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. H 1, 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

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).

A B A pair of causally disconnected atoms

Causal Structure For L>cT, we have [  A,  B ]=0 Therefore U INT =U A ­ U B (LO)  E Total =0, but  E AB >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 Window Function Interaction: H INT =H A +H B H A =  A (t)(e +i  t  A + +e -i  t  A - )  (x A,t) Initial state: |  (0) i =|+ A i |+ B i|VACi Two-level system

Do not use the rotating frame approximation! Relativistic field + probe Window Function Interaction: H INT =H A +H B H A =  A (t)(e +i  t  A + +e -i  t  A - )  (x A,t) Initial state: |  (0) i =|+ A i |+ B i|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) = Tr F  (4£1)  i p i  A (2£2) ­  B (2£2)

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

Emission < Exchange Off resonance Vacuum “window function” |++i + h X AB |VACi |** i “+”… X AB

Characteristic Behavior 1) Exponentially decrease: E¼ e -L 2. Super-oscillatory window functions. (Aharonov(88), Berry(94)). 2) Increasing probe frequency  ¼ L 2. 3) Bell inequalities? Entanglement 9 Bell ineq. Violation. (Werner(89)).

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

Comment {

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

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

Comment Where does Ent. “come from”? }

H chain ! H scalar field  chain / e -q i Q -1 q j /4 is a Gaussian state. ! Exact calculation. A B 1 Circular chain of coupled Harmonic oscillators.

“Mode-Wise” structure Botero, Reznik Giedke, Eisert, Cirac, Plenio, AB AB  AB =  11  22 …  kk (  k+1  k+2 …)  kk /  e -  k n |ni|ni (are 1£1 Gaussian states.) qiqipipiqiqipipi QiQiPiPiQiQiPiPi  AB =  c i |A i i|B i i Scmidth decomposition Mode-Wise decomposition. 1

Mode Participation q i ! Q i =  u i q i p i ! P i =  v i p i A B q i, p i Quantifies the participation of the local (q i, p i ) oscillators in the collective coordinates (Q i,P i ) “normal modes” within each block. Local

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.)