Quantum information Theory: Separability and distillability

Quantum information Theory: Separability and distillability

SFB Coherent Control€U TMR

J. Ignacio CiracInstitute for Theoretical PhysicsUniversity of Innsbruck

KIAS, November 2001

Entangled statesEntangled 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

j0i

j1i

c0j0i + c1j1i

j0i A j0i B

j1i A j1i B

c0j0i A j0i B + c1j1i A j1i B

If the systems can be in

or

then they can also be in

A B

Entangled states possess non-local (quantum) correlations:

A BThe 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):

Alice Bob

01110

01110

No eavesdropper present Send secret messages

jÁi = j0; 1i + j1; 0i

Given an entangled pair, secure secret communication is possible

ApplicationsApplications

Computation.

A quantum computer can perform ceratin tasks more efficiently

A quantum computer can do the same as a classical computer ... and more

quantumprocessor

input

ouput

jª in i

jª o u t i

jª o u t i = U jª i n i

- Factorization (Shor).

- Database search (Grover).

- Quantum simulations.

Precission measurements:

Efficient communication:

AliceBob

0

1010 1

1

AliceBob

+We can use less resources

Entangled state

0

1010 1

1

We can measure more precisely

environment

j©iA B jE iE ! jª iA B E

½A B = trE (jª i hª j) 6= j©ih©j

Problem: DecoherenceProblem: Decoherence

A B

The systems get entangled with the environment.

Reduced density operator:

Solution: Entanglement distillationSolution: Entanglement distillation

environment

......

local operation local operation

(classical communication)

Idea:

Distillation:

½

½

½

½

j0; 1i + j1; 0i

Fundamental problems in Quantum Infomation: Separability and distillability

Fundamental problems in Quantum Infomation: Separability and distillability

A B

Are these systems entangled?

½ ...½

½

½

j©i = j0; 1i + j1; 0i

SEPARABILITY DISTILLABILITY

½

Can we distill these systems?

Additional motivations: ExperimentsAdditional motivations: Experiments

jÁi

Long distance Q. communication?

Ion traps

Atomic ensembles

Cavity QED

NMR Quantum dotsJosephson junctions

Optical lattices

Magnetic traps

Distillability: quantum communication.

Separability:

Quantum Information

Th. PhysicsMathematics

Computer Science Th. PhysicsExp. Physics

Physical implementations:Algorithms, etc:

Basic properties:

Q. OpticsCondensed MatterNMR

Separability

Distillability

This talk

OutlineOutline

Separability.

Distillability.

Gaussian states.

Separability.

Distillability.

Multipartite case:

1. Separability1. Separability

1.1 Pure states1.1 Pure states

Product states are those that can be written as:

Otherwise, they are entangled.

Entangled states cannot be created by local operations.

j©i = jai ­ jbi

jai ­ jbi ! ja(t)i ­ jb(t)i

j0i A j0i B

c0j0i A j0i B + c1j1i A j1i B

Examples:Product state:

Entangled state:

Are these systems entangled?

½

Separable states are those that can be prepared by LOCC out of a product state.

Otherwise, they are entangled.

A state is separable iff ½=X

k

pk jak i hak j ­ jbk i hbk j pk ¸ 0where

(Werner 89)

1.2 Mixed states1.2 Mixed states

½

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 ( )

½

½=X

k

¸ k jª k i hª k j

=X

k

qk j©k i h©k j

= : : :

½=X

k

pk jak i hak j ­ jbk i hbk j

hª k jª j i =±k;j

h©k j©j i 6=±k;j

½=15

(j0; 0i h0; 0j + j1; 1i h1; 1j) +25

j+; +ih+; +j

+110

(j0; ¡ i h0; ¡ j + j1; ¡ i h1; ¡ j)

j§ i =1

p2

(j0i § j1i )

where

½ =120

0

BB@

7 1 2 21 3 2 22 2 3 12 2 1 7

1

CCA

H = C 2 ­ C 2

00 01 10 11

A linear map is called positiveL : A (H ) A (H )½ ¸ 0 ! L (½) ¸ 0

A

B

½

A

½ ½

B

Extensions

state state

state ?

A

B

1.3 Separability: positive maps1.3 Separability: positive maps

: need not be positive, in general

A postive map is completely positive if: ½A B ¸ 0 ! (L ­ 1)(½A B ) ¸ 0

is separable iff for all positive maps ½ ( L ­ 1 ) ( ½) ¸ 0(Horodecki 96)

However, we do not know how to construct all positive maps.

Example: Any physical action.

½ ¸ 0 L (½) ¸ 0

A

B

½

state

A

B

state

Any physical action can be described in terms of a completely positive map.

Example: transposition (in a given basis)

½ =X

i ;j

½i ;j ji i hj j

T (½) =X

i ;j

½i ;j jj i hi j

;

15

µ2 1 + i

1 ¡ i 3

¶!

15

µ2 1 ¡ i

1 + i 3

¶

Extension: partial transposition.

½=X

½i j kl ji ; j i hk; l j (T ­ 1)(½) =X

½i j kl jk; j i hi ; l j

0

BB@

1

CCA

transposes the blocks

Example:

Is called

partial transposition, then

12

1 0 0 10 0 0 00 0 0 01 0 0 1

!12

BB

1 0 0 00 0 1 00 1 0 00 0 0 1

CC

0

@

1

A

0

BB@

1

CCA

Partial transposition is positive, but not completely positive.

A

B

A

B

Is positive

What is known?What is known?

?

SEPARABLE ENTANGLED

PPT NPT

2x2 and 2x3

SEPARABLE ENTANGLED

PPT NPT

(Horodecki and Peres 96)

Gaussian statesSEPAR

ABLE ENTANGLED

PPT NPT

(Giedke, Kraus, Lewenstein, Cirac, 2001)

- Low rank

- Necessary or sufficient conditions

(Horodecki 97)

In general

2. Distillability2. Distillability

...

½

½

½

j©i = j0; 1i + j1; 0i

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 states2.1 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

½A B

random the same random

with ~½= (A ­ B )½(A ­ B )y

U U

and still has non-positive partial transposition.

Zd¹ U (U ­ U )~½(U ­ U )y = aP + + bP ¡ where b =

1d¡

tr(P ¡ ~½)

½TA ¸ 0

·Zd¹ U (U ­ U )~½(U ­ U )y

TA

¸ 0

(IBM, Innsbruck 99)

Qubits:

We consider the (unnormalized) family of states:

x3

one can easily find A, B such that (A ­ B )½­ N (A ­ B )y ! j©ih©j

Higher dimensions:

x2 3distillable?

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

½(x) = P + + xP ¡

H = C 2 ­ C 2

H = C 3 ­ C 3

½TA ¸ 0

½TA ¸ 0

C 2 ­ C 2 ½TA ¸ 0

C 2 ­ C 2 ½TA ¸ 0

½TA ¸ 0

½TA ¸ 0

NPT

distillable

What is known?What is known?

?

Non-DISTILLABLE DIS

TILL

ABLE

PPT NPT

2xN

Non-DISTILLABLE

PPT NPT

(Horodecki 97, Dur et al 2000)

Gaussian states(Giedke, Duan, Zoller, Cirac, 2001)

In general

DIS

TILL

ABLE

Non-DISTILLABLE

PPT NPT

DIS

TILL

ABLE

3. Gaussian states3. Gaussian states

Light source: squeezed states:(2-mode approximation)

Decoherence: photon absorption, phase shifts Gaussian state:

jª i = e¸ (ayby¡ ab) jvaci =1X

n=0

¸ n jn; ni

½= e¡ H

where

is at most quadratic in

H = H (X a ; P a; X b; Pb)

X a =ay + a

p2

; P a = iay ¡ a

p2

Atomic ensembles:Internal levels can be approximated

by continuous variables in Gaussian

states

Optical elements:

- Beam splitters:- Lambda plates:- Polarizers, etc.

Gaussian Gaussian

Measurements:

- Homodyne detection:

localoscillator

X, P

A B

n modes m modes

We consider:

½= e¡ H ! ½0= e¡ H 0

½Gaussian

Is separable and/or distillable?½

H = [L 2(R )]­ n ­ [L 2(R )]­ m

3.1 What is known?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)

½ ½T A = ( T ­ 1 ) ( ½) ¸ 0

2nX2n

3.2 Separability3.2 Separability

All the information about is contained in:

For valid density operators:

R = (X a1 ; P a1 ; X a2 ; : : : ; X b1 ; P b1 ; : : : )

½

° ®;¯ = 2R e h(R ® ¡ d®)(R ¯ ¡ d¯ )i the „correlation matrix“.

where d®= hR ®i ! 0

° ¸ i J

J = J 2 ©J 2 ©: : : ©J 2where

and J 2 =µ

0 ¡ 11 0

¶

is the „symplectic matrix“

°=µ

A CCT 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)?°

° 0 ´ ° ° 1 ° 2 ... ° N

° N ° N +1

° N +1

° N +1

° N

° ° N ! ° 1 ½1 = ½A ­ ½B

(for which one can readily see that is separable)

Facts:

A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]

C N +1 = ¡ I m[C N (B N ¡ i J ) ¡ 1C T ]

A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]

C N +1 = ¡ I m[C N(B N ¡ i J ) ¡ 1C T ]N

A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]

C N +1 = ¡ I m[C N (B N ¡ i J ) ¡ 1C T ]

A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]

C N +1 = ¡ I m[C N(B N ¡ i J ) ¡ 1C T ]N

N

(G. Giedke, B. Kraus, M. Lewenstein, and Cirac, 2001)

Map for CM‘s:

Map for density operators:

Non-linear

½

½

Gaussian

separable

density operators

° N ! ° N +1

½N = e¡ H N ! ½N +1 = e¡ H N +1

(½N +1)A B = trB ~B f [(½N )A B ­ (½N ) ~A ~B ]X B ~B g

A~A

B~B

A~A

CONNECTION WITH POSITIVE MAPS?

3.3 Distillability3.3 Distillability

Idea: take such that

Two modes: N=M=1:

Symmetric states:

distillable state.

A B

A B

Non-symmetric states:

A B A B

General case: N,M

A B

A B

symmetric state.

two modes

° A B = ° B A

½ ½T ¸ 0

½Ts ¸ 0 ½T ¸ 0

~½Ts ¸ 0 ½T

s ¸ 0

½TN ;M ¸ 0 ½T

1;1 ¸ 0

is distillable if and only if ½ ½T ¸ 0

There are no NPT Gaussian states.

(Giedke, Duan, Zoller, and Cirac, 2001)

4. Multipartite case.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.

½

C

½=NX

k=1

pk jak i A hak j ­ jbk i B hbk j ­ jck i C hck j

We can also consider partitions:

Separable A-(BC) Separable B-(AC) Separable C-(AB)

A B

C

A B

C

A B

C

NX

k=1

pk jak i A hak j ­ j' k i B C h' k jNX

k=1

pk jbk i B hbk j ­ j' k i A C h' k jNX

k=1

pk jck i C hck j ­ j' k i A B h' k j

4.1 Bound entangled states.4.1 Bound entangled states.

Consider

A B

C

A B

C

½=NX

k= 1

pk jak i A hak j ­ j' k i B C h' k j =NX

k= 1

pk jbk i B hbk j ­ j' k i A C h' k j

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.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.

N o t d is t illa b le

D is ti lla b le

N o t-d is t illa b le

N o t-d is t illa b le

42

85

137

6

42

85

137

6

4

2 85

137

6

42

85

137

6

N o t-d is t il la b le

D is ti lla b le

N o t-d is t il la b le

42

8

1 2

13

7

6

115

91 0

4

8

1 2

7

6

11

91 0

42

8

1 2

13

7

6

11

59

1 0

2

13

5

2

1

4

3

2

1

4

3

Distillable iff two groups3 and 5 particles

Distillable iff two groups35-45% 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 states4.3 Family of states

where

There are parameters.2N ¡ 1

Define:

Any state can be depolarized to this form.

5. Conclusions5. 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 PhysicsInstitute 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. ZollerJ. 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