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Entanglement-enhanced communication over a correlated-noise channel

Squeezing eigenmodes in parametric down-conversion. Entanglement-enhanced communication over a correlated-noise channel. Andrzej Dragan Wojciech Wasilewski Czes ław Radzewicz Warsaw University Jonathan Ball University of Oxford. Konrad Banaszek

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Entanglement-enhanced communication over a correlated-noise channel

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  1. Squeezing eigenmodesin parametric down-conversion Entanglement-enhanced communication overa correlated-noise channel Andrzej Dragan Wojciech Wasilewski Czesław Radzewicz Warsaw University Jonathan BallUniversity of Oxford Konrad Banaszek Nicolaus Copernicus University Toruń, Poland Alex Lvovsky University of Calgary National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland

  2. Receiver Sender Mutual information: Channel capacity: All that jazz

  3. Independently of the averaged output state has the form: 1/2 H Capacity of coding in the polarization state of a single photon: 1/2 1/2 V 1/2 Depolarization in an optical fibre Photon in a polarization state Random polarizationtransformation V H

  4. Sender: Probabilities of measurement outcomes: 2/3 H&H, V&V 1/3 Capacity per photon pair: 1/3 H&V, V&H 2/3 Information coding V H V H

  5. Probabilities of measurement outcomes: 1 2&0, 0&2 Capacity: 1/2 1&1 1/2 Collective detection

  6. Probabilities of measurement outcomes: 1 2&0, 0&2 Capacity: 1 1&1 Entangled states are useful!

  7. Separable ensemble: 1 2&0, 0&2 1/2 1&1 1/2 Entangled ensemble: 1 2&0, 0&2 1 1&1 Proof-of-principle experiment 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)

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

  9. K. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Phys. Rev. Lett. 92, 257901 (2004) Experimental setup Triplet events: D1 & D2 D3 & D4 Singlet events: D1 & D3 D2 & D3 D2 & D3 D2 & D4

  10. Experimental results

  11. Dealing with collective depolarization • 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). • 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).

  12. 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?

  13. General transformation: Mathematical model 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 W in a single time bin We will decompose with and

  14. Ordering Fock states in a single time bin according to the combined number of photons l: Schwinger representation ... Representation of W: ... Here is (2j+1)-dimensional representation of SU(2). Consequently has the explicit decomposition in the form:

  15. 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) Decomposition into irreducible representations: Decomposition Integration over a removes coherence between subspaces with different total photon number L. Also, no coherence is left between subspaces with different j. Biggest decoherence-free subsystems have usually hybrid character!

  16. 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?

  17. Brutal reality (still simplified): [See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band,Opt. Comm. 221, 337 (2003)] Multimode squeezing SHG Single-mode model: PDC –

  18. The wave function up to the two-photon term: Perturbative regime 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) 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

  19. 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) 

  20. The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields: Squeezing modes 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

  21. The End

  22. Theory Everything that emerges are Werner states One-dimensional optimization problem for the Holevo bound What about phase encoding?

  23. If we subtract one time bin: N bins, L photons ... L–L′photons ... N-1 bins, L′ photons Recursion formula Decompostion of the corresponding su(2) algebra:

  24. Therefore the algebra for L photons in N time bins can be written as a triple direct sum: The product of two angular momentum algebras has the standard decomposition as: Direct sum

  25. 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! Decoherence-free subsystems Rearranging the summation order finally yields:

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