon polarisation
3  
2 
   3
1 
2
  3
2
Minimum error probability
is 1/3.
This corresponds to a POM
with elements
2
ˆ
 j   j  j
3
How can we do a polarisation
measurement with these three
possible results?
Polarisation interferometer - Sasaki et al, Clarke et al.
1 / 6  2 / 3 1 / 6 




 0  0  0 




 0  0  0 




1 / 6   1 / 6   2 / 3 
PBS

2
 0  0  0 
 


 0  3 / 2  3 / 2 
PBS
 1  1 / 2   1 / 2 
 


 0  3 / 2  3 / 2 

2
11/ 2 1/ 2
   

0 0   0 
PBS
 0  0  0 




 2 / 3 1 / 6   1 / 6 
 1 / 3 1 / 12   1 / 12 




 2 / 3  1 / 6   1 / 6 




Unambiguous discrimination
The existence of a minimum error does not mean that error-free
or unambiguous state discrimination is impossible. A von Neumann
measurement with
Pˆ1   1  1


ˆ
P1   1  1
will give unambiguous identification of  2 :
result
1  2
error-free
result
1  ?
inconclusive
There is a more symmetrical approach with
ˆ1 
ˆ 2 
1
1 1  2
1
1 1  2
 2  2
 1  1
ˆ ?  Î  ˆ1  ˆ 2
Result 1
Result 2
Result ?
State  1
1 1  2
0
1  2
State  2
0
1 1  2
1  2
How can we understand the IDP measurement?
Consider an extension into a 3D state-space


b

a
a
b
Unambiguous state discrimination - Huttner et al, Clarke et al.
a
?
b
a
b
A similar device
allows minimum
error discrimination
for the trine states.
Maximum confidence measurements seek to maximise the
conditional probabilities
P( i | i )
for each state.
For unambiguous discrimination these are all 1.
Bayes’ theorem tells us that
pi  i ˆi  i
pi P(i |  i )
P( i | i ) 

P(i )
k pk  k ˆi  k
so the largest values of those give us maximum confidence.
The solution we find is
ˆi  ˆ ˆ j ˆ
1
1
where
ˆ j   j  j
ˆ   pi ˆ i
i
Croke et al Phys. Rev. Lett. 96, 070401 (2006)
Example
• 3 states in a 2dimensional
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:
Maximum confidence
Minimum error
Conclusions
• 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 ...