Vacuum Entanglement
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Vacuum Entanglement. Recent Developments in Quantum Physics Asher Peres’ 70’th Birthday Honour of In 1-2, 2004 February. B. Reznik (Tel Aviv Univ.) Alonso Botero (Los Andes Univ. Columbia) Alex Retzker (Tel Aviv Univ.) Jonathan Silman (Tel Aviv Univ.). A. B. Vacuum Entanglement. - PowerPoint PPT Presentation
Transcript of Vacuum Entanglement
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(.
