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Manipulating Continuous Variable Photonic Entanglement

Imperial College London. Krynica, 15th June 2005. Manipulating Continuous Variable Photonic Entanglement. Martin Plenio Imperial College London Institute for Mathematical Sciences & Department of Physics. Sponsored by:. Royal Society Senior Research Fellowship. QUPRODIS.

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Manipulating Continuous Variable Photonic Entanglement

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  1. Imperial College London Krynica, 15th June 2005 Manipulating Continuous Variable Photonic Entanglement Martin Plenio Imperial College London Institute for Mathematical Sciences&Department of Physics Sponsored by: Royal Society Senior Research Fellowship QUPRODIS

  2. Imperial College London Krynica, 15th June 2005 The vision . . . Prepare and distribute pure-state entanglement Local preparation A B Entangled state between distant sites

  3. Imperial College London Krynica, 15th June 2005 Weakly entangled state A B . . . and the reality Decoherence will degrade entanglement Local preparation Noisy channel Can Alice and Bob ‘repair’ the damaged entanglement? They are restricted to Local Operations and Classical Communication

  4. Imperial College London Krynica, 15th June 2005 The three basic questions of a theory of entanglement Provide efficient methods to • decide which states are entangled and which are disentangled(Characterize)

  5. Imperial College London Krynica, 15th June 2005 The three basic questions of a theory of entanglement Provide efficient methods to • decide which states are entangled and which are disentangled(Characterize) • decide which LOCC entanglement manipulations are possible and provide the protocols to implement them(Manipulate)

  6. Imperial College London Krynica, 15th June 2005 The three basic questions of a theory of entanglement Provide efficient methods to • decide which states are entangled and which are disentangled(Characterize) • 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)

  7. Imperial College London Krynica, 15th June 2005 Practically motivated entanglement theory Theory of entanglement is usually purely abstract All results assume availability of unlimited experimental resources For example: accessibility of all QM allowed operations BUT Doesn’t match experimental reality very well! Develop theory of entanglement under experimentally accessible operations

  8. Imperial College London Krynica, 15th June 2005 Basics of continuous-variable systems • Consider n harmonic oscillators • Canonical coordinates

  9. Imperial College London Krynica, 15th June 2005 Lets go quantum • Harmonic oscillators, light modes or cold atom gases.

  10. Imperial College London Krynica, 15th June 2005 Lets go quantum • Harmonic oscillators, light modes or cold atom gases. • canonical commutation relations whereis a real 2n x 2n matrix is the symplectic matrix

  11. Imperial College London Krynica, 15th June 2005 Characteristic function • Characteristic function(Fourier transform of Wigner function) Simplest example: Vacuum state = Gaussian function

  12. Imperial College London Krynica, 15th June 2005 Arbitrary CV states too general: Restrict to Gaussian states • A state is called Gaussian, if and only if its characteristic function (or its Wigner function) is a Gaussian

  13. Imperial College London Krynica, 15th June 2005 Arbitrary CV states too general: Restrict to Gaussian states • A state is called Gaussian, if and only if its characteristic function (or its Wigner function) is a Gaussian • 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)

  14. Imperial College London Krynica, 15th June 2005 Arbitrary CV states too general: Restrict to Gaussian states • A state is called Gaussian, if and only if its characteristic function (or its Wigner function) is a Gaussian • 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

  15. Imperial College London Krynica, 15th June 2005 First Moments • First moments (local displacements in phase space): Local displacement Local displacement

  16. Imperial College London Krynica, 15th June 2005 Uncertainty Relations • The covariance matrix embodies the second moments • Heisenberg uncertainty principle g represents a physical Gaussian state iff the uncertainty relations are satisfied.

  17. Imperial College London Krynica, 15th June 2005 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).

  18. Imperial College London Krynica, 15th June 2005 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 state Impractical: Experimentally, cannot access all operations

  19. Imperial College London Krynica, 15th June 2005 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 state Impractical: Experimentally, cannot access all operations Programme: Develop theory of what you can and cannot do under Gaussian entanglement under Gaussian operations.

  20. Imperial College London Krynica, 15th June 2005 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 experimentto • 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. A66, 032316 (2002) B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math. Phys.2, 161 (1977)

  21. Imperial College London Krynica, 15th June 2005 Gaussian operations can be implemented ‘easily’! • Gaussian operations: Map any Gaussian state to a Gaussian state • In a quantum optical setting • Application of linear optical elements: • Beam splitters • Phase plates • Squeezers Addition of vacuum modes Measurements: • Homodyne measurements

  22. Imperial College London Krynica, 15th June 2005 Characterization of Gaussian operations Optical elements and additional field modes Homodyne measurement Vacuum detection Transformation: Transformation: Transformation: with where Schur complement of real, symmetric real

  23. Imperial College London Krynica, 15th June 2005 Gaussian manipulation of entanglement • What quantum state transformations can be implemented under Gaussian local operations?

  24. Imperial College London Krynica, 15th June 2005 Gaussian manipulation of entanglement • Apply Gaussian LOCC to the initial state r

  25. Imperial College London Krynica, 15th June 2005 Gaussian manipulation of entanglement • Can one reachr’, ie is there a Gaussian LOCC mapsuch that ?

  26. Imperial College London Krynica, 15th June 2005 Normal form for pure state entanglement Gaussian local unitary A B A B 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)

  27. Imperial College London Krynica, 15th June 2005 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) A B A B G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

  28. Imperial College London Krynica, 15th June 2005 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) A B A B G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

  29. Imperial College London Krynica, 15th June 2005 Comparison General LOCC Gaussian LOCC G. Giedke, J. Eisert, J.I. Cirac and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

  30. Imperial College London Krynica, 15th June 2005 Comparison General LOCC Gaussian LOCC 4 Cannot compress Gaussian pure state entanglement with Gaussian operations ! G. Giedke, J. Eisert, J.I. Cirac and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)

  31. Imperial College London Krynica, 15th June 2005 Gaussian entanglement distillation on mixed states Homodyne measurements General local unitary Gaussian operations (any array of beam splitters, phase shifts and squeezers) A1 B1 A2 B2 Symmetric Gaussian two-mode states r • Characterised by 20 real numbers • When can the degree of entanglement be increased?

  32. Imperial College London Krynica, 15th June 2005 Gaussian entanglement distillation on mixed states • The optimal iterative Gaussian distillation protocol can be identified:

  33. Imperial College London Krynica, 15th June 2005 Gaussian entanglement distillation on mixed states • The optimal iterative Gaussian distillation protocol can be identified: Donothing at all (then at least no entanglement is lost)! J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett.89, 137903 (2002)

  34. Imperial College London Krynica, 15th June 2005 Gaussian entanglement distillation on mixed states • The optimal iterative Gaussian distillation protocol can be identified: Donothing at all (then at least no entanglement is lost)! J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett.89, 137903 (2002) • 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. A66, 032316 (2002)

  35. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel (Gaussian) mixed states

  36. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states

  37. Imperial College London Krynica, 15th June 2005 Procrustean Approach

  38. Imperial College London Krynica, 15th June 2005 Procrustean Approach PD PD Yes/No detector

  39. Imperial College London Krynica, 15th June 2005 Procrustean Approach • 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 .

  40. Imperial College London Krynica, 15th June 2005 Procrustean Approach • Probability of Success depends on q and T: • Example: • Initial supply with q = 0.01 Entanglement Success Probability

  41. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states Iterative Gaussifier (Gaussian operations)

  42. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states Iterative Gaussifier (Gaussian operations)

  43. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states Iterative Gaussifier (Gaussian operations)

  44. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states Iterative Gaussifier (Gaussian operations) (Gaussian) two-mode squeezed states

  45. Imperial College London Krynica, 15th June 2005 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states Iterative Gaussifier (Gaussian operations) (Gaussian) two-mode squeezed states 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)

  46. Imperial College London Krynica, 15th June 2005 Gaussification Yes/No Yes/No A1 B1 50/50 50/50 50/50 A2 B2

  47. Imperial College London Krynica, 15th June 2005 Yes/No Yes/No Yes/No Yes/No Yes/No Yes/No Yes/No Yes/No A1 A1 A1 A1 B1 B1 B1 B1 50/50 50/50 50/50 50/50 50/50 50/50 50/50 50/50 50/50 50/50 50/50 50/50 A2 A2 A2 A2 B2 B2 B2 B2 Procrustean Approach • 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)

  48. Imperial College London Krynica, 15th June 2005 Procrustean Approach Initial Supply ProcrusteanStep Gaussification Final State

  49. Imperial College London Krynica, 15th June 2005 Procrustean Approach • Example:

  50. Imperial College London Krynica, 15th June 2005 Procrustean Approach • Example:

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