Introduction to entanglement theory & Detection of multipartite entanglement close to symmetric Dicke states

Transcription

1 Introduction to entanglement theory & Detection of multipartite entanglement close to symmetric Dicke states G. Tóth 1,2,3 Collaboration: Entanglement th.: G. Vitagliano 1, I. Apellaniz 1, I.L. Egusquiza 1, Cold gas exp.: B. Lücke 4, J. Peise 4, J. Arlt 4, L. Santos 4, C. Klempt 4 1 Theoretical Physics, University of the Basque Country UPV/EHU, Bilbao, Spain 2 IKERBASQUE, Basque Foundation for Science, Bilbao, Spain 3 Wigner Research Centre for Physics, Budapest, Hungary 4 Institut für Quantenoptik, Leibniz Universität Hannover, Hannover, Germany BCAM, Bilbao, 27 January / 53

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3 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 3 / 53

4 Why multipartite entanglement is important? Full tomography is not possible, we still have to say something meaningful. Claiming entanglement is not sufficient for many particles. Many experiments are aiming to create entangled states with many atoms. Only collective quantities can be measured.

5 Notation Qubit: Ψ = α 0 + β 1, Density matrix: ϱ = k p k Ψ k Ψ k, Expectation value: A = Tr(ϱA). Variance: ( A) 2 = Tr(ϱA 2 ) Tr(ϱA) 2.

6 Defintion of Entanglement A state is (fully) separable if it can be written as p k ϱ (k) ϱ (k)... ϱ (k) 1 2 N. k If a state is not separable then it is entangled. R.F. Werner, Phys. Rev. A 1989

7 Questions for multipartite entanglement For many particles, it is not sufficient to say entangled / not entangled. We have to have more than these two levels.

8 k-producibility/k-entanglement A pure state is k-producible if it can be written as Φ = Φ 1 Φ 2 Φ 3 Φ 4... where Φ l are states of at most k qubits. A mixed state is k-producible, if it is a mixture of k-producible pure states. [ e.g., O. Gühne and GT, New J. Phys ] If a state is not k-producible, then it is at least (k + 1)-particle entangled.

9 Genuine multipartite entanglement N-particle entanglement genuine multipartite entanglement.

10 Usefulness of entanglement Entangled states are useful for quantum cryptography, for quantum teleportation. Entangled states outperform separable ones in very general tasks in quamtum metrlogy. Note: Entanglement cannot be obtained from separable states with local operations and classical communications (LOCC). It is a resource.

11 Entanglement detection It is a very hard task to decide whether a quantum state is separable or not. Solved for small systems: 2 2, 2 3. (Peres-Horodecki condition, 1997.) In general, necessary conditions for separability exist. If they are violated, then we know that the state is entangled

12 Examples Examples Two entangled states of four qubits: GHZ 4 = 1 ( ), 2 Ψ B = 1 2 ( ) = ( ). The first state is genuine multipartite entangled, the second state is biseparable.

13 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 13 / 53

14 Theory of quantum entanglement Separable states form a convex set. ρ 1 ρ 2 ρ' Entangled states Separable states

15 Theory of quantum entanglement II A more accurate picture: Boundary: Density matrices with less than full rank All quantum states (convex set) Not only curved boundaries

16 Theory of quantum entanglement III Together with the set of separable states: Pure product states are at the boundary of both sets Separable states All quantum states

17 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 17 / 53

18 Witnesses based on correlations Definition An entanglement witness W is an operator that is positive on all separable (biseparable) states. Thus, Tr(Wϱ) < 0 signals entanglement (genuine multipartite entanglement). [ Horodecki 1996; Terhal 2000; Lewenstein, Kraus, Cirac, Horodecki 2002 ] There are two main goals when searching for entanglement witnesses: Large robustness to noise Optimization Few measurements

19 Convex sets for the entanglement witnesses Entanglement witnesses in the convex set picture Quantum states detected by the witness Entangled states Separable states

20 Witnesses based on correlations Example Witness with Heisenberg interaction W xyz = σ (1) x σ (2) x + σ (1) y σ (2) y + σ (1) z σ (2) z. Proof. For product states of the form Ψ = Ψ 1 Ψ 2, we have σ x σ x + σ y σ y + σ z σ z = l=x,y,z σ l Ψ1 σ l Ψ2 1. Here, we used the Cauchy-Schwarz inequality. Due to convexity, the inequality is also true for separable states. The minimum for quantum states is 3. Such a maximum is obtained for the state 1 2 ( ).

21 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 21 / 53

22 Variance-based criteria For a bipartite system, for both parties ( A k ) 2 + ( B k ) 2 L k. For product states of the form Ψ = Ψ 1 Ψ 2, we have ( (A 1 + A 2 )) 2 = (A 1 + A 2 ) 2 A 1 + A 2 2 = ( A 1 ) 2 Ψ 1 + ( A 2 ) 2 Ψ 2 since for product states Hence, A 1 A 2 A 1 A 2 = 0. ( (A 1 + A 2 )) 2 + ( (B 1 + B 2 )) 2 L 1 + L 2. This is also true for separable states due to the convexity of separable states. [ See Gühne, Phys. Rev. Lett. (2004) for an exhaustive study.]

23 Variance-based criteria II Example: we have Hence, ( x) 2 ( p) ( x) 2 + ( p) 2 1. Then, for two-mode separable states ( (x 1 + x 2 )) 2 + ( (p 1 p 2 )) 2 2. Any state violating this is entangled. [ Generalization: L.M. Duan, G. Giedke, J.I. Cirac, P. Zoller, Phys. Rev. Lett (2000); R. Simon, Phys. Rev. Lett (2000).]

24 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 24 / 53

25 Experiment with photons A photon can have a horizontal (H) and a vertical (V) polarization. H/V can take the role of 0 and 1. Problem: photons do not interact with each other.

26 Photons II MPQ, Munich. Experiments with 6 photons. [ W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, Phys. Rev. Lett ] D (3) 6 = 1 20 ( ).

27 Photons III

28 Photons IV 6-qubit Quantum state tomography [ C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, Phys. Rev. Lett 113, (2014). ]

29 Photons VI Entanglement witness for Detecting genuine multipartite entanglement close to Dicke states W D := D(3) 6 D(3) 6, where D (3) 6 = 1 20 ( ). If we have W D < 0 then the state is genuine multipartite entangled. [G. Tóth, JOSAB 2007.]

30 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 30 / 53

31 Many-particle systems for j=1/2 For spin- 1 2 particles, we can measure the collective angular momentum operators: where l = x, y, z and σ (k) l J l := 1 2 N k=1 σ (k) l, a Pauli spin matrices. We can also measure the variances ( J l ) 2 := J 2 l J l 2.

32 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 32 / 53

33 The standard spin-squeezing criterion The spin squeezing parameter is defined as ξs 2 ( J z ) 2 = N J x 2 + J y 2. [A. Sørensen, L.M. Duan, J.I. Cirac, P. Zoller, Nature 409, 63 (2001).] If ξ 2 s < 1 then the state is entangled. States detected are like this: Variance of J z is small J x is large x y z

34 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 34 / 53

35 Generalized spin squeezing criteria for j = 1 2 Let us assume that for a system we know only J := ( Jx, J y, J z ), K := ( J 2 x, J 2 y, J 2 z ). Then any state violating the following inequalities is entangled: J 2 x + J 2 y + J 2 z N(N+2) 4, ( J x ) 2 + ( J y ) 2 + ( J z ) 2 N 2, J 2 k + J2 l (N 1)( J m ) 2 + N 2, (N 1) [ ( J k ) 2 + ( J l ) 2] J 2 m + N(N 2) 4, where k, l, m take all the possible permutations of x, y, z. [GT, C. Knapp, O. Gühne, and H.J. Briegel, PRL 99, (2007)]

36 Generalized spin squeezing criteria for j = 1 2 II Separable states are in the polytope We set J l = 0 for l = x, y, z.

37 Spin squeezing criteria Two-particle correlations All quantities needed can be obtained with two-particle correlations J l = N j l 1 ϱp2 ; J 2 l = N 4 + N(N 1) j l j l ϱp2. Here, the average 2-particle density matrix is defined as ϱ 2p = 1 ϱ mn. N(N 1) n m Still, we can detect states with a separable ϱ p2. Still, as we will see, we can even detect multipartite entanglement!

38 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 38 / 53

39 Dicke states Symmetric Dicke states with J z = 0 (simply Dicke states in the following) are defined as D N = ( N N 2 ) 1 2 k P k ( 0 N 2 1 N 2 ). E.g., for four qubits they look like D 4 = 1 6 ( ). [photons: Kiesel et al., Phys. Rev. Lett. 2007; Prevedel et al., Phys. Rev. Lett 2007; Wieczorek et al., Phys. Rev. Lett. 2009] [cold atoms: Lücke et al., Science 2011; Hamley et al., Science 2011]

40 Dicke states are useful because they possess strong multipartite entanglement, like GHZ states. [GT, JOSAB 2007.]... are optimal for quantum metrology, similarly to GHZ states. [Hyllus et al., PRA 2012;Lücke et al., Science 2011.] [GT, PRA 2012; GT and Apellaniz, J. Phys. A, special issue for 50 year of Bell s theorem, 2014.]... are macroscopically entangled, like GHZ states. [Fröwis, Dür, PRL 2011]

41 Spin Squeezing Inequality for Dicke states Let us rewrite the third inequality J 2 k + J2 l N 2 (N 1)( J m) 2. It detects states close to Dicke states since Jx 2 + Jy 2 = N ( ) N = max., J 2 z = 0.

42 Spin Squeezing Inequality for Dicke states II Based on the above inequality, we define a new spin squeezing parameter ξos 2 = RHS LHS = (N 1) ( J z ) 2 Jx 2 + J2 y N. 2 [Vitagliano, Apellaniz, Egusquiza, GT, PRA (2014)] For our Dicke state, the numerator is minimal, the denominator is maximal, ξ 2 os = 0. The original spin squeezing parameter would not detect the Dicke states, since ξs 2 ( J z ) 2 = N J x 2 + J y 2 = N ( J z) 2 =. 0

43 Fully polarized states Relation between the second moments and the expectation value J 2 x = J x 2 + ( J x ) 2 J x 2. For states polarized in the x-direction and spin squeezed along the z-direction, for N 1, we have J 2 x J x 2 N. Hence, for fully polarized states ξos 2 ( J z ) 2 = (N 1) Jx 2 + J2 y N 2 ξs 2 ( J z ) 2 = N J x 2 + J y 2.

44 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 44 / 53

45 Multipartite entanglement in spin squeezing We consider pure k-producible states of the form Ψ = M n=1 ψ(n), where ψ (n) is the state of at most k qubits. The spin-squeezing criterion for k-producible states is J x 2 + J y 2 ( J z ) 2 J max F k 2 J max, where J max = N 2 and we use the definition F j (X) := 1 j min ( j z ) 2. jx j =X [Sørensen and Mølmer, Phys. Rev. Lett. 86, 4431 (2001); experimental test: C. Gross et al., Nature 464, 1165 (2010).]

46 Multipartite entanglement around Dicke states Measure the same quantities as before ( J z ) 2 and J 2 x + J 2 y. In contrast, for the original spin-squeezing criterion we measured ( J z ) 2 and J x 2 + J y 2. Pioneering work: linear condition of Luming Duan, Phys. Rev. Lett. (2011). See also Zhang, Duan, arxiv (2014).

47 Due to convexity properties of the expression, this is also valid to mixed separable states. Multipartite entanglement - Our condition Sørensen-Mølmer condition for k-producible states Combine it with ( J z ) 2 J max F k 2 J x 2 + J y 2 J max J 2 x + J 2 y J max ( k 2 + 1) + J x 2 + J y 2, which is true for pure k-producible states. (Remember, J max = N 2.) Condition for entanglement detection around Dicke states J ( J z ) 2 x 2 + J2 y J max( k 2 + 1) J max F k 2 J max..

48 Concrete example N = 8000 particles, and J eff = Jx 2 + J2 y. Red curve: boundary for 28-particle entanglement. Blue dashed line: linear condition given in [L.-M. Duan, Phys. Rev. Lett. 107, (2011).] Red dashed line: tangent of our curve.

49 Outline 1 Motivation Why multipartite entanglement is important? 2 Quantum Entanglement Geometry of quantum states Linear entanglement witnesses Non-linear entanglement witnesses Experiments 3 Spin squeezing and entanglement Collective measurements The original spin-squeezing criterion Generalized criteria for j = Spin squeezing for Dicke states Entanglement detection close to Dicke states Detection of multipartite entanglement close to Dicke states Our conditions are stronger than the original conditions 49 / 53

50 Our condition is stronger Consider spin squeezed states as ground states of H(Λ) = J 2 z ΛJ x. For Λ =, the ground state is fully polarized. For Λ = 0, it is the symmetric Dicke state. Our condition vs. original condition for N=4000 and p=0.05 [ Lücke et al., Phys. Rev. Lett. 112, (2014). ]

51 Experimental results Bose-Einstein condensate, 8000 particles. 28-particle entanglement is detected. (a) (c) separable (b) Δ J z J z ( Jz) 2 J x J y J eff < J ^2 / J 2 eff max > [ Lücke et al., Phys. Rev. Lett. 112, (2014). ]

52 Project participants G. Vitagliano C. Klempt, B. Lücke, J. Mahnke W. Ertmer, I. Geisel, J. Peise, S. Coleman I. Apellaniz L. Santos Hannover I.L. Egusquiza Bilbao

53 Summary Detection of multipartite entanglement close to Dicke states, by measuring collective quantities only. The condition detects all entangled states that can be detected based on the measured quantities (i.e., it is optimal). Vitagliano, Apellaniz, Egusquiza, GT, PRA (2014). Lücke, Peise, Vitagliano, Arlt, Santos, GT, Klempt, PRL 112, (2014) (synopsis at physics.aps.org, Revista Española de Física). THANK YOU FOR YOUR ATTENTION! FOR TRANSPARENCIES, PLEASE SEE