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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.
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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.)
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.)
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).
A B (I) Are A and B entangled? (II) Are Bells' inequalities violated? (III) Where does ent. “come from”?
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).
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.)
Relativistic field + probe Interaction: HINT=HA+HB HA=A(t)(e+i tA+ +e-i tA-) (xA,t) Window Function Two-level system Initial state: |(0) i =|+Ai |+Bi|VACi
Relativistic field + probe Interaction: HINT=HA+HB HA=A(t)(e+i tA+ +e-i tA-) (xA,t) Window Function Two-level system Initial state: |(0) i =|+Ai |+Bi|VACi Do notuse the rotating frame approximation!
Probe Entanglement AB(4£ 4) = TrF(4£1) i piA(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.
Emission < Exchange XAB |++i + hXAB|VACi |**i “+”…
Emission < Exchange XAB |++i + hXAB|VACi |**i “+”… Off resonance Vacuum “window function”
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 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)
{ Comment
{ Comment Asher Peres Lecture Notes on GR. (200?).
Accelerated probes Time QFT Red Shift ! A&B perceive |VACi as a thermal state. (Unruh effect) |VACi= N (n e- n |ni|ni) A B Space Final A&B state becomes entangled. Special case: complementary regions. Summers & Werner (85).
} Comment Where does Ent. “come from”?
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.
“Mode-Wise” structure A B A B Qi Pi qi pi 1 AB = ci|Aii|Bii AB= 1122…kk ( k+1k+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.
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.
Mode Shapes Botero, Reznik (2004).
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.)