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Vacuum Entanglement

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.

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Vacuum Entanglement

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

  2. A B Vacuum Entanglement Motivation: QI Fundamentals: SRQM 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: 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. A B (I) Are A and B entangled? (II) Are Bells' inequalities violated? (III) Where does ent. “come from”?

  5. A B (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).

  6. A pair of causally disconnected atoms A B

  7. CausalStructure 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 notuse 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 + hXAB|VACi |**i “+”…

  12. Emission < Exchange XAB |++i + hXAB|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 Maximal Ent. N () 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 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. } Comment Where does Ent. “come from”?

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

  20. “Mode-Wise” structure A 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. A B Mode Participation Local qi! Qi=ui qi pi! Pi=vipi qi, pi 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.)

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