Vacuum Entanglement. A. Retzker (Tel-Aviv Univ.) J. I. Cirac (Max Planck Inst., Garching.) B. Reznik (Tel-Aviv Univ.) J. Silman (Tel-Aviv Univ.

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1 Vacuum Entanglement. Retzker (Tel-viv Univ.) J. I. Cirac (Max Planck Inst., Garching.). Reznik (Tel-viv Univ.) J. Silman (Tel-viv Univ.) International summer school on Quantum Information ugust 9 September 30,005

2 Vacuum Entanglement Motivation: QI: natural set up to study Ent causal structure! LO. many body Ent. Q. Phys.: Can Ent. shed light on quantum effects? (low temp. Q. coherences, Q. phase transitions, DMRG, Entropy rea law.)

3 ackground Continuum results: H Entanglement entropy: Unruh (76), ombelli et. l. (86), Srednicki (93), Callan & Wilczek (94). lgebraic Field Theory: Summers & Werner (85), Halvarson & Clifton (00),Verch & Werner (004). Discrete models: Spin chains: Wootters (01), Nielsen (0), Latorre et. al. (03).

4 (I) re and entangled? Yes, for arbitrary separation. ("tom probes ). (II) re ell s inequalities violated? Yes, for arbitrary separation. (Filtration, hidden non-locality). (III) Where does it come from? Localization, shielding. (Harmonic Chain). (IV) Can we detect it? Entanglement Swapping. (Linear Ion trap).

5 Outline : (1). Field entanglement: local probes. (). Linear Ion trap: detection of ground state ent.

6 Probing Field Entanglement RFT Causal structure QI : LOCC L> ct pair of causally disconnected localized detectors

7 Causal Structure + LO For L>cT, we have [φ,φ ]=0 Therefore U INT =U U E Total =0, but E >0. (Ent. Swapping) LO x t = 0 x + t = 0 Vacuum ent! Detectors ent. Lower bound.

8 Field Detectors Interaction Interaction: E 1 E 0 E=Ω H INT =H +H H =ε (t)(e +iω t σ + +e -iω t σ - ) φ(x,t) Two-level system Window Function Initial state: Ψ ( 0) = 0 Note: we do not use the rotating wave approximation. Unruh (76),. Dewitt (76), particle-detector models.

9 Probe Entanglement? ρ (4 4) i p = i Tr ρ F ρ ρ ( total ) Calculate to the second order (in ε) 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.

10 1 U = U U = iε dth ε T dtdt H H Interaction (1 ' )(...) Ψ ( T) = U Interaction 0

11 1 U = U U = i dth T dtdt H H Interaction (1 ε ε ' )(...) ρ Ψ ( T) = U Interaction X X 0 X 5 ( T) = + O( ε ) E E E E E E X E = Φ = Φ Φ 0 = 0 1 = 0 or photon photons γ ε + ε ε + γ γ

12 1 U = U U = i dth T dtdt H H Interaction (1 ε ε ' )(...) ρ Ψ ( T) = U Interaction X X 0 X 5 ( T) = + O( ε ) E E E P.T. P.T, E E E X E = Φ = Φ Φ 0 = 0 1 = 0 or photon photons γ ε + ε ε + γ γ

13 Emission < Exchange E E < 0 X γ ε ε < ε γ γ X dω ω ωε [ ( Ω + ω)] < Sin( ω L) ε ( Ω+ ω ) ε ( Ω ω) L % d 0 0 Off resonance Vacuum window function Superocillatory functions (haronov (88), erry(94)).

14 Entanglement for every separation We can tailor a superoscillatory window function for every L to resonate with the vacuum window function sin( Lω ) Superocillatory window function Vacuum Window function Exchange term exp(-f(l/t))

15 ell s Inequalities N (ρ) Maximal Ent. Filtered No violation of ell s inequalities. ut, by applying local filters Negativity Ω > + hx VC> > + η > > + hx VC> > > + M (ρ) Maximal violation CHSH ineq. Violated iff M (ρ)>1, (Horokecki (95).) Ω Hidden non-locality. Popescu (95). Gisin (96).

16 Summary (1) 1) Vacuum entanglement can be distilled! ) Lower bound: E e -(L/T) (possibly e -L/T ) 3) High frequency (UV) effect: Ω= L. 4) ell inequalities violation for arbitrary separation maximal hidden non-locality. Reznik, Retzker, Silman PR 71, 04104

17 Can we detect Vacuum Entanglement?

18 Detection of Vacuum Entanglement in a Linear Ion Trap Internal levels H=H 0 +H int H 0 =ω z (σ z +σ z )+ ν n a n a n H int =Ω(t)(e -iφ σ + (k) +e iφ σ - (k) )x k Paul Trap 1/ω z << T<<1/ν 0. Retzker, J. I. Cirac,. Reznik PRL 94,

19 Entanglement in a linear trap vac = c d βn e n n n Entanglement between symmetric groups of ions as a function of the total number (left) and separation of finite groups (right).

20 Causal Structure U =U U + O([x (0),x (T)])

21 Two trapped ions Swapping spatial internal states vac U=(e iα x σ x -e iβ p σ y)-... ( e β ) χ + E formation (ρ final ) accounts for 97% of the calculated Entangtlement: E( vaci>)=0.136 e-bits. Final internal state

22 Long Ion Chain ut how do we check that ent. is not due to non-local interaction?

23 Long Ion Chain ut how do we check that ent. is not due to non-local interaction? H H truncated =H H We compare the cases with a truncated and free Hamiltonians

24 Long Ion Chain L=6,15, N=0 L=10,11 N=0 η=exchange/emission >1, signifies entanglement. δ denotes the detuning, L the locations of and.

25 Summary tom Probes: Vacuum Entanglement can be swapped to detectors. ell s inequalities are violated ( hidden non-locality). Ent. reduces exponentially with the separation. High probe frequencies are needed for large separation. Linear ion trap: - proof of principle of the general idea is experimentally feasible for two ions. -One can entangle internal levels of two ions without performing gate operations.