Vacuum Entanglement

B. Reznik (Tel-Aviv Univ.)

A. Botero (Los Andes. Univ. Columbia.)J. I. Cirac (Max Planck Inst., Garching.)A. Retzker (Tel-Aviv Univ.)J. Silman (Tel-Aviv Univ.)

Quantum Information Theory: Present Status and Future DirectionsAug. 23-27 2004, INI, Cambridge

 

Vacuum Entanglement

Motivation: QI

Fundamentals: SR QM QI: natural set up to study Ent.causal structure ! LO.H1, many body Ent.

.Q. Phys.: Can Ent. shed light on “quantum effects”? (low temp. Q. coherences, Q. phase transitions, DMRG, Entropy Area law.)

A

B

See also Latorre’s, Verstraete’s & Plenio’s talks

Continuum results :BH Entanglement entropy:Unruh (76), Bombelli et. Al. (86), Srednicki (93), Callan & Wilczek (94). Albebraic Field Theory :

Summers & Werner (85), Halvarson & Clifton (00).Entanglement probes:

Reznik (00), Reznik, Retzker & Silman (03).

Discrete models:Harmonic chains: Audenaert et. al (02), Botero & Reznik (04).Spin chains: Wootters (01), Nielsen (02), Latorre et. al. (03).Linear Ion trap: Retzker, Cirac & Reznik (04).

Background

AA

BB

)I (Are A and B entangled?

)II (Are Bell’s inequalities violated?

)III (Where does ent. “come from?”

)IV (Can we detect it?

AA

BB

)I (Are A and B entangled? Yes, for arbitrary separation.

") Atom probes.(”

)II (Are Bell’s inequalities violated? Yes, for arbitrary separation .

) Filtration, “hidden” non-locality .(

)III (Where does it “come from?” Localization, shielding.

(Harmonic Chain).

)IV (Can we detect it? Entanglement Swapping.

)Linear Ion trap.(

Plan:

)1 .(Field entanglement: local probes. Reznik (00), Reznik, Retzker, Silman (03).

)2 .(Harmonic chain: spatial structure of ent. Botero, Reznik (04).

)3 .(Linear Ion trap: detection of ground state ent. Retzker, Cirac, Reznik (04).

A

B

A pair of causally disconnected localized detectorsA pair of causally disconnected localized detectors

Probing Field Entanglement

RFT! Causal structure

QI ! LOCC

L> cT

B. Reznik quant-ph/0008006,0112044B. Reznik, A. Retzker & J. Silman quant-ph/0310058

Causal Structure + LO

For L>cT, we have [A,B]=0Therefore UINT=UA­ UB

ETotal =0, butEAB >0. (Ent. Swapping)

Vacuum ent ! Detectors’ ent .Lower bound.

LO

0E

1EE

Note: we do not use the rotating wave approximation.

Field – Detectors Interaction

Window Function

Interaction:

HINT=HA+HB

HA=A(t)(e+i t A+ +e-i tA

-) (xA,t)

Initial state:

|(0) i =|+Ai |+Bi|VACi

Two-level system

Unruh (76), B. Dewitt (76), particle-detector models .

Probe Entanglement

Calculate to the second order (in the final state,and evaluate the reduced density matrix.

Finally, we use Peres’s (96) partial transposition criterion to check inseparability and use the Negativity as a measure.

?

AB(4£ 4) = TrF (4£1)

i pi A(2£2)­B

(2£2)

21(1 ' )(...)

2Interaction A B A A AU U U i dtH T dtdt H H

( ) 0Interaction A BT U

21(1 ' )(...)

2Interaction A B A A AU U U i dtH T dtdt H H

( ) 0Interaction A BT U

2

52

2

1 0

0( ) ( )

AB

AB AB

AB

A A B

B A B

X

X XT O

E E E

E E E

0 0 2,

0 1,

AB A B

A A

X or photons

E photon

** ++ +* *+

21(1 ' )(...)

2Interaction A B A A AU U U i dtH T dtdt H H

( ) 0Interaction A BT U

2

52

2

1 0

0( ) ( )

AB

AB AB

AB

A A B

B A B

X

X XT O

E E E

E E E

0 0 2,

0 1,

AB A B

A A

X or photons

E photon

P.T,

P.T.

** ++ +* *+

Emission < Exchange

|++i + h XAB|VACi |**i…”+“ XAB

||EA||||EB||< ||h0|XABi||

2

0 0[ ( )] (( ) ) ( )A BSin L

dd

L

Off resonance Vacuum “window function”

Superocillatory functions (Aharonov (88), Berry(94)) .

We can tailor a superoscillatory window function for every Lto resonate with the vacuum “window function”

Entanglement for every separation

sin( )L

Superocillatory window function

VacuumWindow function

Exchange term / exp(-f(L/T))

Bell’s Inequalities

No violation of Bell’s inequalities .

But, by applying local filters

Maximal EntMaximal Ent..

Maximal violationMaximal violation

N­N­(())

M­M­(())

| ++i + h XAB|VACi |**i…”+“ ! |+i|+i + h XAB|VACi|*i|*i…”+“

CHSH ineq. Violated iffM­()>1, (Horokecki (95).)

“Hidden” non-locality. Popescu (95). Gisin (96).

NegativityNegativity

FilteredFiltered

Reznik, Retzker, SilmanReznik, Retzker, Silman 0310058

Summary (1)

1 (Vacuum entanglement can be distilled!

2( Lower bound: E ¸ e-(L/T)2

) possibly e-L/T(

3 (High frequency (UV) effect: ¼ L2 .

4 (Bell inequalities violation for arbitrary separation maximal “hidden” non-locality .

Reznik, Retzker, Silman Reznik, Retzker, Silman 0310058

Spatial structure of entanglement in theHarmonic Chain

A. Botero & B. Reznik 0403233

The Harmonic Chain model

Hchain! Hscalar field

=s (x)2+(5)2+m22(x))dx

chain/ e-qi Q-1 qj/4

I) Gaussian state ! Exact

calculation of Ent.

II) Mode-wise decomposition !

Identify spatial Ent. Structure

AA BB

11

Circular Harmonicchain

A. Botero & B. Reznik 0403233

Gaussian (pure) Entanglement

A Covariance Matrix M Second moments h qi qji, h pi pji

SymplecticSpectrum

EntanglementEntropy

i

The reduced density matrix of a Gaussian stateIs a Gaussian density matrix.

AB

E= (+1/2)log(+1/2) – (-1/2)log(-1/2)

Entanglement: block vs. the rest

))Log-2 scaleLog-2 scale((

weak

strong

E' 1/3 lnNb +Ec(,Nb)

c=1, bosonic 1-d FT

Mode-Wise decomposition theorem

Botero, Reznik 02090260209026 (bosonic modes)Botero, Reznik 0404176 0404176 (fermionic modes)

A B A B

AB= 1122…kk 0..,

kk / e-k n|ni|ni

Two modes squeezed state

qi

pi

Qi

Pi

AB = ci|Aii|Bii

Schmidt Mode-Wise decomposition

Local U

Local U

Spatial Entanglement Structure

qi ! Qm= ui qi

pi !­­­­ Pm= vi pi

AA BB

qqii, p, pii

quantifies the contribution of local (qi, pi) oscillators

to the collective coordinates (Qi,Pi)

Circular Harmonicchain

local collective

Participation function : Pi=ui vi, Pi=1

Site Participation Function

Weakcoupling

Strongcoupling

N=32+48 osc. Modes are ordered in decreasingEnt. Contribution, from front to back.

Mode Shapes

strongweak

Outer modes

Inner modes

continuum

Solid – u Dashed -v

Entanglement Contribution

Entanglement as a function of mode number decays exponentially.

Logarithmic dependence in the continuum limit with the overall 1/3 coefficient as predicted by conformal field theory .

Inclusion of an ultraviolet cutoff leads to Localization of the highest frequency inner modes .

Mode shape hierarchy with distinctive layered structure ,with exponential decreasing contribution of the innermost modes.

Summary (2)

Can we detect Vacuum Entanglement?

Detection of Vacuum Entanglement

in a Linear Ion Trap

Paul Trap

AB

Internal levels

H0=z(zA+z

B)+n any an

Hint=(t)(e-i+(k)+ei-

(k))xk

H=H0+Hint

A. Retzker, J. I. Cirac, B. Reznik 0408059

1/z << T<<1/0

Entanglement in a linear trap

Entanglement between symmetric groups of ions as a function of the total number (left) and separation of finite groups (right).

|vaci=|0ci|0bi/ n e- n|niA|niB

Causal Structure

UAB=UA­ UB + O([xA(0),xB(T)])

Two trapped ions

Final internal state

Eformation(final) accounts for 97% of the calculatedEntantlement: E(|vaci)=0.136 e-bits.

|vaci|++i­ !­­|i(|**i+ e-|++i)

“Swapping” spatial ! internal states

U=(ei x x ­ ei p y) ...

U

Long Ion Chain

A B

But how do we check that ent. is not due to “non-local” interaction?

Long Ion Chain

A B

But how do we check that ent. is not due to “non-local” interaction?

Htruncated=HA© HB

We compare the cases with a truncated and free Hamiltonians

HAB !

Long Ion Chain

L=6,15, N=20 L=10,11 N=20

denotes the detuning, L the locations of A and B.

exchange/emission >1 , signifies entanglement.

Summary

Atom Probes :Vacuum Entanglement can be “swapped” to detectors.Bell’s inequalities are violated (“hidden” non-locality).Ent. reduces exponentially with the separation .High probe frequencies are needed for large separation.

Harmonic Chain:Persistence of ent. for large separation is linked with localization of the interior modes. This seems to provide a mechanism for

“shielding” entanglement from the exterior regions .

Linear ion trap: -A proof of principle of the general idea is experimentally feasible for

two ions .-One can entangle internal levels of two ions without performing gate

operations.