Entanglement in Quantum Information Science

1. Imperial College London Abingdon, 12th July 2003 Sponsored by: QUPRODIS Royal Society Senior Research Fellowship Entanglement in Quantum Information Science Local Collaborators: D. Browne,J. Hartley,S. Scheel, S. Virmani Non-local UK collaborators: K. Audenaert (Bangor) S.F. Huelga (Hertfordshire) I. Walmsley, C. Silberhorn (Oxford) Non-local spatially separated collaborators 02/03: J. Eisert (Potsdam), J.I. Cirac (München), R.F. Werner (Braunschweig) Martin Plenio Imperial College London

2. Imperial College London Abingdon, 12th July 2003 Areas we are thinking about • Mathematical methods in quantum information science • Identify and develop of useful tools from Matrix Analysis, Optimization Theory Collaborations outside IC : Audenaert (Bangor), Eisert (Potsdam), Werner (Braunschweig)

3. Imperial College London Abingdon, 12th July 2003 Areas we are thinking about • Mathematical methods in quantum information science • Identify and develop of useful tools from Matrix Analysis, Optimization Theory Collaborations outside IC : Audenaert (Bangor), Eisert (Potsdam), Werner (Braunschweig) • Theory of entanglement as a resource • Manipulate, Quantify + Provide abstract protocols • All of the above for infinite dimensional systems  Light modes, Cold gases, condensed matter systems … Collaborations outside IC: Cirac (Munich), Eisert (Potsdam)

4. Imperial College London Abingdon, 12th July 2003 Areas we are thinking about • Mathematical methods in quantum information science • Identify and develop of useful tools from Matrix Analysis, Optimization Theory Collaborations outside IC : Audenaert (Bangor), Eisert (Potsdam), Werner (Braunschweig) • Theory of entanglement as a resource • Manipulate, Quantify + Provide abstract protocols • All of the above for infinite dimensional systems  Light modes, Cold gases, condensed matter systems … Collaborations outside IC: Cirac (Munich), Eisert (Potsdam) • Practical implementation of quantum information processing • System Ion traps, CQED specific: Light modes Condensed matter systems • General: Novel non-standard approaches to QIP eg QIP supported by noise Collaborations outside IC: Eisert (Potsdam), Huelga (Hertfordshire), Walmsley (Oxford)

5. Imperial College London Abingdon, 12th July 2003 Areas we are thinking about • Mathematical methods in quantum information science • Identify and develop of useful tools from Matrix Analysis, Optimization Theory Collaborations outside IC : Audenaert (Bangor), Eisert (Potsdam), Werner (Braunschweig) • Theory of entanglement as a resource • Manipulate, Quantify + Provide abstract protocols • All of the above for infinite dimensional systems  Light modes, Cold gases, condensed matter systems … Collaborations outside IC: Cirac (Munich), Eisert (Potsdam) • Practical implementation of quantum information processing • System Ion traps, CQED specific: Light modes Condensed matter systems • General: Novel non-standard approaches to QIP eg QIP supported by noise Collaborations outside IC: Eisert (Potsdam), Huelga (Hertfordshire), Walmsley (Oxford) • Applications of quantum information science to other areas of physics • Statistical physics, condensed matter systems, QFT, black holes Collaborations outside IC: Eisert (Potsdam), Werner (Braunschwieg)

6. Imperial College London Abingdon, 12th July 2003 The vision . . . Prepare and distribute pure-state entanglement Local preparation A B Entangled state between distant sites

7. Imperial College London Abingdon, 12th July 2003 Weakly entangled state A B . . . and the reality Decoherence will degrade entanglement Local preparation Noisy channel Can Alice and Bob ‘repair’ the damaged entanglement?

8. Imperial College London Abingdon, 12th July 2003 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)

9. Imperial College London Abingdon, 12th July 2003 . . . and what about experiments? 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

10. Imperial College London Abingdon, 12th July 2003 Discrete systems If you can implement a particular single qubit rotations then you can generally do any single bit rotation. Only single qubit operations possible Single qubit rotations + A two-qubit operation No entanglement Everything is possible Its difficult to find an interesting intermediate class that is experimentally well motivated.

11. Imperial College London Abingdon, 12th July 2003 From discrete systems . . . . . . to infinite dimensional, continuous-variable systems

12. Imperial College London Abingdon, 12th July 2003 Quantum Continuous Variable Systems • Harmonic oscillators, light modes or cold atom gases. • canonical variables with commutation relations

13. Imperial College London Abingdon, 12th July 2003 Characteristic function • Characteristic function (Fourier transform of Wigner function)

14. Imperial College London Abingdon, 12th July 2003 General CV states too general: Restrict to Gaussian states • A state is called Gaussian, iff 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 Abingdon, 12th July 2003 CV entanglement of Gaussian states • Separability + Distillability Necessary and sufficient criterion known for M x N systems Simon, PRL 84, 2726 (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 I can make any state Impractical: Experimentally, cannot access all operations Programme: Develop theory of entanglement under Gaussian operations.

16. Imperial College London Abingdon, 12th July 2003 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 and M.B. Plenio, Phys. Rev. Lett.89, 097901 (2002) J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89, 137902 (2002) J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett.89, 137903 (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)

17. Imperial College London Abingdon, 12th July 2003 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 • Photon detection (vacuum outcome) • Applications of Gaussian states and operations: • Unconditional teleporation • Continuous-variable quantum key distribution

18. Imperial College London Abingdon, 12th July 2003 Gaussian manipulation of entanglement • What quantum state transformations can be implemented under natural constraints?

19. Imperial College London Abingdon, 12th July 2003 Gaussian manipulation of entanglement • What quantum state transformations can be implemented under natural constraints?

20. Imperial College London Abingdon, 12th July 2003 Gaussian manipulation of entanglement • Is there a local quantum operation such that ?

21. Imperial College London Abingdon, 12th July 2003 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)

22. Imperial College London Abingdon, 12th July 2003 What can you do without squeezers? Question: Given a Gaussian state of n modes, described by covariance matrix g, is there an array of beamsplitters and phase plates such that it can be turned into an entangled state. Answer: M.M. Wolf, J. Eisert and M.B. Plenio, Phys. Rev. Lett. 90, 047904 (2003) Question: Given a mixed Gaussian state of 2 modes, described by covariance matrix g, when can it be transformed into a state with covariance matrix g’, by Gaussian local operations. Answer: Necessary and sufficient conditions can be given. J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89, 097901 (2002)

23. Imperial College London Abingdon, 12th July 2003 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?

24. Imperial College London Abingdon, 12th July 2003 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) • 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)

25. Imperial College London Abingdon, 12th July 2003 The end of the story? But… … are there feasible Gaussian operations that map non-Gaussian states onto (approximately) • pure • entangled • Gaussian • states in an iterative (all-optical) procedure? D. Browne, J. Eisert, S. Scheel, and M.B. Plenio, Phys. Rev. A 68, …. (2003) or quant-ph/0211173

26. Imperial College London Abingdon, 12th July 2003 Distillation by leaving the Gaussian regime once (Gaussian) two-mode squeezed states Transmission through noisy channel Initial step: non-Gaussian state (Gaussian) mixed states Gaussifier (Gaussian) two-mode squeezed states

27. Imperial College London Abingdon, 12th July 2003 Initial Non-Gaussian step: The Procrustean chop Photon detectors: distinguish vacuum state (‘ no click’)from the rest (‘click’) A1 B1 A2 B2 • Starting from a two-mode squeezed states usingbeam splittersand photon detectors, but keeping the non-vacuum output contribution

28. Imperial College London Abingdon, 12th July 2003 Gaussification: A single step Photon detectors: distinguishing the vacuum state (‘ no click’)from the rest (‘click’) 50:50 beam splitters A1 B1 A2 B2 • The state is kept in case of the vacuum outcome, otherwise discarded • This output state is the input for the next step

29. Imperial College London Abingdon, 12th July 2003 Gaussification: A single step For all the details watch out for the Oxford talks of: Jens Eisert on Tuesday Dan Browne on Thursday

30. Imperial College London Abingdon, 12th July 2003 Summary and Conclusions • Reviewed theory of quantum entanglement both for discrete and continuous systems. • Standard approach unconcerned with practical feasibility • For discrete systems no separation between feasible and infeasible operations that is natural and interesting exists • In CV systems such a separation exists and I presented the development of such a theory Future: Develop entanglement theory of Gaussian CV systems Apply to theoretical problems and support work on a possible experimental demonstration of Gaussifier.