Entanglement-enhanced communication overa correlated-noise channel

Andrzej DraganWojciech Wasilewski

Czesław RadzewiczWarsaw University

Jonathan BallUniversity of Oxford

Konrad BanaszekNicolaus Copernicus University Toruń, Poland

Alex LvovskyUniversity of Calgary

Squeezing eigenmodesin parametric down-conversion

National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland

All that jazz

Send

er

Rece

ive

r

Mutual information:

Channel capacity:

Depolarization in an optical fibre

Photon in a polarization state

H

V

H

V

1/2

1/2

1/21/2

2

1

Independently of the averaged output state has the form:

Capacity of coding in the polarization state of a single photon:

Random polarization transformation

Information coding

H

H

V

VSender:

Probabilities of measurement outcomes:

H&H, V&V

H&V, V&H

2/3

2/3

1/31/3

Capacity per photon pair:

Collective detection

Probabilities of measurement outcomes:

2&0, 0&2

1&1

1

1/2

1/2Capacity:

Entangled states are useful!

Probabilities of measurement outcomes:

2&0, 0&2

1&1

1

1Capacity:

Proof-of-principle experiment

2&0, 0&2

1&1

1

1

Entangled ensemble:

2&0, 0&2

1&1

1

1/2

1/2

Separable ensemble:

These are optimal ensembles for separable and entangled inputs (assuming collective detection), which follows from optimizing the Holevo bound.J. Ball, A. Dragan, and K.Banaszek,Phys. Rev. A 69, 042324 (2004)

Source of polarization-entangled pairsP. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999)

For a suitable polarization of the pump pulses, the generated two-photon state has the form:

With a half-wave plate in one arm it can be transformed into:

or

Experimental setupK. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Phys. Rev. Lett. 92, 257901 (2004)

Triplet events: D1 & D2 D3 & D4

Singlet events:D1 & D3 D2 & D3 D2 & D3 D2 & D4

Experimental results

Dealing with collective depolarization

1) Phase encoding in time bins: fixed input polarization, polarization-independent receiver.J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett. 82, 2594 (1999).

2) Decoherence-free subspacesfor a train of single photons.J.-C. Boileau, D. Gottesman, R. Laamme,D. Poulin, and R. W. Spekkens, Phys. Rev. Lett. 92, 017901 (2004).

General scenario

Physical system:• arbitrarily many photons• N time bins that encompass two orthogonal polarizations

• How many distinguishable states can we send via the channel?• What is the biggest decoherence-free subspace?

Mathematical modelGeneral transformation:

where:

– the entire quantum state of light across N time bins– element of U(2) describing the transformation of the polarization modes in a single time bin.– unitary representation of in a single time bin

We will decompose withand

Schwinger representation

...

Ordering Fock states in a single time bin according to the combined number of photons l:

Here is (2j+1)-dimensional representation of SU(2). Consequently has the explicit decomposition in the form:

Representation of

...

DecompositionDecomposition into irreducible representations:

Integration over removes coherence between subspaces with different total photon number L. Also, no coherence is left between subspaces with different j.

tells us: • how many orthogonal states can be sent in the subspace j• dimensionality of the decoherence-free subsystem

Recursion formula for :J. L. Ball and K. Banaszek,quant-ph/0410077;Open Syst. Inf. Dyn. 12, 121 (2005)

Biggest decoherence-free subsystems have usually hybrid character!

Questions

• Relationship to quantum reference frames for spin systems [S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. Lett. 91, 027901 (2003)]

• Partial correlations?• Linear optical implementations?• How much entanglement is needed for implementing decoherence-free subsystems?• Shared phase reference?• Self-referencing schemes? [Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 91, 087901 (2003)]

• Other decoherence mechanisms, e.g. polarization mode dispersion?

Multimode squeezing

–

Single-mode model:

SHG

PDC

Brutal reality (still simplified):

[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band,Opt. Comm. 221, 337 (2003)]

Perturbative regime

Schmidt decomposition for a symmetric two-photon wave function:C. K. Law, I. A. Walmsley, and J. H. Eberly,Phys. Rev. Lett. 84, 5304 (2000)

We can now define eigenmodes which yields:

The spectral amplitudes characterize pure squeezing modes

The wave function up to the two-photon term:

W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997);T. E. Keller and M. H. Rubin, Phys. Rev A 56, 1534 (1997)

Decomposition for arbitrary pump

As the commutation relations for the output field operators must be preserved, the two integral kernels can be decomposed using the Bloch-Messiah theorem:

S. L. Braunstein, quant-ph/9904002;see also R. S. Bennink and R. W. Boyd,Phys. Rev. A 66, 053815 (2002)

Squeezing modesThe Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields:

which evolve according to

• describe modes that are described by pure squeezed states • tell us what modes need to be seeded to retain purity

Some properties:• For low pump powers, usually a large number of modes becomes squeezed with similar squeezing parameters• Any superposition of these modes (with right phases!) will exhibit squeezing• The shape of the modes changes with the increasing pump intensity! This and much more in a poster by Wojtek

Wasilewski

The End

Theory

Everything that emerges are Werner states

One-dimensional optimization problem for the Holevo bound

What about phase encoding?

Recursion formulaDecompostion of the corresponding su(2) algebra:

If we subtract one time bin:

N bins, L photons

N-1 bins, L′ photons

......

L–L′photons

Direct sumThe product of two angular momentum algebras has the standard decomposition as:

Therefore the algebra for L photons in N time bins can be written as a triple direct sum:

Decoherence-free subsystems

Rearranging the summation order finally yields:

Underlined entries with correspond to pure phase encoding (with all the input pulses having identical polarizations)– in most cases we can do better than that!