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.)

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.)

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).

(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? (Filtration, “hidden” non-locality). (III) Where does it “come from”? Localization, shielding. (Harmonic Chain). A B

A pair of causally disconnected atoms B

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.)

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

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!

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.

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

Emission < Exchange XAB |++i + h XAB|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 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)

{ Comment

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

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).

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

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.

“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.

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.

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.)