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

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

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

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

5. The three basic questions of a theory of entanglement  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) Provide efficient methods to Krynica, 15th June 2005Imperial College London

6. The three basic questions of a theory of entanglement  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) Provide efficient methods to Krynica, 15th June 2005Imperial College London

7. 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 Krynica, 15th June 2005Imperial College London

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

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

10.  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. Krynica, 15th June 2005Imperial College London

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

12.  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 Krynica, 15th June 2005Imperial College London

13.  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) Arbitrary CV states too general: Restrict to Gaussian states Krynica, 15th June 2005Imperial College London

14.  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) Arbitrary CV states too general: Restrict to Gaussian states coherent states squeezed states (one and two modes) thermal states Krynica, 15th June 2005Imperial College London

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

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

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

18. 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 Krynica, 15th June 2005Imperial College London

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

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

21.  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 Krynica, 15th June 2005Imperial College London

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

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

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

25. Gaussian manipulation of entanglement  Can one reach  ’, ie is there a Gaussian LOCC map such that ? Krynica, 15th June 2005Imperial College London

26. Normal form for pure state entanglement ABAB 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) Krynica, 15th June 2005Imperial College London

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

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

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

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

31. A1B1 A2B2 Homodyne measurements General local unitary Gaussian operations (any array of beam splitters, phase shifts and squeezers) Symmetric Gaussian two-mode states   Characterised by 20 real numbers  When can the degree of entanglement be increased? Gaussian entanglement distillation on mixed states Krynica, 15th June 2005Imperial College London

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

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

34. 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)!  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) Krynica, 15th June 2005Imperial College London

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

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

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

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

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

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

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

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

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

45. Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Initial step: non-Gaussian state (Gaussian) mixed states Transmission through noisy channel (Gaussian) two-mode squeezed states Imperial College London Theory: DE Browne, J Eisert, S Scheel, MB Plenio Phys. Rev. A 67, 062320 (2003); J Eisert, DE Browne, S Scheel, MB Plenio, Annals of Physics NY 311, 431 (2004) Iterative Gaussifier (Gaussian operations) Krynica, 15th June 2005

46. Gaussification Imperial College London Krynica, 15th June 2005 A1B1 A2B2 50/50 Yes/No

47. Procrustean Approach Imperial College London Krynica, 15th June 2005 A1B1 A2B2 50/50 Yes/No A1B1 A2B2 50/50 Yes/No A1B1 A2B2 50/50 Yes/No A1B1 A2B2 50/50 Yes/No  Can prove that this converges to a Gaussian state for |  0 | > |  1 |  The Gaussian state to which it converges is the two-mode squeezed 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. Procrustean Approach Imperial College London Krynica, 15th June 2005 Initial Supply Procrustean Step Gaussification Final State

49. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: EntanglementFidelityProbability Initial state0.00150.805

50. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: EntanglementFidelityProbability Initial state0.00150.805 Procrustean (T=0.017) 0.820.9320.0004

51. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: EntanglementFidelityProbability Initial state0.00150.805 Procrustean (T=0.017) 0.820.9320.0004 Gaussification 10.970.9330.75

52. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: EntanglementFidelityProbability Initial state0.00150.805 Procrustean (T=0.017) 0.820.9320.0004 Gaussification 10.970.9330.75 21.110.9670.74

53. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: EntanglementFidelityProbability Initial state0.00150.805 Procrustean (T=0.017) 0.820.9320.0004 Gaussification 10.970.9330.75 21.110.9670.74 31.240.9870.71

54. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: EntanglementFidelityProbability Initial state0.00150.805 Procrustean (T=0.017) 0.820.9320.0004 Gaussification 10.970.9330.75 21.110.9670.74 31.240.9870.71 41.330.9960.69

55. Procrustean Approach Imperial College London Krynica, 15th June 2005 Example: Probability Fidelity w.r.t. Gaussian target state

56. Finite Detector Efficiency Imperial College London EntanglementMixedness h 1-Tr[  2 ] h 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

57. Improving the Procrustean Step Imperial College London Krynica, 15th June 2005 Source T Fibre-loop detector with loss

58. Photon Number Resolving Detectors Imperial College London Krynica, 15th June 2005 APD 50/50 (2 m )L L 2 m+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 input pulse Principle: photons separated into distributed modes input pulse APDs linear network © Walmsley

59. Detector Efficiency Imperial College London Krynica, 15th June 2005 fi

60. Photon Number Resolution Imperial College London Krynica, 15th June 2005 Entanglement Increase 0.05 0.060.070.080.090.100.110.120.130.140.15 Number of loops Conditioned on two photons

61. Summary Imperial College London Krynica, 15th June 2005 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