Presentation on theme: "Entanglement-enhanced communication over a correlated-noise channel"— Presentation transcript:
1Entanglement-enhanced communication over a correlated-noise channel Squeezing eigenmodes in parametric down-conversionEntanglement-enhanced communication over a correlated-noise channelAndrzej DraganWojciech WasilewskiCzesław RadzewiczWarsaw UniversityJonathan Ball University of OxfordKonrad BanaszekNicolaus Copernicus University Toruń, PolandAlex LvovskyUniversity of CalgaryNational Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland
2All that jazzSenderReceiverMutual information:Channel capacity:
3Depolarization in an optical fibre Independently of the averaged output state has the form:Photon in a polarization stateRandom polarization transformationVHJeżeli każda transformacja unitarna może się zdarzyć z równym prawdopodobieństwem (a matematycy wiedzą co to znaczy, zbiór transformacji tworzy grupę Liego a ta ma kanoniczną miarę/rozkład prawdopodobieństwa), to na wyjściu mamy to co mamy...HV1/2Capacity of coding in the polarization state of a single photon:
4Information coding Sender: V H V Probabilities of measurement outcomes:H&H, V&VH&V, V&H2/31/3HCapacity per photon pair:
6Entangled states are useful! Probabilities of measurement outcomes:2&0, 0&21&11Capacity:
7Proof-of-principle experiment 2&0, 0&21&111/2Separable 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, (2004)2&0, 0&21&11Entangled ensemble:
8Source of polarization-entangled pairs P. 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:Stare (jeden wiek) odwieczne pytanie: powinnismy dodawac prawdopodobieństwa (procesy rozróżnialne) czy tez amplitudy prawdopodobieństwa (procesy nierozróżnialne). Chcielibyśmy jakoś testować spójność tej superpozycji.With a half-wave plate in one arm it can be transformed into:or
9Experimental setup Triplet events: D1 & D2 D3 & D4 Singlet events: K. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Phys. Rev. Lett. 92, (2004)Triplet events:D1 & D D3 & D4Singlet events:D1 & D3 D2 & D3 D2 & D3 D2 & D4
11Dealing 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 subspaces for a train of single photons. J.-C. Boileau, D. Gottesman, R. Laamme, D. Poulin, and R. W. Spekkens, Phys. Rev. Lett. 92, (2004).
12General scenario Physical system: arbitrarily many photons N time bins that encompass two orthogonal polarizationsHow many distinguishable states can we send via the channel?What is the biggest decoherence-free subspace?
13Mathematical model General 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 W in a single time binWe will decompose withand
14Schwinger representation Ordering Fock states in a single time bin according to the combined number of photons l:...Representation of W:...Here is (2j+1)-dimensional representation of SU(2). Consequently has the explicit decomposition in the form:
15Decomposition Decomposition into irreducible representations: Integration over a 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 jdimensionality of the decoherence-free subsystemRecursion formula for :J. L. Ball and K. Banaszek, quant-ph/ ; Open Syst. Inf. Dyn. 12, 121 (2005)Biggest decoherence-free subsystems have usually hybrid character!
16QuestionsRelationship to quantum reference frames for spin systems [S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. Lett. 91, (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, (2003)]Other decoherence mechanisms, e.g. polarization mode dispersion?
17Multimode squeezing – Single-mode model: SHGSingle-mode model:PDCBrutal reality (still simplified):[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band, Opt. Comm. 221, 337 (2003)]–
18Perturbative regime 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)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
19Decomposition 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/ ; see also R. S. Bennink and R. W. Boyd, Phys. Rev. A 66, (2002)
20Squeezing modesThe Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields:which evolve according todescribe modes that are described by pure squeezed statestell us what modes need to be seeded to retain puritySome properties:For low pump powers, usually a large number of modes becomes squeezed with similar squeezing parametersAny superposition of these modes (with right phases!) will exhibit squeezingThe shape of the modes changes with the increasing pump intensity!This and much more in a poster by Wojtek Wasilewski
22Theory Everything that emerges are Werner states One-dimensional optimization problem for the Holevo boundWhat about phase encoding?
23Recursion formula N bins, L photons ... L–L′ photons Decompostion of the corresponding su(2) algebra:If we subtract one time bin:N bins, L photonsN-1 bins, L′ photons...L–L′ photons
24Direct 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:
25Decoherence-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!