manipulating continuous variable photonic entanglement martin plenio imperial college london...

Manipulating Continuous Variable Photonic Entanglement

Martin Plenio

Imperial College LondonInstitute for Mathematical Sciences

&Department of Physics

Imperial College London Krynica, 15th June 2005

Sponsored by:Royal Society Senior Research Fellowship

Local preparation

A BEntangled state between distant sites

The vision . . .

Prepare and distribute pure-state entanglement

Krynica, 15th June 2005Imperial College London

. . . and the reality

A BWeakly entangled state

Noisy channel

Local preparation

Decoherence will degrade entanglement

Can Alice and Bob ‘repair’ the damaged entanglement?


They are restricted to Local Operations and Classical Communication

The three basic questions of a theory of entanglement

decide which states are entangled and which are disentangled (Characterize)

Provide efficient methods to




decide which LOCC entanglement manipulations are possible and provide the protocols to implement them (Manipulate)





decide which LOCC entanglement manipulations are possible and provide the protocols to implement them (Manipulate)

decide how much entanglement is in a state and how efficient entanglement manipulations can be (Quantify)



Practically motivated entanglement theory

Theory of entanglement is usually purely abstract

For example: accessibility of all QM allowed operations

Doesn’t match experimental reality very well!

All results assume availability of unlimited experimental resources

Develop theory of entanglement under experimentally accessible operations

BUT


Consider n harmonic oscillators

nn PXPXPX , , , 2211

Canonical coordinates ),,...,,(),,...,,( 1121221 nnnn PXPX OOOO

Basics of continuous-variable systems


Lets go quantum

Harmonic oscillators, light modes or cold atom gases.


canonical commutation relations

where is a real 2n x 2n matrix is the symplectic matrix

Lets go quantum

Harmonic oscillators, light modes or cold atom gases.


Characteristic function (Fourier transform of Wigner function)

Characteristic function

Simplest example: Vacuum state = Gaussian function


A state is called Gaussian, if and only if its characteristic function (or its Wigner function) is a Gaussian

Arbitrary CV states too general: Restrict to Gaussian states



Gaussian states are completely determined by their first and second moments

Are the states that can be made experimentally with current technology (see in a moment)




Gaussian states are completely determined by their first and second moments

Are the states that can be made experimentally with current technology (see in a moment)


coherent states

squeezed states(one and two modes)

thermal states


First moments (local displacements in phase space):

First Moments


Local displacement Local displacement

The covariance matrix embodies the second moments

Heisenberg uncertainty principle

Uncertainty Relations


represents a physical Gaussian state iff the uncertainty relations are satisfied.

CV entanglement of Gaussian states

Separability + Distillability Necessary and sufficient criterion known for M x N systems Simon, PRL 84, 2726 (2000); Duan, Giedke, Cirac Zoller, PRL 84, 2722 (2000); Werner and Wolf, PRL 86, 3658 (2001); G. Giedke, Fortschr. Phys. 49, 973 (2001)

These statements concern Gaussian states, but assume the availability of all possible operations (even very hard ones).





Inconsistent:With general operations one can make any stateImpractical: Experimentally, cannot access all operations





Develop theory of what you can and cannot do under Gaussian entanglement under Gaussian operations.

Programme:

Inconsistent:With general operations one can make any stateImpractical: Experimentally, cannot access all operations


Characterization of Gaussian operations

For all general Gaussian operations, a ‘dictionary’would be helpful that links the

physical manipulation that can be done in an experiment to

the mathematical transformation law

J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002)J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89, 097901 (2002)J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89, 137902 (2002)G. Giedke and J.I. Cirac, Phys. Rev. A 66, 032316 (2002)B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math. Phys. 2, 161 (1977)


In a quantum optical setting

Application of linear optical elements: Beam splitters Phase plates Squeezers

Gaussian operations can be implemented ‘easily’!

Measurements: Homodyne measurements

Addition of vacuum modes

Gaussian operations: Map any Gaussian state to a Gaussian state


Characterization of Gaussian operations

Optical elements and additional field modes

Vacuum detection Homodyne measurement

C1 C3

C3T C2

AAT G

C1 C3(C2 1) 1C3T TCCC 3

121 )(

)0,1,...,0,1(diag

G i iAT A 0

Transformation: Transformation: Transformation:

with where

C1 C3

C3T C2 1

Schur complement of

G

Areal, symmetricreal


Gaussian manipulation of entanglement

What quantum state transformations can be implemented under Gaussian local operations?



Apply Gaussian LOCC to the initial state



Can one reach ’, ie is there a Gaussian LOCC map such that

?

'

E () '

E


Normal form for pure state entanglement

A B A B

r1

r2

rN

Gaussian local

unitary

G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)A. Botero and B. Reznik, Phys. Rev. A 67, 052311 (2003)


The general theorem

Necessary and sufficient condition for the transformation of pure Gaussian states under Gaussian local operations (GLOCC):

under GLOCC

if and only if (componentwise)

r r '

G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

A B

r1

r2

rN

A B

1'r


2'r

Nr '

The general theorem

Necessary and sufficient condition for the transformation of pure Gaussian states under Gaussian local operations (GLOCC):

under GLOCC

if and only if (componentwise)

r r '

G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

A B

r1

r2

rN

A B

11 'rr


22 'rr

NN rr '

Comparison


General LOCC

r1

r2

01 r

2'

2 rr

Gaussian LOCC

r1

r2

01 r

2'

2 rr

G. Giedke, J. Eisert, J.I. Cirac and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

Comparison


General LOCC

r1

r2

01 r

2'

2 rr

Gaussian LOCC

r1

r2

01 r

2'

2 rr

G. Giedke, J. Eisert, J.I. Cirac and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

Cannot compress Gaussian pure state entanglement with Gaussian operations !

A1 B1

A2 B2

Homodyne measurements

General local unitary Gaussianoperations (any array of beam splitters, phase shifts and squeezers)

SymmetricGaussian two-modestates

Characterised by 20 real numbers When can the degree of entanglement be increased?

Gaussian entanglement distillation on mixed states



The optimal iterative Gaussian distillation protocol can be identified:




Do nothing at all (then at least no entanglement is lost)!

J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002)




Do nothing at all (then at least no entanglement is lost)!

Subsequently it was shown that even for the most general scheme with N-copy Gaussian inputs the best is to do nothing

Challenge for the preparation of entangled Gaussian states over large distances as there are no quantum repeaters based on Gaussian operations (cryptography).

G. Giedke and J.I. Cirac, Phys. Rev. A 66, 032316 (2002)

J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002)


Distillation by leaving the Gaussian regime once

(Gaussian) two-mode squeezed states

(Gaussian) mixed states

Transmission through noisy channel




Initial step: non-Gaussian state




Procrustean Approach




PD

PD

Yes/No detector



• Simple protocol to generate non-Gaussian states of higher entanglement from a weakly squeezed 2-mode squeezed state.

If both detector click – keep the state.

If |q|¿1 the remaining state has essentially the form:

Choose transmittivity T of the beam splitter to get desired .



• Probability of Success depends on q and T:• Example:

– Initial supply with q = 0.01

Entanglement Success Probability




Iterative Gaussifier (Gaussian operations)










Imperial College London

Theory: DE Browne, J Eisert, S Scheel, MB PlenioPhys. Rev. A 67, 062320 (2003);J Eisert, DE Browne, S Scheel, MB Plenio, Annalsof Physics NY 311, 431 (2004)


Krynica, 15th June 2005

Gaussification


A1 B1

A2 B2

50/5050/50 50/50

Yes/No Yes/No



A1 B1

A2 B2

50/5050/50 50/50

Yes/No Yes/No

A1 B1

A2 B2

50/5050/50 50/50

Yes/No Yes/No

A1 B1

A2 B2

50/5050/50 50/50

Yes/No Yes/No

A1 B1

A2 B2

50/5050/50 50/50

Yes/No Yes/No

Can prove that this converges to a Gaussian state for |0| > |1|

The Gaussian state to which it converges is the two-modesqueezed state with q= 1/0.

For rigorous proof see Browne, Eisert, Scheel, Plenio Phys. Rev. A 67, 062320 (2003);Eisert, Browne, Scheel, Plenio, Annals of Physics NY 311, 431 (2004)



Initial Supply

Procrustean Step

Gaussification

Final State



• Example:

Entanglement Fidelity Probability

Initial state 0.0015 0.805



• Example:



Procrustean (T=0.017)

0.82 0.932 0.0004



• Example:




0.82 0.932 0.0004

Gaussification 1 0.97 0.933 0.75



• Example:




0.82 0.932 0.0004


2 1.11 0.967 0.74



• Example:




0.82 0.932 0.0004


2 1.11 0.967 0.74

3 1.24 0.987 0.71



• Example:




0.82 0.932 0.0004


2 1.11 0.967 0.74

3 1.24 0.987 0.71

4 1.33 0.996 0.69



• Example:

Probability Fidelity w.r.t. Gaussian target state

Finite Detector Efficiency

Imperial College London

Entanglement Mixedness

1-Tr[2]

log. neg.

1

2

NG 1

2

Input: Weakly entangled two-mode squeezed state (logneg <0.1) Non-Gaussian step Two Gaussification steps Plot resulting entanglement and mixedness versus detector efficiency

Krynica, 15th June 2005

Improving the Procrustean Step


Source

T

Fibre-loop detector with loss

Photon Number Resolving Detectors


APD

50/50

(2m)LL

2m+1 Light pulses

D. Achilles, Ch. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, Opt. Lett. 28, 2387 (2003).

Fiber based experimental implementation

realization of time-multiplexing with passive linear elements & two APDs

inputpulse

Principle: photons separated into distributed modes

ˆ U •••

inputpulse

APDs

linear network

•••

© W

alm

s ley

Detector Efficiency


fi

Photon Number Resolution


Enta

ngle

ment

Incr

ease

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

Number of loops

Conditioned on two photons

Summary


• Gaussian operations on Gaussian states cannot distill entanglement

• Single non-Gaussian step allows for subsequent distillation by Gaussian operations

• Fibre loop detector based schemes robust against against finite detector efficiencies and low number resolution.

• Robustness suggests experimental feasibility

manipulating continuous variable photonic entanglement martin plenio imperial college london...

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