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Probing Vacuum Entanglement

Probing Vacuum Entanglement. Benni Reznik Tel-Aviv University In collaboration with: A. Botero, J. I. Cirac, A. Retzker, J. Silman. Eilat, Feb 27, 2006. Tel Aviv University. Vacuum Entanglement. Motivation: QI Fundamental: SR QM QI : natural set up to study Ent.

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Probing Vacuum Entanglement

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  1. Probing Vacuum Entanglement Benni Reznik Tel-Aviv University In collaboration with: A. Botero, J. I. Cirac, A. Retzker, J. Silman.

  2. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement Motivation: QI Fundamental: SRQM QI: natural set up to study Ent. causal structure ! LO. H1, many body Ent. Q. Physics New “quantum effects”? Q. phase transitions, Entropy Area law. A B

  3. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory Background Continuum results: BH Entanglement entropy: Unruh (76), Bombelli et. Al. (86), Srednicki (93), Callan & Wilczek (94) . Albebraic Field Theory: Summers & Werner (85), Halvarson & Clifton (00). Entanglement probes: Reznik (00), Reznik, Retzker & Silman (03). Verch & Werner (04). Discrete models: Harmonic chains: Audenaert et. al (02), Botero & Reznik (04). Spin chains: Wootters (01), Nielsen (02), Latorre et. al. (03). Linear Ion trap: Retzker, Cirac & Reznik (04).

  4. Eilat, Feb 27, 2006 Tel Aviv University Entanglement in the vacuum • Vacuum Entanglement in field theory. • Probing vacuum Entanglement B. Reznik, Found. of Phys, 33, 167 (2003). B. Reznik, J. Silman, A. Retzker, Phys. Rev. A 71, 042104 (2005). J. Silman, B. Reznik, Phys. Rev. A. 71, 032322 ( 2006). • Vacuum entanglement in a the harmonic lattice. • Properties of entanglement • Spatial structure of entanglement in 1D and 2D lattices. A. Botero, B. Reznik, Phys. Rev. A 67, 052311 (2003). A. Botero, B. Reznik, Phys. Rev. A 70, 052329 (2004) . A. Botero, B. Reznik, Phys. Lett. A 331, 39, (2004). • Detection of “Vacuum” entanglement in an ion trap. A. Retzker, I. J. Cirac, B. Reznik Phys. Rev. Lett. 93, 056402 (2005)

  5. Eilat, Feb 27, 2006 Tel Aviv University Entanglement in the vacuum • Vacuum Entanglement in field theory. • Probing vacuum Entanglement B. Reznik, Found. of Phys, 33, 167 (2003). B. Reznik, J. Silman, A. Retzker, Phys. Rev. A 71, 042104 (2005). J. Silman, B. Reznik, Phys. Rev. A. 71, 032322 ( 2006). • Vacuum entanglement in a the harmonic lattice. • Properties of entanglement • Spatial structure of entanglement in 1D and 2D lattices. A. Botero, B. Reznik, Phys. Rev. A 67, 052311 (2003). A. Botero, B. Reznik, Phys. Rev. A 70, 052329 (2004) . A. Botero, B. Reznik, Phys. Lett. A 331, 39, (2004). • Detection of “Vacuum” entanglement in an ion trap. A. Retzker, I. J. Cirac, B. Reznik Phys. Rev. Lett. 93, 056402 (2005)

  6. Eilat, Feb 27, 2006 Tel Aviv University A B Vacuum Entanglement in Field Theory Are A and B entangled? Yes, for arbitrary separation. ("Atom probes”). Are Bell’s inequalities violated? Yes, for arbitrary separation. Can we detect it? Entanglement Swapping. (Linear Ion trap).

  7. Eilat, Feb 27, 2006 Tel Aviv University A B L> cT Vacuum Entanglement in Field Theory RQFT! Causal structure A pair of causally disconnected localized detectors

  8. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory CausalStructure + LO Local operations: For L>cT, we have [A,B]=0 Therefore UINT=UA­ UB ETotal =0, we can have: EAB >0. (Ent. Swapping) Detectors’ ent.  Vacuum ent. Lower bound.

  9. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory Field – Detectors Interaction 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 Note: we do notuse the rotating wave approximation. Unruh (76), B. Dewitt (76), particle-detector models.

  10. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory 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 criterion to check inseparability and use the Negativity as a measure.

  11. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory **++ *+ +*

  12. Eilat, Feb 27, 2006 Tel Aviv University Off resonanceVacuum “window function” Vacuum Entanglement in Field Theory Emission < Exchange The inequality can be satisfied for every finite L and T. Lower bound : negativity decays like e-L2/T2.

  13. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory Violation of Bell’s Inequalities N () 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). Gisin (96). 

  14. Eilat, Feb 27, 2006 Tel Aviv University A C B Vacuum Entanglement in Field Theory Multi-Partite entanglement The same method can be used to study Entanglement between N regions. Is there genuine 3-party nonlocality? For N=3, the correlations violate the Sveltichny inequalites. (a hybrid local-nonlocal hidden-variable model). G. Svetlichny, Phys. Rev. D 35, 3066 (1987). 000  W =(|001i+|010i+|100i)/31/2 J. Silman, B. Reznik, Phys. Rev. A. 71, 032322 ( 2006).

  15. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Field Theory Summary (1) • Entanglement can be extracted from arbitrary separated • regions in vacuum. • Lower bound : decays like e-(L/T)2 (possibly e-L/T) • Bell inequalities violation for arbitrary separation: • “hidden” non-locality.

  16. Eilat, Feb 27, 2006 Tel Aviv University Entanglement in the vacuum • Vacuum Entanglement in field theory. • Probing vacuum Entanglement B. Reznik, Found. of Phys, 33, 167 (2003). B. Reznik, J. Silman, A. Retzker, Phys. Rev. A 71, 042104 (2005). J. Silman, B. Reznik, Phys. Rev. A. 71, 032322 ( 2006). • Vacuum entanglement in a the harmonic lattice. • Properties of entanglement • Spatial structure of entanglement in 1D and 2D lattices. A. Botero, B. Reznik, Phys. Rev. A 67, 052311 (2003). A. Botero, B. Reznik, Phys. Rev. A 70, 052329 (2004) . A. Botero, B. Reznik, Phys. Lett. A 331, 39, (2004). • Detection of “Vacuum” entanglement in an ion trap. A. Retzker, I. J. Cirac, B. Reznik Phys. Rev. Lett. 93, 056402 (2005)

  17. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Harmonic lattices Hfield=s2 + r2+m22 Hlattice =h i,ji pi2 +qi2 -  qiqj Single dimensionless parameter: 0<<1 ground-state/ e- qT G q /4 Gaussian ground state:

  18. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Harmonic lattices How to calculate entanglement of Gaussian states • The wave function can be fully described by second moments q-q and p-p and q-p • correlations  the covariance matrix M2N£ 2N. • To calculate a reduced density matrix, we just take the relevant sub-block! • To calculate the von-Neumann entropy we obtain the Symplectic Spectrum, • bring M to a diagonal form, then : E= (+1/2)log(+1/2) – (-1/2)log(-1/2) • For mixed state entanglement, Peres criteria means p -p and <1/2 R. Simon, Phys. Rev. Lett. 84, 2726 (2000). See also talks by Klaus Moelmer and Sam Braunstein

  19. Eilat, Feb 27, 2006 Tel Aviv University Universality! 1/3 =(c+ c)/6 , c=1, bosonic 1D CFT c=1/2 fermionic 1D CFT Vacuum Entanglement in Harmonic lattices Two complementary regionsPure state entanglement E'1/3 lnN Callan & Wilczek 1994. -- Conformal field theory. Vidal et. al. PRL, 2003. -- critical Spin chains. Botero, Reznik, PRA 2004, -- critical Harmonics chain.

  20. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Harmonic lattices Harmonic 1D chain Entangled (non-vanishing negativity)

  21. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Harmonic lattices Two oscillators Entangled

  22. Eilat, Feb 27, 2006 Tel Aviv University Vacuum Entanglement in Harmonic lattices Two oscillators Not Entangled Even for a critical model! (Joe Eberly: “Entanglement sudden Death” in space) • Correlations between two sites decay with the distance. • Entanglement vanishes for separation of few sites! Osterloh et. al. Nature, 2002. (spin chains)

  23. Eilat, Feb 27, 2006 Tel Aviv University A B Maximal separation increases with the number of oscillators in the block For N!1 , ent¼ exp(-L/D) Vacuum Entanglement in Harmonic lattices Two blocks (Slightly slower decay compared with the lower bound obtained by using detector probes in the vacuum. ) A. Retzker, B. Reznik, work in progress.

  24. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement How can we understand the persistence of entanglement in the continuum? “Where” does entanglement come from? • Modewise decomposition • Participation function • Some examples in 1D and 2D.

  25. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement Mode-Wisedecomposition theorem A B A B Local U Qi Pi qi pi Schmidt decomposition Mode-Wise decomposition AB= 1122…kk 0,.. AB = ci|Aii|Bii kk/ e-k n|ni|ni Two modes squeezed state ETotal =k Ek Botero, Reznik0209026(bosonic modes) Botero, Reznik0404176 (fermionic modes) Entanglement is always 1£1 !

  26. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement Participation function local collective qi! Qm=ui qi pi! Pm=vipi Participation function: Pi=ui vi,  Pi=1 Quantifies the local contribution of qi, pi to the collective coordinates Qi,Pi Invariant under local rescaling.

  27. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement Participation Function: circular 1D chain Weak coupling Strong coupling N=32+48 osc. Modes are ordered in decreasing Ent. Contribution, from front to back.

  28. Eilat, Feb 27, 2006 Tel Aviv University Outer modes Inner modes Spatial structure of entanglement Mode Shapes Solid – u Dashed -v Weak coupling strong continuum

  29. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement 2-D Harmonic lattice Entanglement / Area Entanglement arises from surface modes

  30. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement 2D Harmonic lattice E Mode number First surface mode Entanglement arises from surface modes Entanglement / Area

  31. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement Mode shapes: 2D Lattice Mode Number 1

  32. Eilat, Feb 27, 2006 Tel Aviv University Spatial structure of entanglement Mode shapes: 2D Lattice

  33. Eilat, Feb 27, 2006 Tel Aviv University Entanglement in Harmonic lattices Summary (2) • Entanglement truncates to zero after a finite separation. In the continuum limit it decays exponentially rather then as a power law, even at criticality. • Mode shape hierarchy with distinctive layered structure, • with exponential decreasing contribution of the innermost modes. • (linked with the area law). • This is possibly related with the effect of Localization • of the inner modes.

  34. Can we detect Vacuum Entanglement?

  35. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement Linear Ion Trap Ions internal levels H=H0+Hint H0=z(zA+zB)+n any an B Hint=(t)(e-i+(k)+ei-(k))xk A 1/z << T<<1/0 Paul Trap A. Retzker, J. I. Cirac, B. Reznik, PRL, 2005.

  36. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement One block Two ions Entanglement between symmetric groups of ions as a function of the total number (left) and separation of finite groups (right).

  37. Eilat, Feb 27, 2006 Tel Aviv University  UAB=UA­ UB + O([xA(0),xB(T)]) Detection of vacuum entanglement Approximate Causal Structure Classical Quantum

  38. Two trapped ions

  39. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement population of first two levels is 99% The entanglement is 0.136 e-bits The available operations: is a unit vector in the x-y plane

  40. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement Swap external to internal levels External degrees of freedom Internal degrees of freedom

  41. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement Swap operation To realize the p coupling we use two kicks in opposite directions:

  42. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement Final two ions internal state “Swapping” spatial internal states U=(ei x x­ei p y)­... Eformation(final) accounts for 97% of the calculated Entangtlement: E(|vac>)=0.136 e-bits. Final internal state

  43. Eilat, Feb 27, 2006 Tel Aviv University vs. Htruncated = HA + HB HAB We compare the cases with a truncated and free Hamiltonians Detection of vacuum entanglement Entanglement between two regions A B How do we check that ent. is not due to “non-local” interaction?

  44. Eilat, Feb 27, 2006 Tel Aviv University Detection of vacuum entanglement L=6,15, N=20 L=10,11 N=20 =exchange/emission >1 , signifies entanglement.  denotes the detuning, L the locations of A and B.

  45. Eilat, Feb 27, 2006 Tel Aviv University Summary Atom Probes: Vacuum Entanglement can extracted to local probes. Entanglement reduces exponentially with the separation. Bell’s inequalities are violated (“hidden” non-locality). Harmonic Chain: Persistence of entanglement for large separation is possibly linked with localization of the interior modes and a “shielding” effect. Linear ion trap: A proof of principle of the general idea is experimentally feasible. Entangling internal levels of two ions without performing gates.

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