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

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

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

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

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

environment Problem: Decoherence AB The systems get entangled with the environment. Reduced density operator:

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

Fundamental problems in Quantum Infomation: Separability and distillability AB Are these systems entangled?... SEPARABILITYDISTILLABILITY Can we distill these systems?

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

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

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

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

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

Problem: given, there are infinitely many decompositions spectral decomposition need not be orthogonal Example: two qubits ( ) where

A linear map is called positive A B A B Extensions state ? A B 1.3 Separability: positive maps : 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.

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

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

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

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

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

Qubits: We consider the (unnormalized) family of states: x 3 one can easily find A, B such that Higher dimensions: x 23 distillable ? there is a strong evidence that they are not distillable: for any finite N, all projections onto have Idea: find A, B such that they project onto with NPT distillable

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

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

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

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

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

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

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

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

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

4.1 Bound entangled states. Consider A B C A B C but such that it is not separable C-(AB). 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. QUESTIONS:

4.2 Activation of BES. A B C A B C but A and B can act jointly A B C singlets Consider (Dür and Cirac, 1999) Then they may be able to distill GHZ 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. Two parties can distill iff the other join If two particles remain separated not distillable. Superactivation (Shor and Smolin, 2000) A B C Two copies ACTIVATION OF BOUND ENTANGLED STATES

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

5. Conclusions Maybe we can use the methods developed here to attack the general problem. 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. Multipartite systems: New behavior regarding separability and bound entanglement. Family of states which display new activation properties. Gaussian states: Solved the separability and distillability problem for two systems. Solved the separability problem for three (1-mode) systems

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

Institute for Theoretical Physics € 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. P. Zoller J. I. Cirac 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