Optimizing
linear optics
quantum gates
J. Eisert
University of Potsdam, Germany
Entanglement and transfer of quantum information
Cambridge, September 2004
Quantum computation with
linear optics
Effective non-linearities
 Photons are relatively prone to decoherence, precise state
control is possible with linear optical elements
 Universal quantum computation can be done using optical
systems only
 The required non-linearities can be effectively obtained …
Input
Output
Optical
network
Effective non-linearities
 Photons are relatively prone to decoherence, precise state
control is possible with linear optical elements
 Universal quantum computation can be done using optical
systems only
 The required non-linearities can be effectively obtained …
Input
Output
Optical
network
Effective non-linearities
 Photons are relatively prone to decoherence, precise state
control is possible with linear optical elements
 Universal quantum computation can be done using optical
systems only
 The required non-linearities can be effectively obtained …
Input
?
Optical
network
Output
Effective non-linearities
 Photons are relatively prone to decoherence, precise state
control is possible with linear optical elements
 Universal quantum computation can be done using optical
systems only
 The required non-linearities can be effectively obtained …
Input
Output
Linear optics
network
Auxiliary modes,
Auxiliary photons
 by employing appropriate measurements
Measurements
KLM scheme
Knill, Laflamme, Milburn (2001): Universal quantum computation
is possible with
 Single photon sources
 linear optical networks
 photon counters, followed by postselection and feedforward
Input
Output
Linear optics
network
Auxiliary modes,
Auxiliary photons
Measurements
E Knill, R Laflamme, GJ Milburn, Nature 409 (2001)
TB Pittman, BC Jacobs, JD Franson, Phys Rev A 64 (2001)
JL O’ Brien, GJ Pryde, AG White, TC Ralph, D Branning, Nature 426 (2003)
Non-linear sign shifts
 At the foundation of the KLM contruction is a
non-deterministic gate,
- the non-linear sign shift gate, acting as
 0   1   2 
  0   1   2
(0)
(1)
(2)
(0)
(1)
(2)
 Using two such non-linear sign shifts, one can construct a
control-sign and a control-not gate
NSS
NSS
Success probabilities
 At the foundation of the KLM contruction is a
non-deterministic gate,
- the non-linear sign shift gate, acting as
 0   1   2 
  0   1   2
(0)
(1)
(2)
(0)
(1)
(2)
 Using teleportation, the overall scheme can be uplifted to a
scalable scheme with close-to-unity success probability,
using a significant overhead in resources
 To efficiently use the gates, one would like to implement
them with as high a probability as possible
Central question of the talk
 How well can the elementary gates be performed with
- static networks of arbitrary size,
- using any number of auxiliary modes and photons,
- making use of linear optics and photon counters,
followed by postselection?
 Meaning, what are the optimal success probabilities
of elementary gates?
Central question of the talk
 How well can the elementary gates be performed with
- static networks of arbitrary size,
- using any number of auxiliary modes and photons,
- making use of linear optics and photon counters,
followed by postselection?
 Meaning, what are the optimal success probabilities
of elementary gates?
Seems a key question for two reasons:
 Quantity that determines the necessary overhead in
resources
 For small-scale applications such as quantum repeaters,
high fidelity of the quantum gates may often be the
demanding requirement of salient interest (abandon some
of the feed-forward but rather postselect)
Networks for the non-linear sign shift
Input:
 (0) 0   (1) 1   (2) 2
Output:
 (0) 0   (1) 1   (2) 2
Networks for the non-linear sign shift
Input:
 (0) 0   (1) 1   (2) 2
Output:
 (0) 0   (1) 1   (2) 2
 Success probability
Network of linear optics elements
popt  0
(obviously, as the
non-linearity is not
available)
Networks for the non-linear sign shift
Input:
 (0) 0   (1) 1   (2) 2
Auxiliary
mode
Output:
 (0) 0   (1) 1   (2) 2
Photon
counter
 Success probability
Network of linear optics elements
popt  0
(the relevant constraints
cannot be fulfilled)
Networks for the non-linear sign shift
Input:
 (0) 0   (1) 1   (2) 2
Auxiliary
modes
Output:
 (0) 0   (1) 1   (2) 2
Photon
counters
 Success probability
Network of linear optics elements
popt  1/ 4??
(the best known scheme
has this success
probability
Networks for the non-linear sign shift
Input:
Output:
 (0) 0   (1) 1   (2) 2
 (0) 0   (1) 1   (2) 2
0
0
1
1
E Knill, R Laflamme, GJ Milburn, Nature 409 (2001)
 Success probability
Alternative schemes:
S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A
68 (2003)
TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A
65 (2001)
popt  1/ 4??
(the best known scheme
has this success
probability
Networks for the non-linear sign shift
Input:
 (0) 0   (1) 1   (2) 2
Auxiliary
modes
Output:
 (0) 0   (1) 1   (2) 2
Photon
counters
 Success probability
Network of linear optics elements
popt  ??
Networks for the non-linear sign shift
Input:
 (0) 0   (1) 1   (2) 2
Auxiliary
modes
Output:
 (0) 0   (1) 1   (2) 2
Photon
counters
 Success probability
Network of linear optics elements
popt  ???
Short history of the problem for the non-linear sign-s
 Knill, Laflamme, Milburn/Ralph, White, Munro, Milburn, Scheel, Knight
(2001-2003):
Construction of schemes that realize a non-linear sign shift
with success probability 1/4
 Knill (2003):
Any scheme with postselected linear optics cannot succeed
with a higher success probability than 1/2
 Reck, Zeilinger, Bernstein, Bertani (1994)/
Scheel, Lütkenhaus (2004):
Network can be written with a single beam splitter
communicating with the input
Conjectured that probability 1/4 could already be optimal
 Aniello (2004)
Looked at the problem with exactly one auxiliary photon
E Knill, R Laflamme, GJ Milburn, Nature 409 (2001)
TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A 65 (2001)
S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A 68 (2003)
M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994)
S Scheel, N Luetkenhaus, New J Phys 3 (2004)
(A late) overview over the talk
 Finding optimal success probabilities of elementary gates
within the paradigm of postselected linear optics
 Why is this a difficult problem?
J Eisert, quant-ph/0409156
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
J Eisert, M Curty, M Luetkenhaus, work in progress
WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress
(A late) overview over the talk
 Finding optimal success probabilities of elementary gates
within the paradigm of postselected linear optics
 Why is this a difficult problem?
 Help from an unexpected side: Methods from
semidefinite programming and convex optimization
as practical analytical tools
J Eisert, quant-ph/0409156
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
J Eisert, M Curty, M Luetkenhaus, work in progress
WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress
(A late) overview over the talk
 Finding optimal success probabilities of elementary gates
within the paradigm of postselected linear optics
 Why is this a difficult problem?
 Help from an unexpected side: Methods from
semidefinite programming and convex optimization
as practical analytical tools
 Formulate strategy: will develop a general recipe to give
rigorous bounds on success probabilities
 Look at more general settings, work in progress
J Eisert, quant-ph/0409156
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
J Eisert, M Curty, M Luetkenhaus, work in progress
WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress
(A late) overview over the talk
 Finding optimal success probabilities of elementary gates
within the paradigm of postselected linear optics
 Why is this a difficult problem?
 Help from an unexpected side: Methods from
semidefinite programming and convex optimization
as practical analytical tools
 Formulate strategy: will develop a general recipe to give
rigorous bounds on success probabilities
 Look at more general settings, work in progress
 Finally: stretch the developed ideas a bit further:
 Experimentally accessible entanglement witnesses for
imperfect photon detectors
 Complete hierarchies of tests for entanglement
J Eisert, quant-ph/0409156
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
J Eisert, M Curty, M Luetkenhaus, work in progress
WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress
Quantum gates
Input:
in   (0) 0  ...  (N ) N
Output:
out   (0)ei 0  ...  (N )ei N
0
N
 These are the quantum gates we will be looking at in the
following (which include the non-linear sign shift)
Quantum gates
Input:
in   (0) 0  ...  (N ) N
Output:
out   (0)ei 0  ...  (N )ei N
0
Arbitrary number
of additional field
modes
auxiliary photons
(Potentially complex) networks of linear optics elements
N
Quantum gates
Input:
in   (0) 0  ...  (N ) N
Output:
out   (0)ei 0  ...  (N )ei N
0
Arbitrary number
of additional field
modes
auxiliary photons
(Potentially complex) networks of linear optics elements
N
Quantum gates
Input:
in   (0) 0  ...  (N ) N
Output:
out   (0)ei 0  ...  (N )ei N
0
N
Arbitrary number
of additional field
modes
auxiliary photons
(Potentially complex) networks of linear optics elements
M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994)
S Scheel, N Luetkenhaus, New J Phys 3 (2004)
The input is linked only once to the auxiliary modes
Input:
Output:
out   (0)ei 0  ...  (N )ei N
in   (0) 0  ...  (N ) N
0
N
t C
 State vector of
auxiliary modes
“preparation”
1
 “measure2
ment”
3

n
   xk k  k
k0
3
3
3
(Potentially complex) networks of linear optics elements
M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994)
S Scheel, N Luetkenhaus, New J Phys 3 (2004)
Finding the optimal success probability
Input:
Output:
 State vector of
 “measure-
auxiliary modes
ment”
“preparation”
N
 U1
n
n
p  out    x k  (V12 (t)  13 ) j  k   k
1/ 2
( j)
j 0 k  0
Finding the optimal success probability
 Single beam splitter, characterized by
complex transmittivity t  C
nˆ1 r * aˆ2 aˆ1
 State vector of V12(t)  t e
auxiliary modes
e
r aˆ1 aˆ 2 nˆ 2
t
 “measurement”
“preparation”
N
 U1
n
n
p  out    x k  (V12 (t)  13 ) j  k   k
1/ 2
( j)
j 0 k  0
Finding the optimal success probability
 Arbitrarily many ( n) states of
arbitrary or infinite dimension
 State vector of
 “measure-
auxiliary modes
ment”
“preparation”
N
 U1
n
n
p  out    x k  (V12 (t)  13 ) j  k   k
1/ 2
( j)
j 0 k  0
Finding the optimal success probability
 Arbitrarily many ( n) states of
arbitrary or infinite dimension
 State vector of
 “measure-
auxiliary modes
ment”
“preparation”
 Weights
N
 U1
n
n
p  out    x k  (V12 (t)  13 ) j  k   k
1/ 2
( j)
j 0 k  0
 Non-convex function (exhibiting many local minima)
The problem with non-convex problems
 This innocent-looking problem of finding the optimal success
probability may be conceived as an optimization problem, but
one which is
- non-convex and
- infinite dimensional,
as we do not wish to restrict the number of
- photons in the auxiliary modes
- auxiliary modes
- linear optical elements
The problem with non-convex problems
 This innocent-looking problem of finding the optimal success
probability may be conceived as an optimization problem, but
one which is
- non-convex and
- infinite dimensional,
as we do not wish to restrict the number of
- photons in the auxiliary modes
- auxiliary modes
- linear optical elements
Infinitely many
local maxima
The problem with non-convex problems
Infinitely many
local maxima
The problem with non-convex problems
Infinitely many
local maxima
The problem with non-convex problems
Infinitely many
local maxima
The problem with non-convex problems
 Somehow, it would be good to arrive from the “other side”
Infinitely many
local maxima
The problem with non-convex problems
 Somehow, it would be good to arrive from the “other side”
Infinitely many
local maxima
The problem with non-convex problems
 Somehow, it would be good to arrive from the “other side”
Infinitely many
local maxima
The problem with non-convex problems
 Somehow, it would be good to arrive from the “other side”
 This is what we will be trying to do…
Infinitely many
local maxima
Convex optimization? Can it help?
Convex optimization problems
 What is a convex optimization problem again?
 Find the minimum of a convex function over a convex set
Convex optimization problems
 What is a convex optimization problem again?
 Find the minimum of a convex function over a convex set
Function
Set
Convex optimization problems
 What is a convex optimization problem again?
 Find the minimum of a convex function over a convex set
Function
Set
Semidefinite programs
 Class of convex optimization problems that we will make use of
- is efficiently solvable (but we are now not primarily dealing
with numerics),
- and is a powerful analytical tool:
 So-called semidefinite programs
Function
Set
Semidefinite programs
 Class of convex optimization problems that we will make use of
- is efficiently solvable (but we are now not primarily dealing
with numerics),
- and is a powerful analytical tool:
 So-called semidefinite programs
Linear function
Vector
T
c x
x Fi  0
Minimize the linear multivariate function
N
subject to the constraint
0
i1 i
F 
Set
 We will see in a second why they are so helpful
Matrices
Yes, ok, …
… but why should this help us to
assess the performance of quantum gates
in the context of linear optics?
1. Recasting the problem
 Again, the output of the quantum network, depending on
preparations and measurements, can be written as
N
n
p  out    x  f
1/ 2
( j)
( j)
k k k
j
j 0 k  0
 Functioning of the gate requires that
n
x
( j)
k k k
 Here
f
1/ 2 i j
p e
for all
j  0,...,N
k 0
k   k   k ,
f k  j k V1,2 (t) j k
( j)
J Eisert, quant-ph/0409156
1. Recasting the problem
 After all, the
(i) success probability should be maximized,
(ii) provided that the gate works
 Functioning of the gate requires that
n
x
( j)
k k k
 Here
f
1/ 2 i j
p e
for all
j  0,...,N
k 0
k   k   k ,
f k  j k V1,2 (t) j k
( j)
J Eisert, quant-ph/0409156
1. Recasting the problem
 But then, the problem is a non-convex infinite dimensional
problem, involving polynomials of arbitrary order in the
transmittivity t
 The strategy is now the following…
(one which can be applied to a number of contexts)
 Functioning of the gate requires that
n
x
( j)
k k k
 Here
f
1/ 2 i j
p e
for all
j  0,...,N
k 0
k   k   k ,
f k  j k V1,2 (t) j k
( j)
J Eisert, quant-ph/0409156
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
2. - transmittivities | t | 1
- complex scalar products 1,...,n 1
Isolate very difficult part of
the problem
Maximize over all
- weights x1,...,x n 1 ,
x x 1
T
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
2. - transmittivities | t | 1
- complex scalar products 1,...,n 1
Isolate very difficult part of
the problem
Relax to make
convex
Maximize over all
- weights x1,...,x n 1 ,
x x 1
T
3. Writing it as a semidefinite problem
 Then for each (t, k )
the problem is found to be one with
matrix constraints the elements of which are polynomials of
arbitrary degree in
t
 The resulting problem may look strange, but it is actually a
semidefinite program
 Why does this help us?
4. Lagrange duality
 Because we can exploit the (very helpful) idea
of Lagrange duality
 Primal problem
Lagrange duality
 Dual problem
 That is, for each problem, one can construct a so-called
“dual problem”
 Both are semidefinite problems
4. Lagrange duality
Original (primal)
problem
Globally optimal point
4. Lagrange duality
Dual problem
Original (primal)
problem
Globally optimal point
4. Lagrange duality
Educated guess
Dual problem
Original (primal)
problem
Globally optimal point
 Every solution (!) of the dual problem (any educated guess)
is a bound for the optimal solution of the primal problem
“Approaching the problem from the other side”
4. Lagrange duality
Educated guess
Dual problem
Original (primal)
problem
Globally optimal point
 Every solution (!) of the dual problem (any educated guess)
is a bound for the optimal solution of the primal problem
“Approaching the problem from the other side”
4. Lagrange duality
Educated guess
Dual problem
Original (primal)
problem
Globally optimal point
 Every solution (!) of the dual problem (any educated guess)
is a bound for the optimal solution of the primal problem
“Approaching the problem from the other side”
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
Isolate very difficult part of
the problem
2. - transmittivities | t | 1
Maximize over all
- weights x1,...,x n 1 ,
3.
Semidefinite program in
- matrix Z
- complex scalar products 1,...,n 1
“
4. - transmittivities | t | 1
- complex scalar products 1,...,n 1
x x 1
T
Make use of idea of Lagrange duality
“approaching from the other side”
Minimize dual over all
- matrices M  0
The entries of the
5. Construction of a solutionmatrices
for the dual
 The dual can be shown to be of the form
T
(1,...,1) z
minimize
subject to
0

 z1

F0  




z2

0


 2N 2
 0


  v a Fa 
 a1

...



zn 2 


F j, j  1,...,2N  1
are polynomials of degree
in the transmittivity
n
t
W




 F2N 2  0




as optimization problem (still infinite dimensional) in vectors
and
z
v
5. Construction of a solution for the dual
 The dual can be shown to be of the form
T
(1,...,1) z
minimize
subject to
0

 z1

F0  




z2

0


 2N 2
 0


  v a Fa 
 a1

...



zn 2 


W




 F2N 2  0




 Finding a solution now means “guessing” a matrix W
 Can be done, even in a way such that the unwanted
dependence on preparation and measurement
eliminated
 Instance of a problem that can explicitly solved
 k  can be
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
Isolate very difficult part of
the problem
2. - transmittivities | t | 1
Maximize over all
- weights x1,...,x n 1 ,
3.
Semidefinite program in
- matrix Z
- complex scalar products 1,...,n 1
“
4. - transmittivities | t | 1
- complex scalar products 1,...,n 1
x x 1
T
Make use of idea of Lagrange duality
“approaching from the other side”
Minimize dual over all
- matrices M  0
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
Isolate very difficult part of
the problem
2. - transmittivities | t | 1
Maximize over all
- weights x1,...,x n 1 ,
3.
Semidefinite program in
- matrix Z
- complex scalar products 1,...,n 1
“
4. - transmittivities | t | 1
- complex scalar products 1,...,n 1
5.
x x 1
T
Make use of idea of Lagrange duality
“approaching from the other side”
Minimize dual over all
- matrices M  0
Construct explicit solution,
independent from | t | 1, 1,...,n 1
The strategy
1. Maximize over all
- transmittivities | t | 1
- complex scalar products 1,...,n 1
T
- weights x1,...,x n 1 , x x  1
2. - transmittivities
- complex scalar products
3.
“
4. - transmittivities
- complex scalar products
5.
Isolate very difficult part of
the problem
Maximize over all
- weights
Semidefinite program in
- matrix
Make use of idea of Lagrange duality
“approaching from the other side”
Minimize dual over all
- matrices
Construct explicit solution,
independent from
This gives a general bound
for the original problem
(optimal success probability)
6. Done!
 For the non-linear sign shift, e.g., one can construct a
solution for the dual problem for each t,
t 1
 This solution delivers in each case
1.0
0.75
Educated guess
0.5
0.25
0
Optimal point
6. Done!
 For the non-linear sign shift, e.g., one can construct a
solution for the dual problem for each t,
t 1
 This solution delivers in each case pmax 1/4
using the argument of Lagrange duality, we are done!
1.0
0.75
Educated guess
0.5
0.25
0
Optimal point
6. Done!
 For the non-linear sign shift, e.g., one can construct a
solution for the dual problem for each t,
t 1
 This solution delivers in each case pmax 1/4
using the argument of Lagrange duality, we are done!
 So, this pmax 1/4 gives a bound for the original problem …
… and one of which we know it is optimal
Educated guess
0.5
0.25
0
Optimal point
Realizing a non-linear sign shift gate
 (0) 0   (1) 1   (2) 2
Auxiliary
modes
 (0) 0   (1) 1   (2) 2
Photon
counters
J Eisert, quant-ph/0409156
Realizing a non-linear sign shift gate
 (0) 0   (1) 1   (2) 2
 (0) 0   (1) 1   (2) 2
Photon
counters
Auxiliary
modes
 No matter how hard we try, there is within the paradigm of
linear optics, photon counting, followed by postselection,
no way to go beyond the optimal success probability of
pmax 1/4
J Eisert, quant-ph/0409156
Realizing a non-linear sign shift gate
 (0) 0   (1) 1   (2) 2
Auxiliary
modes
 (0) 0   (1) 1   (2) 2
Photon
counters
 Surprisingly: any additional resources in terms of modes/photons
than two auxiliary modes/photons do not lift up the success
probability at all
J Eisert, quant-ph/0409156
Success probabilities of other sign gates
 The same method can be immediately applied to other
quantum gates, e.g., to the sign-shift with phase

i (2)
 0   1   2 
  0   1  e 
(0)
(1)
(2)
(0)
(1)
2
Success probabilities of other sign gates
 The same method can be immediately applied to other
quantum gates, e.g., to the sign-shift with phase

i (2)
 0   1   2 
  0   1  e 
(0)
(1)
(2)
(0)
(1)
pmax
1
0.75
0.5
0.25
0
0


2
Success probabilities of other sign gates
 The same method can be immediately applied to other
quantum gates, e.g., to the sign-shift with phase

i (2)
 0   1   2 
  0   1  e 
(0)
(1)
(2)
(0)
(1)
“do nothing”
pmax
Non-linear sign
shift gate
1
0.75
0.5
0.25
0
0

2

Higher photon numbers
 The same method can be immediately applied to other
quantum gates, such as those involving higher photon
numbers
(0)
(1)
(2)
i (3)
 (0) 0   (1) 1   (2) 2   (3) 3 
  0   1   2  e  3
“do nothing”
pmax
1
0.75
0.5
0.25
0
0


Non-linear sign shift with one step of feed-forward
 Assessing success probabilities with single rounds of
classical feedback
Work in progress with WJ Munro, P Kok, K Nemoto, S Scheel
Input:
Output:
 0  1   2
(0)
(1)
(2)
 (0) 0   (1) 1   (2) 2
Non-linear sign shift with one step of feed-forward
 Assessing success probabilities with single rounds of
classical feedback
Work in progress with WJ Munro, P Kok, K Nemoto, S Scheel
Input:
Output:
 (0) 0   (1) 1   (2) 2
 0  1   2
(0)
(1)
(2)
success
Potentially
correctable
failures
incorrectable
failures
success
failure
Feed-forward seems not to help so much
 Then, it turns out that whenever we choose an optimal gate
in the first run, succeeding with
pmax 1/4 …
 … then any classical feedforward follows by a correction network
can increase the success probability to at most
pmax  0.3
 That is, single rounds of feed-forward at the level of
individual gates do not help very much at all!
Finally,
… extending these ideas to find
other tools relevant to optical settings
Joint work with P Hyllus, O Gühne, M Curty, N Lütkenhaus
Stretch these ideas further to get practical tools
What was the point of the method before?

We developed a strategy to make methods from convex
optimization applicable to solve a
- non-convex and
- infinite-dimensional problem
to assess linear optical schemes
Stretch these ideas further to get practical tools
What was the point of the method before?

We developed a strategy to make methods from convex
optimization applicable to solve a
- non-convex and
- infinite-dimensional problem
to assess linear optical schemes
Can such strategies also formulated to find
 good experimentally accessible witnesses to detect
entanglement, which work for
- weak pulses and
- finite detection efficiencies?
 Practical tools to construct complete hierarchies of
criteria for multi-particle entanglement?
Entanglement witnesses
 Scenario 1: (experimentally) detecting entanglement directly
“Yes, it is entangled!”
Unknown state

Entanglement
witness
“Hm, I don’t know”
W
 An entanglement witness is an observable W = W
+
with
tr[ W ] 1
“Yes, it is entangled!”
tr[ W ] 1
“Hm, I don’t know”
Theory:
M & P & R Horodecki, Phys Lett A 232 (1996)
BM Terhal, Phys Lett A 271 (2000)
G Toth, quant-ph/0406061
Experiment:
M Barbieri et al, Phys Rev Lett 91 (2003)
M Bourennane et al, Phys Rev Lett 92 (2004)
Entanglement witnesses
 Entanglement witnesses are important tools
- if complete state tomography is inaccessible/expensive
- in quantum key distribution:
necessary for the positivity of the intrinsic information
is that quantum correlations can be detected
U Maurer, S Wolf, IEEE Trans Inf Theory 45 (1999)
N Gisin, S Wolf, quant-ph/0005042
M Curty, M Lewenstein, N Luetkenhaus, Phys Rev Lett 92 (2004)
Entanglement witnesses
Problem: a number of entanglement witnesses are known - for
the two-qubit case all can be classified - but,
 It is difficult (actually NP) to find optimal entanglement
witnesses in general
 Include non-unit detection efficiencies:
detectors do not click although they should have
Imperfect detectors
modeled by perfect
detectors, preceded by
beam splitter
Entanglement witnesses
 Starting from any entanglement witness, one always gets
an optimal one if one knows
  min a,b W a,b  tr[ a,b a,bW ]
subject to
a,b
being a product state vector
 Then W  1 is an optimal witness
 But: how does one find this global minimum?
We have to be sure, otherwise we do not get an entanglement
witness
The strategy
1. Minimize
a,b M a,b  tr[ a,b a,b M]
over all state vectors
a,b
The strategy
1. Minimize
a,b M a,b  tr[ a,b a,b M]
over all state vectors
2. Minimize
over all

Equivalently
a,b
tr[ M]
with
tr[  A ]  tr[ A ]  1
2
3
tr[  B ]  tr[ B ]  1
2
3
The strategy
1. Minimize
Equivalently
over all state vectors
2. Minimize
over all
with
 Any operator 
that satisfies
tr[  ]  tr[ ]  1
2
3
is necessarily a pure state (polynomial
characterization of pure states)
NS Jones, N Linden, quant-ph/0407117
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
The strategy
1. Minimize
a,b M a,b  tr[ a,b a,b M]
over all state vectors
2. Minimize
over all

Equivalently
a,b
tr[ M]
with
tr[  A ]  tr[ A ]  1
2
3
tr[  B ]  tr[ B ]  1
2
3. Write this as a hierarchy of
semidefinite programs
3
Use methods from relaxation
theory of polynomially
constrained problems
The strategy
1. Minimize
over all state vectors
2. Minimize
over all
Equivalently
 Any polynomially constrained problem
with
typically computationally hard NP problems
can be relaxed to efficiently solvable
semidefinite programs
3. Write this as a hierarchy of
semidefinite programs
 One can even find hierarchies that
approximate the solution to arbitrary
accuracy
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
JB Lasserre, SIAM J Optimization 11 (2001)
The strategy
1. Minimize
a,b M a,b  tr[ a,b a,b M]
over all state vectors
2. Minimize
over all

Equivalently
a,b
tr[ M]
with
tr[  A ]  tr[ A ]  1
2
3
tr[  B ]  tr[ B ]  1
2
3
Use methods from relaxation
theory of polynomially
constrained problems
3. Write this as a hierarchy of
semidefinite programs
4.
Since each step gives a lower
bound, each step gives an
entanglement witness
Entanglement witnesses for finite detection efficiencies
 In this way, one can obtain good entanglement witnesses
for imperfect detectors
 = min tr[ a,b a,b i A WAi ]
i
subject to
a,b
being a product state vector
 Starting from a witness in an error-free setting, one gets
a new optimal witness as
 A WA
i
i

i
 1
J Eisert, M Curty, N Luetkenhaus, work in progress
Hierarchies of criteria for multi-particle entanglement
 Scenario 2: complete hierarchies of sufficient criteria for
multi-particle entanglement: Testing whether a known state
(i.e., from state tomography) is multi-particle entangled
“I don’t
know”
Known state

(e.g., from
state
tomography)
1st sufficient
criterion for
entanglement
“Yes, it is
entangled!”
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156
Hierarchies of criteria for multi-particle entanglement
 Scenario 2: complete hierarchies of sufficient criteria for
multi-particle entanglement: Testing whether a known state
(i.e., from state tomography) is multi-particle entangled
“I don’t
know”
Known state

(e.g., from
state
tomography)
“I don’t
know”
1st sufficient
criterion for
entanglement
2nd sufficient
criterion for
entanglement
“Yes, it is
entangled!”
“Yes, it is
entangled!”
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156
Hierarchies of criteria for multi-particle entanglement
 Scenario 2: complete hierarchies of sufficient criteria for
multi-particle entanglement: Testing whether a known state
(i.e., from state tomography) is multi-particle entangled
“I don’t
know”
Known state

(e.g., from
state
tomography)
“I don’t
know”
1st sufficient
criterion for
entanglement
2nd sufficient
criterion for
entanglement
“Yes, it is
entangled!”
“Yes, it is
entangled!”
 Every entangled state is necessarily detected in some
step of the hierarchy
J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135
Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156
The lessons to learn
Physically:
 We have developed a strategy which is applicable to
assess/find optimal linear optical schemes
 Non-linear sign shift with linear optics, photon counting
followed by postselection cannot be implemented with
higher success probability than 1/4
 First tight general upper bound for success probability
- Good news: the most feasible protocol is already the
optimal one
- Single rounds of feedforward do not help very much
- But: also motivates the search for hybrid methods,
leaving the strict framework of linear optics (see
following talk by Bill Munro)
The lessons to learn
Formally:
the methods of convex optimization

are powerful when one intends to assess the maximum
performance of linear optics schemes without
restricting the allowed resources

in a situation where it is hard to conceive that one can
find an direct solution to the original problem
 And finally, we had a look at where very much related
ideas can be useful to detect entanglement in optical
settings