J. Eisert University of Potsdam, Germany Entanglement and transfer of quantum information Cambridge, September 2004 Optimizing linear optics quantum gates

J. Eisert University of Potsdam, Germany

Entanglement and transfer of quantum informationCambridge, September 2004

Optimizing linear optics quantum gates

Quantum computation withlinear optics

Effective non-linearities

OutputOpticalnetwork

Input

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






Input

?





Output

Auxiliary modes, Auxiliary photons

Linear opticsnetwork

Input

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

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)

Output

Auxiliary modes, Auxiliary photons

Linear opticsnetwork

Input

Measurements

Non-linear sign shifts

At the foundation of the KLM contruction is a non-deterministic gate,

- the non-linear sign shift gate, acting as

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2 ⏐ → ⏐ ξ(0) 0 +ξ(1) 1 −ξ(2) 2

NSS

NSS

Using two such non-linear sign shifts, one can construct a control-sign and a control-not gate

Success probabilities

At the foundation of the KLM contruction is a non-deterministic gate,

- the non-linear sign shift gate, acting as

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

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2 ⏐ → ⏐ ξ(0) 0 +ξ(1) 1 −ξ(2) 2

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

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)

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?

Networks for the non-linear sign shift

Input: Output:

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2


Input: Output:

Network of linear optics elements

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

Success probability

(obviously, as the non-linearity is not available)

€

popt =0

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2


Input: Output:

Auxiliary mode

Success probability

(the relevant constraints cannot be fulfilled)

€

popt =0

Photoncounter


€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2


Input: Output:

Auxiliary modes

Success probability

(the best known scheme has this success probability

€

popt =1/4??

Photoncounters


€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2


Input: Output:

Success probability

(the best known scheme has this success probability

€

popt =1/4??

€

0

€

1

€

0

€

1

E Knill, R Laflamme, GJ Milburn, Nature 409 (2001)

Alternative schemes:

S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A68 (2003)TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A65 (2001)

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2


Input: Output:

Auxiliary modes

Success probability

€

popt =??

Photoncounters


€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2


Input: Output:

Auxiliary modes

Success probability

€

popt =???

Photoncounters


€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2

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 photonE 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/0409156J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135J Eisert, M Curty, M Luetkenhaus, work in progressWJ Munro, S Scheel, K Nemoto, J Eisert, work in progress




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






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 entanglementJ Eisert, quant-ph/0409156J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135J Eisert, M Curty, M Luetkenhaus, work in progressWJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

Quantum gates

€

ψin =ξ(0) 0 +...+ξ(N) NInput:

€

ψout =ξ(0)eiϕ0 0 +...+ξ(N)eiϕN NOutput:

These are the quantum gates we will be looking at in the following (which include the non-linear sign shift)

Quantum gates

Arbitrary number of additional fieldmodes

auxiliary photons

(Potentially complex) networks of linear optics elements

Input: Output:

€

ψin =ξ(0) 0 +...+ξ(N) N

€

ψout =ξ(0)eiϕ0 0 +...+ξ(N)eiϕN N

Quantum gates


auxiliary photons

Input: Output:

€

ψin =ξ(0) 0 +...+ξ(N) N

€



Quantum gates


auxiliary photons

Input: Output:

€

ψin =ξ(0) 0 +...+ξ(N) N

€



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

€

t∈C

€

ω = xk kk=0

n

∑ ⊗ ωk

€

η

State vector of auxiliary modes

“preparation”

€

2

€

3

€

3

€

3

€

3

€

1

Input: Output:

“measurement”

€

ψin =ξ(0) 0 +...+ξ(N) N

€



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

€

η =U 1⊗n

€

3


“preparation”

“measurement”

€

p1/2 ψout = ξ( j )xk η (V12(t)⊗13) j ⊗ kk=0

n

∑ ⊗ ωkj=0

N

∑

Input: Output:

€

ψin =ξ(0) 0 +...+ξ(N) N

€



€

η =U 1⊗n

€

t∈C

€

ω = xk kk=0

n

∑ ⊗ ωk

€

2

€

3

€

3

€

1

€


n

∑ ⊗ ωkj=0

N

∑

€

η


“preparation”

“measurement”

Single beam splitter, characterized by complex transmittivity

€

V12(t) =t ˆ n 1e−r* ˆ a 2+ˆ a 1er ˆ a 1+ˆ a 2t−ˆ n 2

€

t∈C

€

ψin =ξ(0) 0 +...+ξ(N) N

€



€

η =U 1⊗n

€

t∈C

€

ω = xk kk=0

n

∑ ⊗ ωk

€

2

€

3

€

3

€

1

Arbitrarily many ( ) states of arbitrary or infinite dimension

€

n

€

η


“preparation”

“measurement”

€


n

∑ ⊗ ωkj=0

N

∑

€

ψin =ξ(0) 0 +...+ξ(N) N

€



€

η =U 1⊗n

€

t∈C

€

ω = xk kk=0

n

∑ ⊗ ωk

€

2

€

3

€

3

€

1

€

ψout =ξ0eiϕ0 0 +...+ξNeiϕN N

Arbitrarily many ( ) states of arbitrary or infinite dimension

Non-convex function (exhibiting many local minima)

€

n

€

η


“preparation”

“measurement”

€


n

∑ ⊗ ωkj=0

N

∑

Weights

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 manylocal maxima

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









Somehow, it would be good to arrive from the “other side”










This is what we will be trying to do…

Convex optimization? Can it help?

Convex optimization problems

Find the minimum of a convex function over a convex set

What is a convex optimization problem again?




Function

Set


Function

Set



Semidefinite programs

Function

Set

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

Semidefinite programs

Set

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

€

cT xMinimize the linear multivariate function

subject to the constraint

€

F0 + xiFi ≥0i=1

N∑

Linear function

Matrices

Vector

We will see in a second why they are so helpful

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

€

εk = η k ⊗ ωk ,

€

p1/2 ψout = ξ( j )xkεk fk( j )

k=0

n

∑j=0

N

∑ j

€

xkεk fk( j )

k=0

n

∑ =p1/2eiϕ j

€

fk( j ) = j kV1,2(t) j k

Again, the output of the quantum network, depending on preparations and measurements, can be written as

for all

€

j =0,...,N

Here

J Eisert, quant-ph/0409156

Functioning of the gate requires that


€


€

xkεk fk( j )

k=0

n

∑ =p1/2eiϕ j

€

fk( j ) = j kV1,2(t) j k

for all

€

j =0,...,N

Here



After all, the

(i) success probability should be maximized, (ii) provided that the gate works


€


€

xkεk fk( j )

k=0

n

∑ =p1/2eiϕ j

€

fk( j ) = j kV1,2(t) j k

for all

€

j =0,...,N

Here



But then, the problem is a non-convex infinite dimensional problem, involving polynomials of arbitrary order in the transmittivity

The strategy is now the following…

(one which can be applied to a number of contexts)

€

t

The strategy

Maximize over all- transmittivities- complex scalar products

- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1

1.

The strategy


- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1

- transmittivities- complex scalar products

€

|t |<1

€

ε1,...,εn+1

Maximize over all- weights

€

x1,...,xn+1, xT x =1

Isolate very difficult part of the problem

1.

2.

The strategy


- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1


€

|t |<1

€

ε1,...,εn+1


€

x1,...,xn+1, xT x ≤1


Relax to makeconvex

1.

2.

3. Writing it as a semidefinite problem

Then for each the problem is found to be one with matrix constraints the elements of which are polynomials of arbitrary degree in

The resulting problem may look strange, but it is actually a semidefinite program

Why does this help us?

€

(t, εk{ })

€

t

4. Lagrange duality

Primal problem Dual problem

That is, for each problem, one can construct a so-called “dual problem”

Because we can exploit the (very helpful) idea of Lagrange duality

Lagrange duality

Both are semidefinite problems

4. Lagrange duality

Globally optimal point

Original (primal) problem

4. Lagrange duality



Dual problem

4. Lagrange duality



Dual problem

Every solution (!) of the dual problem (any educated guess) is a bound for the optimal solution of the primal problem

Educated guess

“Approaching the problem from the other side”

4. Lagrange duality



Dual problem



Educated guess

4. Lagrange duality



Dual problem



Educated guess

The strategy


- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1


€

|t |<1

€

ε1,...,εn+1


€

x1,...,xn+1, xT x ≤1



€

|t |<1

€

ε1,...,εn+1

Minimize dual over all - matrices

€

M ≥0

Make use of idea of Lagrange duality “approaching from the other side”

“ Semidefinite program in - matrix

€

Z

1.

2.

3.

4.

The dual can be shown to be of the form

as optimization problem (still infinite dimensional) in vectors and

5. Construction of a solution for the dual

€

(1,...,1)T z

€

F0 +

0

z1

z2

...

zn+2

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

+ vaFa +a=1

2N+2

∑

0

0

W

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

+F2N+2 ≥0

minimize

subject to

€

z

€

v

The entries of the matrices

are polynomials of degree in the transmittivity

€

F j, j =1,...,2N +1

€

n

€

t

The dual can be shown to be of the form

5. Construction of a solution for the dual

€

(1,...,1)T z

€

F0 +

0

z1

z2

...

zn+2

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

+ vaFa +a=1

2N+2

∑

0

0

W

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

+F2N+2 ≥0

minimize

subject to

Finding a solution now means “guessing” a matrix

Can be done, even in a way such that the unwanted dependence on preparation and measurement can be eliminated

Instance of a problem that can explicitly solved

€

W

€

εk{ }

The strategy


- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1


€

|t |<1

€

ε1,...,εn+1


€

x1,...,xn+1, xT x ≤1



€

|t |<1

€

ε1,...,εn+1


€

M ≥0



€

Z

1.

2.

3.

4.

The strategy


- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1


€

|t |<1

€

ε1,...,εn+1


€

x1,...,xn+1, xT x ≤1



€

|t |<1

€

ε1,...,εn+1


€

M ≥0


Construct explicit solution, independent from

€

|t |<1,

€

ε1,...,εn+1


€

Z

1.

2.

3.

4.

5.

The strategy


- weights

€

|t |<1

€

ε1,...,εn+1

€

x1,...,xn+1, xT x =1





€

|t |<1

€

ε1,...,εn+1



Construct explicit solution, independent from

This gives a general boundfor the original problem(optimal success probability)


1.

2.

3.

4.

5.

For the non-linear sign shift, e.g., one can construct a solution for the dual problem for each

This solution delivers in each case

6. Done!

€

t, t ≤1

Educated guess

Optimal point

0.25

0

0.5

0.75

1.0



using the argument of Lagrange duality, we are done!

6. Done!

€

pmax=1/4

€

t, t ≤1

Educated guess

Optimal point

0.25

0

0.5

0.75

1.0

6. Done!

€

pmax=1/4

€

t, t ≤1

Educated guess

Optimal point

0.25

0

0.5



using the argument of Lagrange duality, we are done!

So, this gives a bound for the original problem … … and one of which we know it is optimal

€

pmax=1/4

€

pmax=1/4

Realizing a non-linear sign shift gate

Auxiliary modes

Photoncounters

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2



Auxiliary modes

Photoncounters

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2

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



Auxiliary modes

Photoncounters

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1)1 −ξ(2) 2

Surprisingly: any additional resources in terms of modes/photons than two auxiliary modes/photons do not lift up the success probability at all


The same method can be immediately applied to other quantum gates, e.g., to the sign-shift with phase

Success probabilities of other sign gates

€

ϕ

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2 ⏐ → ⏐ ξ(0) 0 +ξ(1) 1 +eiϕξ(2) 2



€

ϕ0

0.25

0.5

0.75

1

0

€

π

€

pmax

€

ϕ

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2 ⏐ → ⏐ ξ(0) 0 +ξ(1) 1 +eiϕξ(2) 2



€

ϕ0

0.25

0.5

0.75

1

0

€

π

€

pmax

€

ϕ

“do nothing” Non-linear signshift gate

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2 ⏐ → ⏐ ξ(0) 0 +ξ(1) 1 +eiϕξ(2) 2

The same method can be immediately applied to other quantum gates, such as those involving higher photon numbers

Higher photon numbers

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2 +ξ(3) 3 ⏐ → ⏐ ξ(0) 0 +ξ(1) 1 +ξ(2) 2 +eiϕξ(3) 3

€

ϕ0

0.25

0.5

0.75

1

0

€

π

€

pmax“do nothing”

Non-linear sign shift with one step of feed-forward

Input: Output:

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

€

ξ(0) 0 +ξ(1) 1 −ξ(2) 2

Assessing success probabilities with single rounds of classical feedback

Work in progress with WJ Munro, P Kok, K Nemoto, S Scheel

incorrectablefailures

Non-linear sign shift with one step of feed-forward

Input: Output:

€

ξ(0) 0 +ξ(1)1 +ξ(2) 2

Assessing success probabilities with single rounds of classical feedback

Work in progress with WJ Munro, P Kok, K Nemoto, S Scheel

success

failure

success

€

ξ(0) 0 +ξ(1) 1 −ξ(2) 2

Potentiallycorrectablefailures

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 …

… then any classical feedforward follows by a correction network can increase the success probability to at most

That is, single rounds of feed-forward at the level of individual gates do not help very much at all!

€

pmax=1/4

€

pmax ≈0.3

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

€

ρ

Scenario 1: (experimentally) detecting entanglement directly

Unknown state“Yes, it is entangled!”

“Hm, I don’t know”

€

tr[ρW]<1

€

W

€

tr[ρW]≥1

“Yes, it is entangled!”

“Hm, I don’t know”

Entanglement witness

An entanglement witness is an observable with

€

W =W+

Experiment:

M Barbieri et al, Phys Rev Lett 91 (2003)M Bourennane et al, Phys Rev Lett 92 (2004)

Theory:

M & P & R Horodecki, Phys Lett A 232 (1996)BM Terhal, Phys Lett A 271 (2000)G Toth, quant-ph/0406061


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/0005042M Curty, M Lewenstein, N Luetkenhaus, Phys Rev Lett 92 (2004)


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


Starting from any entanglement witness, one always gets an optimal one if one knows

Then 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

€

ε=min a,bW a,b =tr[ a,b a,bW]

€

a,bsubject to being a product state vector

€

W −ε1

The strategy

Minimize

over all state vectors

€

a,b M a,b =tr[ a,b a,bM]

1.

€

a,b

The strategy

Minimize


€

a,b M a,b =tr[ a,b a,bM]Equivalently

1.

€

a,b

Minimize

over all with

€

ρ

2.

€

tr[ρM]

€

tr[ρA2 ]=tr[ρA

3]=1

€

tr[ρB2 ]=tr[ρB

3]=1

The strategy

Minimize

over all state vectors Equivalently

1.

Minimize

over all with

2.

Any operator that satisfies

is necessarily a pure state (polynomial characterization of pure states)

€

ρ

€

tr[ρ2]=tr[ρ3]=1

NS Jones, N Linden, quant-ph/0407117J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135

The strategy

Minimize


€


1.

€

a,b

Minimize

over all with

€

ρ

2.

€

tr[ρM]

€

tr[ρA2 ]=tr[ρA

3]=1

Write this as a hierarchy of semidefinite programs

3.

Use methods from relaxation theory of polynomially constrained problems

€

tr[ρB2 ]=tr[ρB

3]=1

The strategy


3.

Minimize

over all state vectors Equivalently

1.

€

a,b

Minimize

over all with

€

ρ

2.

€

tr[ρA2 ]=tr[ρA

3]=1

€

tr[ρB2 ]=tr[ρB

3]=1

Any polynomially constrained problem

typically computationally hard NP problems

can be relaxed to efficiently solvable 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

Minimize


€


1.

€

a,b

Minimize

over all with

€

ρ

2.

€

tr[ρM]

€

tr[ρA2 ]=tr[ρA

3]=1


3.

Since each step gives a lowerbound, each step gives anentanglement witness

4.

Use methods from relaxation theory of polynomially constrained problems

€

tr[ρB2 ]=tr[ρB

3]=1

Entanglement witnesses for finite detection efficiencies

In this way, one can obtain good entanglement witnesses for imperfect detectors

€

ε=mintr[ a,b a,b Aii∑ WAi

+]

€

a,bsubject to being a product state vector

Starting from a witness in an error-free setting, one gets a new optimal witness as

€

Aii∑ WAi

+−ε1

J Eisert, M Curty, N Luetkenhaus, work in progress

Hierarchies of criteria for multi-particle entanglement

€

ρ

(e.g., from state tomography)

1st sufficientcriterion forentanglement


“I don’t know”

Known state

J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156

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


€

ρ


1st sufficient criterion forentanglement



Known state2nd sufficient criterion forentanglement






€

ρ


1st sufficient criterion forentanglement



Known state2nd sufficient criterion forentanglement



Every entangled state is necessarily detected in some step of the hierarchy



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

Documents

J. Eisert University of Potsdam, Germany Entanglement and transfer of quantum information Cambridge, September 2004 Optimizing linear optics quantum gates