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

2. Vacuum Entanglement Tel Aviv University Eilat, Feb 27, 2006 Motivation: QI Fundamental: SR QM QI: natural set up to study Ent. causal structure ! LO. H 1, many body Ent. Q. Physics New “quantum effects”? Q. phase transitions, Entropy Area law. A B

3. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 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). Background

4. Entanglement in the vacuum Tel Aviv University  Probing vacuum Entanglement  Vacuum Entanglement in field theory. Eilat, Feb 27, 2006  Vacuum entanglement in a the harmonic lattice.  Properties of entanglement  Spatial structure of entanglement in 1D and 2D lattices. A. Retzker, I. J. Cirac, B. Reznik Phys. Rev. Lett. 93, 056402 (2005)  Detection of “Vacuum” entanglement in an ion trap. 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). 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).

5. Entanglement in the vacuum Tel Aviv University  Probing vacuum Entanglement  Vacuum Entanglement in field theory. Eilat, Feb 27, 2006  Vacuum entanglement in a the harmonic lattice.  Properties of entanglement  Spatial structure of entanglement in 1D and 2D lattices. A. Retzker, I. J. Cirac, B. Reznik Phys. Rev. Lett. 93, 056402 (2005)  Detection of “Vacuum” entanglement in an ion trap. 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). 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).

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

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

8. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 Causal Structure + LO Local operations: For L>cT, we have [  A,  B ]=0 Therefore U INT =U A ­ U B  E Total =0, we can have:  E AB >0. (Ent. Swapping) Detectors’ ent.  Vacuum ent. Lower bound.

9. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 Note: we do not use the rotating wave approximation. Field – Detectors Interaction 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 Unruh (76), B. Dewitt (76), particle-detector models.

10. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 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 criterion to check inseparability and use the Negativity as a measure. ?  AB (4£ 4) = Tr F  (4£1)  i p i  A (2£2) ­  B (2£2)

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

12. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 Emission < Exchange Off resonance Vacuum “window function” The inequality can be satisfied for every finite L and T. Lower bound : negativity decays like e -L 2 /T 2.

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

14. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 Multi-Partite entanglement A B C J. Silman, B. Reznik, Phys. Rev. A. 71, 032322 ( 2006). The same method can be used to study Entanglement between N regions. 000  W =(|001i+|010i+|100i)/3 1/2 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).

15. Vacuum Entanglement in Field Theory Tel Aviv University Eilat, Feb 27, 2006 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. Entanglement in the vacuum Tel Aviv University  Probing vacuum Entanglement  Vacuum Entanglement in field theory. Eilat, Feb 27, 2006  Vacuum entanglement in a the harmonic lattice.  Properties of entanglement  Spatial structure of entanglement in 1D and 2D lattices. A. Retzker, I. J. Cirac, B. Reznik Phys. Rev. Lett. 93, 056402 (2005)  Detection of “Vacuum” entanglement in an ion trap. 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). 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).

17. Vacuum Entanglement in Harmonic lattices Tel Aviv University Eilat, Feb 27, 2006 H field =s  2 + r  2 +m 2  2 H lattice =  h i,ji p i 2 +q i 2 -  q i q j  ground-state / e - q T G q /4 Gaussian ground state: Single dimensionless parameter: 0<  <1

18. Vacuum Entanglement in Harmonic lattices Tel Aviv University Eilat, Feb 27, 2006 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 M 2N £ 2N. 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).  To calculate the von-Neumann entropy we obtain the Symplectic Spectrum, bring M to a diagonal form, then :  To calculate a reduced density matrix, we just take the relevant sub-block! See also talks by Klaus Moelmer and Sam Braunstein

19. Vacuum Entanglement in Harmonic lattices Tel Aviv University Eilat, Feb 27, 2006 Two complementary regions Pure 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. Universality! 1/3 =(c+ c)/6, c=1, bosonic 1D CFT c=1/2 fermionic 1D CFT

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

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

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

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

25. Spatial structure of entanglement Tel Aviv University Eilat, Feb 27, 2006 Mode-Wise decomposition theorem 0209026 Botero, Reznik 0209026 (bosonic modes) 0404176 Botero, Reznik 0404176 (fermionic modes) AB AB  AB =  11  22 …  kk  0,..  kk /  e -  k n |ni|ni Two modes squeezed state E Total =  k E k qipiqipi QiPiQiPi  AB =  c i |A i i|B i i Schmidt decompositionMode-Wise decomposition Local U Entanglement is always 1£1 !

26. Spatial structure of entanglement Tel Aviv University Eilat, Feb 27, 2006 Participation function q i ! Q m =  u i q i p i ! P m =  v i p i Quantifies the local contribution of q i, p i to the collective coordinates Q i,P i Invariant under local rescaling. localcollective Participation function: P i =u i v i,  P i =1

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

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

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

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

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

33. Entanglement in Harmonic lattices Tel Aviv University Eilat, Feb 27, 2006  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. Summary (2)

34. Can we detect Vacuum Entanglement?

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

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

37. Detection of vacuum entanglement Tel Aviv University Eilat, Feb 27, 2006 Approximate Causal Structure U AB =U A ­ U B + O([x A (0),x B (T)])  Classical Quantum

38. Two trapped ions

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

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

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

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

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

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

45. Summary Tel Aviv University Eilat, Feb 27, 2006 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.