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

2. Vacuum Entanglement A B 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.)

3. Background Continuum results: Albebraic Field Theory:
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).

4. (I) Are A and B entangled?
(II) Are Bells' inequalities violated?
(III) Where does ent. “come from”? A B

5. (I) Are A and B entangled?
Yes, for arbitrary separation.
("Atom probes”).
(II) Are Bells' inequalities violated?
(Filtration, “hidden” non-locality).
(III) Where does it “come from”?
Localization, shielding (Harmonic Chain). A B

6. A pair of causally disconnected atomsB

7. Causal Structure For L>cT, we have [A,B]=0
Therefore UINT=UA­ UB (LO)
 ETotal =0, but
 EAB >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.)

8. Relativistic field + probe
Interaction:
HINT=HA+HB
HA=A(t)(e+i t A+ +e-i tA-) (xA,t)
Window Function
Two-level system
Initial state:
|(0) i =|+Ai |+Bi|VACi

9. Relativistic field + probe
Interaction:
HINT=HA+HB
HA=A(t)(e+i t A+ +e-i tA-) (xA,t)
Window Function
Two-level system
Initial state:
|(0) i =|+Ai |+Bi|VACi
Do not use the rotating frame approximation!

10. Probe Entanglement AB(4£ 4) = TrF (4£1)  i pi A(2£2)­B(2£2)?
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.

11. Emission < Exchange
XAB
|++i + h XAB|VACi |**i “+”…

12. Emission < Exchange
XAB
|++i + h XAB|VACi |**i “+”…
Off resonance Vacuum “window function”

13. 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)).

14. Bells’ inequalities N () Filtered |++i + h XAB|VACi |**i “+”…
Maximal Ent.
No violation of Bell’s inequalities.
But, by applying local filters
Filtered
|++i + h XAB|VACi |**i “+”…
! 2 |+i|+i + h XAB|VACi|*i|*i “+”…
Negativity 
M ()
Maximal violation
CHSH ineq. Violated iff
M ()>1, (Horokecki (95).)
“Hidden” non-locality.
(Popescu(95).) 
Reznik, Retzker, Silman (2003)

15. {
Comment

16. {
Comment
Asher Peres Lecture Notes on GR. (200?).

17. Accelerated probes Red Shift ! A&B perceive |VACi as a thermal state.
Time
QFT
Red Shift ! A&B perceive |VACi
as a thermal state.
(Unruh effect)
|VACi= ­ N (n e- n |ni|ni) A B
Space
Final A&B state becomes entangled.
Special case: complementary regions.
Summers & Werner (85).

18. Where does Ent. “come from”?}
Comment
Where does Ent. “come from”?

19. Where does Ent. “come from”?1 A B
Hchain! Hscalar field
chain/ e-qi Q-1 qj/4
is a Gaussian state.
! Exact calculation.
Circular chain of coupled
Harmonic oscillators.

20. “Mode-Wise” structureA B A B qi pi Qi Pi 1
AB = ci|Aii|Bii
AB= 1122…kk ( k+1k+2…)
Scmidth decomposition
Mode-Wise decomposition.
kk /  e-k n|ni|ni
(are 1£1 Gaussian states.)
Botero, Reznik 2003.
Giedke, Eisert, Cirac, Plenio, 2003.

21. Mode Participation qi ! Qi= ui qi pi ! Pi= vi pi Local qi, pi A B
Quantifies the participation of the
local (qi, pi) oscillators in the
collective coordinates (Qi,Pi)
“normal modes” within each block.

22. Mode Shapes
Botero, Reznik (2004).

23. Discussion Atom 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.)