Vacuum Entanglement. B. Reznik (Tel-Aviv Univ.)

Transcription

1 Vacuum Entanglement. Reznik (Tel-viv Univ.). otero (Los ndes. Univ. Columbia.) J. I. Cirac (Max Planck Inst., Garching.). Retzker (Tel-viv Univ.) J. Silman (Tel-viv Univ.) Quantum Information Theory: Present Status and Future Directions ug , INI, Cambridge

2 Vacuum Entanglement Motivation: Fundamentals: SR QI QM QI: natural set up to study Ent. causal structure LO. H, many body Ent.. Q. Phys.: Can Ent. shed light on quantum effects? (low temp. Q. coherences, Q. phase transitions, DMRG, Entropy rea law.) See also Latorre s, Verstraete s & Plenio s talks

3 ackground Continuum results: H Entanglement entropy: Unruh (76), ombelli et. l. (86), Srednicki (93), Callan & Wilczek (94). lbebraic Field Theory: Summers & Werner (85), Halvarson & Clifton (00). Entanglement probes: Reznik (00), Reznik, Retzker & Silman (03). Discrete models: Harmonic chains: udenaert et. al (02), otero & Reznik (04). Spin chains: Wootters (01), Nielsen (02), Latorre et. al. (03). Linear Ion trap: Retzker, Cirac & Reznik (04).

4 (I) re and entangled? (II) re ell s inequalities violated? (III) Where does ent. come from? (IV) Can we detect it?

5 (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).

6 Plan : (1). Field entanglement: local probes. Reznik (00), Reznik, Retzker, Silman (03). (2). Harmonic chain: spatial structure of ent. otero, Reznik (04). (3). Linear Ion trap: detection of ground state ent. Retzker, Cirac, Reznik (04).

7 Probing Field Entanglement RFT Causal structure QI LOCC L> ct pair of causally disconnected localized detectors. Reznik quant-ph/ , Reznik,. Retzker & J. Silman quant-ph/

8 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.

9 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) i = i i VCi Note: we do not use the rotating wave approximation. Unruh (76),. Dewitt (76), particle-detector models.

10 Probe Entanglement ρ (4 4) = Tr F ρ (4 )? i p i ρ (2 2) ρ (2 2) 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.

11 1 U = U U = iε dth ε T dtdt H H 2 2 Interaction (1 ' )(...) Ψ ( T) = U Interaction 0

12 1 U = U U = i dth T dtdt H H 2 2 Interaction (1 ε ε ' )(...) ρ X Ψ ( T) = U Interaction X X 2 5 ( T) = 2 + O( ε ) E E E E E E 2 X Φ Φ 0 0or2, photons E Φ 0 1, photon γ ε + ε ε + 2 γ γ

13 1 U = U U = i dth T dtdt H H 2 2 Interaction (1 ε ε ' )(...) ρ X Ψ ( T) = U Interaction X X 2 5 ( T) = 2 + O( ε ) P.T. P.T, E E E E E E 2 X Φ Φ 0 0or2, photons E Φ 0 1, photon γ ε + ε ε + 2 γ γ

14 Emission < Exchange E E < h0 X i γ ε ε < ε 2 γ γ i + h X VCi i + X 2 dω ωdωε [ ( Ω + ω)] < Sin( L) ( ) ω ε Ω+ ω ε ( Ω ω) L 0 0 Off resonance Vacuum window function Superocillatory functions (haronov (88), erry(94)).

15 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))

16 ell s Inequalities N (ρ) Maximal Ent. Filtered No violation of ell s inequalities. ut, by applying local filters Negativity Ω i + h X VCi i + η 2 i i + h X VCi i i + M (ρ) Maximal violation CHSH ineq. Violated iff M (ρ)>1, (Horokecki (95).) Ω Hidden non-locality. Popescu (95). Gisin (96). Reznik, Retzker, Silman

17 Summary (1) 1) Vacuum entanglement can be distilled! 2) Lower bound: E e -(L/T)2 (possibly e -L/T ) 3) High frequency (UV) effect: Ω L 2. 4) ell inequalities violation for arbitrary separation maximal hidden non-locality. Reznik, Retzker, Silman

18 Spatial structure of entanglement in the Harmonic Chain. otero &. Reznik

19 The Harmonic Chain model H chain H scalar field = π(x)2+(5φ) 2 +m 2 φ 2 (x))dx ψ chain e -q i Q -1 qj/4 Circular Harmonic chain. otero &. Reznik I) Gaussian state Exact calculation of Ent. II) Mode-wise decomposition Identify spatial Ent. Structure

20 Gaussian (pure) Entanglement The reduced density matrix of a Gaussian state Is a Gaussian density matrix. Ψ ρ Covariance Matrix M Second moments h q i q j i, h p i p j i Symplectic Spectrum λ i Entanglement Entropy E= (λ+1/2)log(λ+1/2) (λ-1/2)log(λ-1/2)

21 Entanglement: block vs. the rest weak strong E E' 1/3 lnn b +E c (α,n b ) N b (Log-2 2 scale) c=1, bosonic 1-d FT

22 Mode-Wise decomposition theorem q i p i Local U Q i P i ψ = c i i i i i Schmidt Local U ψ = ψ 11 ψ 22 ψ kk ψ 0,.. Mode-Wise decomposition otero, Reznik otero, Reznik (bosonic modes) (fermionic modes) ψ kk e -β k n ni ni Two modes squeezed state

23 Spatial Entanglement Structure q i, p i local collective q i Q m = u i q i p i P m = v i p i Participation function: P i =u i v i, P i =1 Circular Harmonic chain quantifies the contribution of local (q i, p i ) oscillators to the collective coordinates (Q i,p i )

24 Site Participation Function Weak coupling Strong coupling N=32+48 osc. Modes are ordered in decreasing Ent. Contribution, from front to back.

25 Mode Shapes Outer modes m=1 m=2 Solid u Dashed -v m=23 m=24 m=45 m=46 m=73 Inner modes m=74 =0.2 weak =5.0 x 10 ( = 4.4) -8 strong continuum

26 Entanglement Contribution Entanglement (e-bits) Even Odd = = 0.1 = 0.35 = = Mode Depth Entanglement as a function of mode number decays exponentially.

27 Summary (2) Logarithmic dependence in the continuum limit with the overall 1/3 coefficient as predicted by conformal field theory. Inclusion of an ultraviolet cutoff leads to Localization of the highest frequency inner modes. Mode shape hierarchy with distinctive layered structure, with exponential decreasing contribution of the innermost modes.

28 Can we detect Vacuum Entanglement?

29 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

30 Entanglement in a linear trap vaci= 0 c i 0 b i n e -β n ni ni Entanglement between symmetric groups of ions as a function of the total number (left) and separation of finite groups (right).

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

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

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

34 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

35 Long Ion Chain L=6,15, N=20 L=10,11 N=20 η=exchange/emission >1, signifies entanglement. δ denotes the detuning, L the locations of and.

36 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. Harmonic Chain: Persistence of ent. for large separation is linked with localization of the interior modes. This seems to provide a mechanism for shielding entanglement from the exterior regions. 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.