Quantum information Theory: Separability and distillability

1. Quantum information Theory: Separability and distillability J. Ignacio Cirac Institute for Theoretical Physics University of Innsbruck KIAS, November 2001 SFB Coherent Control€U TMR

2. Entangled states Superposition principle in Quantum Mechanics: If the systems can be in or then they can also be in Two or more systems: entangled states If the systems can be in A B or then they can also be in Entangled states possess non-local (quantum) correlations: The outcomes of measurements in A and B are correlated. A B In order to explain these correlations classically (with a realistic theory), we must have non-locality. Fundamental implications: Bell´s theorem.

3. Applications Secret communication. Alice Bob 1. Check that particles are indeed entangled. 2. Measure in A and B (z direction): Alice Bob Correlations in all directions. 0 1 1 1 0 0 1 1 1 0 No eavesdropper present Send secret messages Given an entangled pair, secure secret communication is possible

4. Computation. ouput quantumprocessor input A quantum computer can perform ceratin tasks more efficiently A quantum computer can do the same as a classical computer ... and more - Factorization (Shor). - Database search (Grover). - Quantum simulations.

5. Precission measurements: We can measure more precisely Efficient communication: Bob Alice Bob Alice We can use less resources + Entangled state

6. Problem: Decoherence environment A B The systems get entangled with the environment. Reduced density operator:

7. Solution: Entanglement distillation Idea: ... local operation local operation environment (classical communication) Distillation: ...

8. Fundamental problems in Quantum Infomation: Separability and distillability SEPARABILITY DISTILLABILITY A B ... Can we distill these systems? Are these systems entangled?

9. Additional motivations: Experiments Separability: Ion traps Cavity QED Optical lattices Magnetic traps NMR Quantum dots Josephson junctions Atomic ensembles Distillability: quantum communication. Long distance Q. communication?

10. Basic properties: This talk Separability Th. Physics Mathematics Distillability Quantum Information Algorithms, etc: Physical implementations: Computer Science Th. Physics Exp. Physics Q. Optics Condensed Matter NMR

11. Outline Separability. Distillability. Gaussian states. Separability. Distillability. Multipartite case:

12. 1. Separability Are these systems entangled? 1.1 Pure states Product states are those that can be written as: Otherwise, they are entangled. Examples: Product state: Entangled state: Entangled states cannot be created by local operations.

13. 1.2 Mixed states In order to create an entangled state, one needs interactions. Separable states are those that can be prepared by LOCC out of a product state. Otherwise, they are entangled. where A state is separable iff (Werner 89)

14. Problem: given , there are infinitely many decompositions spectral decomposition need not be orthogonal Example: two qubits ( ) 00 01 10 11 where

15. 1.3 Separability: positive maps A linear map is called positive A B state state Extensions A A B B state : need not be positive, in general ? A postive map is completely positive if: is separable iff for all positive maps (Horodecki 96) However, we do not know how to construct all positive maps.

16. Example: Any physical action. A A B B state state Any physical action can be described in terms of a completely positive map.

17. Example: transposition (in a given basis) Is positive A A Extension: partial transposition. B B Is called partial transposition , then Example: transposes the blocks Partial transposition is positive, but not completely positive.

18. What is known? PPT NPT In general - Low rank SEPARABLE ENTANGLED - Necessary or sufficient conditions ? (Horodecki 97) 2x2 and 2x3 Gaussian states (Horodecki and Peres 96) (Giedke, Kraus, Lewenstein, Cirac, 2001) PPT NPT PPT NPT ENTANGLED ENTANGLED SEPARABLE SEPARABLE

19. 2. Distillability ... Can we distill MES using LOCC? PPT states cannot be distilled. Thus, there are bound entangled states. (Horodecki 97) There seems to be NPT states that cannot be distilled. (DiVincezo et al, Dur et al, 2000)

20. 2.1 NPT states (IBM, Innsbruck 99) We just have to concentrate on states with non-positive partial transposition. Idea: If then there exists A and B, such that with Physically, this means that A B random the same random and still has non-positive partial transposition. Thus, we can concentrate on states of the form: where

21. We consider the (unnormalized) family of states: Qubits: x 3 distillable one can easily find A, B such that Higher dimensions: x 2 3 ? distillable NPT Idea: find A, B such that they project onto with there is a strong evidence that they are not distillable: for any finite N, all projections onto have

22. What is known? PPT NPT In general DISTILLABLE Non-DISTILLABLE ? 2xN Gaussian states (Horodecki 97, Dur et al 2000) (Giedke, Duan, Zoller, Cirac, 2001) PPT NPT PPT NPT DISTILLABLE DISTILLABLE Non-DISTILLABLE Non-DISTILLABLE

23. 3. Gaussian states Light source: squeezed states: (2-mode approximation) Gaussian state: Decoherence: photon absorption, phase shifts where is at most quadratic in Internal levels can be approximated by continuous variables in Gaussian states Atomic ensembles:

24. Optical elements: Measurements: X, P - Homodyne detection: • Beam splitters: • Lambda plates: • Polarizers, etc. Gaussian Gaussian local oscillator We consider: B A m modes n modes Gaussian Is separable and/or distillable?

25. 3.1 What is known? 1 mode + 1 mode: is separable iff (Duan, Giedke, Cirac and Zoller, 2000; Simon 2000) 2 modes + 2 modes: There exist PPT entangled states. (Werner and Wolf 2000)

26. µ ¶ A C ° = T C B 3.2 Separability CORRELATION MATRIX All the information about is contained in: the „correlation matrix“. where 2nX2n 2mX2m For valid density operators: is the „symplectic matrix“ where and

27. Given a CM, : does it correspond to a separable state (separable)? (G. Giedke, B. Kraus, M. Lewenstein, and Cirac, 2001) Idea: define a map ... Facts: is a CM of a separable state iff is too. is no CM or If is a CM of an entangled state, then either is a CM of an entangled state If is separable, then . This last corresponds to (for which one can readily see that is separable)

28. density operators Gaussian separable CONNECTION WITH POSITIVE MAPS? Map for CM‘s: Map for density operators: Non-linear

29. 3.3 Distillability (Giedke, Duan, Zoller, and Cirac, 2001) Idea: take such that Two modes: N=M=1: Symmetric states: Non-symmetric states: distillable state. symmetric state. General case: N,M two modes is distillable if and only if There are no NPT Gaussian states.

30. 4. Multipartite case. C A Are these systems entangled? B Fully separable states are those that can be prepared by LOCC out of a product state. We can also consider partitions: Separable A-(BC) Separable B-(AC) Separable C-(AB) C C C A A A B B B

31. 4.1 Bound entangled states. Consider C C A A B B but such that it is not separable C-(AB). QUESTIONS: Is B entangled with A or C? Is A entangled with B or C? Is C entangled with A or B? Consequence: Nothing can be distilled out of it. It is a bound entangled state.

32. 4.2 Activation of BES. (Dür and Cirac, 1999) Consider C C A A B B but A and B can act jointly C A B singlets Then they may be able to distill GHZ states.

33. ACTIVATION OF BOUND ENTANGLED STATES Distillable iff two groups 3 and 5 particles Distillable iff two groups 35-45% and 65-55% Distillable iff two groups have more than 2 particles. If two particles remain separated not distillable. Superactivation Two parties can distill iff the other join (Shor and Smolin, 2000) C A B Two copies

34. 4.3 Family of states Define: where There are parameters. Any state can be depolarized to this form.

35. 5. Conclusions The separability problem is one of the most challanging problems in quantum Information theory. It is relevant from the theoretical and experimental point of view. Gaussian states: Solved the separability and distillability problem for two systems. Solved the separability problem for three (1-mode) systems Maybe we can use the methods developed here to attack the general problem. Multipartite systems: New behavior regarding separability and bound entanglement. Family of states which display new activation properties.

36. Hannover Innsbruck: M. Lewenstein Geza Giedke Barbara Kraus Collaborations: Wolfgang Dür Guifré Vidal R. Tarrach (Barcelona) P. Horodecki (Gdansk) J.I.C. L.M. Duan (Innsbruck) P. Zoller (Innsbruck) SFB Coherent Control€U TMR EQUIP KIAS, November 2001

37. Institute for Theoretical Physics Postdocs: - L.M. Duan (*) - P. Fedichev - D. Jaksch - C. Menotti (*) - B. Paredes - G. Vidal - T. Calarco Ph D: - W. Dur (*) - G. Giedke (*) - B. Kraus - K. Schulze P. Zoller J. I. Cirac FWF SFB F015: „Control and Measurement of Coherent Quantum Systems“ EU networks:„Coherent Matter Waves“, „Quantum Information“ EU (IST): „EQUIP“ Austrian Industry: Institute for Quantum Information Ges.m.b.H. €