Vacuum Entanglement

B. Reznik (Tel Aviv Univ.)

Alonso Botero (Los Andes Univ. Columbia)Alex Retzker (Tel Aviv Univ.)Jonathan Silman (Tel Aviv Univ.)

Recent Developments in Quantum Physics 

Asher Peres’ 70’th Birthday Honour of In 1-2, 2004February

 

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, BH Ent. Entropy.)

A

B

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

Summers & Werner (85), Halvason & Clifton (2000).Entanglement probes:

Reznik (2000), Reznik, Retzker & Silman (2003).

Discrete models:Harmonic chain: Audenaert et. al (2002), Botero & Reznik (2004).Spin chains: Wootters (2001), Nielsen (2002), Latorre et. al. (2003).

Background

AA

BB

)I (Are A and B entangled?

)II (Are Bells' inequalities violated?

)III (Where does ent. “come from?”

AA

BB

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

") Atom probes.(”

)II (Are Bells' inequalities violated? Yes, for arbitrary separation .

) Filtration, “hidden” non-locality .(

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

(Harmonic Chain).

L cT

A

B

A pair of causally disconnected atomsA pair of causally disconnected atoms

Causal Structure

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

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

Vacuum ent ! Atom ent .Lower bound.

)Why not the use direct approach?simplicity, 4£ 4, vs. 1£ 1

but wait to the second part(.

0E

1EE

Relativistic field + probe

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

0E

1EE

Do not use the rotating frame approximation!

Relativistic field + probe

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

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 criteria to check non-separability and use the Negativity as a measure.

?

AB(4£ 4) = TrF (4£1)

i pi A(2£2)­B

(2£2)

Emission < Exchange

|++i + h XAB|VACi |**i…”+“

XAB

Emission < Exchange

2

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

dd

L

Off resonance Vacuum “window function”

|++i + h XAB|VACi |**i…”+“

XAB

Characteristic Behavior

1( Exponentially decrease: E¼ e-L2.

Super-oscillatory window functions. )Aharonov(88), Berry(94).(

2 (Increasing probe frequency ¼ L2 .

3 (Bell inequalities?

Entanglement 9 Bell ineq. Violation . )Werner(89).(

Bells’ 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)(.

NegativityNegativity

FilteredFiltered

Reznik, Retzker, Silman (2003)Reznik, Retzker, Silman (2003)

Comment {

Comment

Asher Peres Lecture Notes on GRAsher Peres Lecture Notes on GR. (200?). (200?)..

{

Accelerated probes

SpaceSpace

TimeTime

AA BB

Red Shift ! A&B perceive |VACi as a thermal state.

) Unruh effect(

|VACi= ­ N (n e- n |ni|ni)

QFTQFT

Final A&B state becomes entangled .Special case: complementary regions.Summers & Werner (85)Summers & Werner (85)..

Comment

Where does Ent. “come from?”

}

Where does Ent. “come from?”

Hchain! Hscalar field

chain/ e-qi Q-1 qj/4

is a Gaussian state.

! Exact calculation.

AA BB

11

Circular chain of coupledHarmonic oscillators.

“Mode-Wise” structure

Botero, Reznik 2003.Botero, Reznik 2003.Giedke, Eisert, Cirac, Plenio, 2003.Giedke, Eisert, Cirac, Plenio, 2003.

A B A B

AB= 1122…kk ( k+1k+2…)

kk / e-k n|ni|ni

)are 1£1 Gaussian states(.

qqii

ppii

QQii

PPii

AB = ci|Aii|Bii

Scmidth decomposition Mode-Wise decomposition.

11

Mode Participation

qi ! Qi= ui qi

pi ! Pi= vi piAA BB

qqii, p, pii

Quantifies the participation of the local (qi, pi) oscillators in the collective coordinates (Qi,Pi)

“normal modes” within each block.

LocalLocal

Mode Shapes

Botero, Reznik (2004)Botero, Reznik (2004)..

DiscussionAtom Probes :

Vacuum Entanglement can be swapped (In theory) to atoms.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 seem to provide a mechanism for

“shielding” entanglement from exterior regions .

)Therefore in spin or harmonic chains entanglement between single sites truncates(.