Entanglement detection close to multi-qubit Dicke states in photonic experiments (review)

Transcription

1 Entanglement detection close to multi-qubit Dicke states in photonic experiments (review) G. Tóth 1,2,3 1 Theoretical Physics, University of the Basque Country UPV/EHU, Bilbao, Spain 2, Bilbao, Spain 3 Wigner Research Center for Physics, Budapest, Hungary Based on the following papers: GT, JOSAB 2007 GT, C. Knapp, O. Gühne, H. Briegel, PRL 2007 W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, GT, H. Weinfurter, PRL 2009 GT, M.W. Mitchell, NJP 2010 GT, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, PRL 2010 G. Vitagliano, P. Hyllus, I. Egusquiza, GT, PRL 2011, I. Urizar-Lanz, P. Hyllus, I. Egusquiza, GT, arxiv 2012; etc. Mátraháza Workshop, Hungary, May / 42

2

3 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 3 / 42

4 Why Dicke states are important? They naturally arise in symmetric systems. They are highly entangled and possess high level of multipartite entanglement. They are very useful in metrology.

5 Dicke states We will consider mostly symmetric Dicke states defined as ( ) 1 D (n) n N = N 2 k P k ( 1 n 0 (N n) ), where the summation is over all different permutations. For even N, D (n) N is the Dicke state with J z = 0. It appears also in Dicke s original paper as the most superradiant state. Another interesting state is the permutationally invariant singlet ϱ s j = 0, j z = 0, α j = 0, j z = 0, α. α

6 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 6 / 42

7 Genuine multipartite entanglement Definition A state is (fully) separable if it can be written as k p k ϱ (k) ϱ (k)... ϱ (k) 1 2 N. Definition A pure multi-qubit quantum state is called biseparable if it can be written as the tensor product of two multi-qubit states Ψ = Ψ 1 Ψ 2. Here Ψ is an N-qubit state. A mixed state is called biseparable, if it can be obtained by mixing pure biseparable states. Definition If a state is not biseparable then it is called genuine multi-partite entangled.

8 Genuine multipartite entanglement II The following 10-particle state is entangled 1 2 ( ) 0 8 but it is not 10-particle multipartite entangled. We can see only two entangled particles. For a N-qubit experiment, we need that all qubits are entangled with each other.

9 Convex sets for the multipartite case The idea of convex sets also work for the multi-qubit case: A state is biseparable if it can be obtained by mixing pure biseparable states. Genuine multipartite entangled states Witness Biseparable states Separable states

10 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 10 / 42

11 Entanglement close to symmetric Dicke states A state is genuine multipartite entangled if Tr( D (N/2) D (N/2) ϱ) > 1 N N 2 N N 1. The bound is close to 1 2 [ GT, JOSAB 2007 ] for large N. Only GHZ states and cluster states are known to have similar characteristics. Thus, symmetric Dicke states are suitable for creating genuine multipartite entanglement in many-body experiments. For other states, a larger fidelity is required that makes experiments more demanding. Eg., a state is entangled if Tr( D (1) N D(1) N ϱ) > N 1 N.

12 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 12 / 42

13 Complete set of the generalized spin squeezing criteria 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/2 (N 1)( J m ) 2, (N 1) [ ( J k ) 2 + ( J l ) 2] J 2 m + N(N 2)/4. where k, l, m takes all the possible permutations of x, y, z. [ GT, C. Knapp, O. Gühne, and H.J. Briegel, Phys. Rev. Lett ]

14 The polytope The previous inequalities, for fixed J x/y/z, describe a polytope in the J 2 x/y/z space. Separable states correspond to points inside the polytope. Note: Convexity comes up again! J 2 z 10 A z 5 B y 0 0 S 5 J 2 x B x B z 10 0 A x 5 A y 10 J 2 y

15 Detecting Dicke states Entanglement detection close to Dicke states. For separable states, J 2 x + J 2 y N/2 (N 1)( J z ) 2. The Dicke state D (N/2) has N J 2 x + J 2 y = N 2 ( J z ) 2 = 0. ( ) N 2 + 1, Thus, it maximally violates the inequality. Detection of m-partite entanglement close to Dicke states. Such entanglement is present if [ L.-M. Duan, Phys. Rev. Lett ] J 2 x + J2 y N(1/4 + ( J z ) 2 ) 1 > m.

16 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 16 / 42

17 Modified quantities for j > 1 2 For the j = 1 2 case, the spin squeezing inequalities were developed based on the first and second moments and variances of the such collective operators. For the j > 1 2 case, we define the modified second moment Ã2 k := A2 k (a (n) k )2 = a (n) k a(m) k and the modified variance ( A k ) 2 := ( A k ) 2 n n m n (a (n) k )2.

18 The inequalities for j> 1 2 with the angular momentum coordinates II For fully separable states of spin-j particles, all the following inequalities are fulfilled J 2 x + J 2 y + J 2 z Nj(Nj + 1), ( J x ) 2 + ( J y ) 2 + ( J z ) 2 Nj, J 2 k + J 2 l N(N 1)j 2 (N 1)( J m ) 2, (N 1) [ ( J k ) 2 + ( J l ) 2] J 2 m N(N 1)j 2, where k, l, m take all possible permutations of x, y, z. Violation of any of the inequalities implies entanglement.

19 Dicke states for j>1/2 Entanglement condition for Dicke states J 2 k + J 2 l N(N 1)j 2 (N 1)( J m ) 2. A state maximally violating the inequality is ( ) 1 D (n,j) n N = N 2 k P k ( + j N j (N n) ).

20 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 20 / 42

21 Symmetric Dicke states with J z = 0 are optimal for metrology The small parameter θ must be estimated by making measurements on the output state : ϱ U (θ )=exp( iaθ ) ϱ output Cramér-Rao bound with the quantum Fisher information F Q is θ F usual Q 1. [ϱ, A] [C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976); A. S. Holevo, Probabilistic and Statistical Aspect of Quantum Theory (North-Holland,

22 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 22 / 42

23 Properties of the Quantum Fisher information For obtaining entanglement conditions, it is sufficient to know that following two relations. 1 For a pure state ϱ, we have F[ϱ, J l ] = 4( J l ) 2 ϱ. 2 F[ϱ, J l ] is convex in the state, that is F[p 1 ϱ 1 + p 2 ϱ 2, J l ] p 1 F[ϱ 1, J l ] + p 2 F[ϱ 2, J l ]. From these two statements, it also follows that F[ϱ, J l ] 4( J l ) 2 ϱ. [C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976); A. S. Holevo, Probabilistic and Statistical Aspect of Quantum Theory (North-Holland, Amsterdam, 1982); S.L. Braunstein and C.M. Caves, Phys. Rev. Lett. 72, 3439 (1994); L. Pezzé and A. Smerzi, Phys. Rev. Lett. 102, (2009). ]

24 Quantum Fisher information and entanglement I Observation 1 For N-qubit separable states, the values of F Q [ϱ, J l ] for l = x, y, z are bounded as F Q [ϱ, J l ] N. The GHZ state maximally violates this with N 2. The D (N/2) state gives roughly half of the maximal value, still N very good. [L. Pezzé and A. Smerzi, Phys. Rev. Lett. 102, (2009).]

25 Quantum Fisher information and entanglement II Observation 2 For N-qubit separable states, the values of F Q [ϱ, J l ] for l = x, y, z are bounded as F Q [ϱ, J l ] 2N. l=x,y,z Both the GHZ state and the D (N/2) state maximally violate it N with N(N + 2). [GT, Phys. Rev. A (2012); P. Hyllus et al., Phys. Rev. A 85, (2012).]

26 Dicke states and metrology In summary, when a single metrological tasks is considered, is close to optimal. D (N/2) N When three metrological tasks are considered, D (N/2) is optimal. N

27 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 27 / 42

28 An experiment: Dicke state with photons 28 / 42 [ W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, GT, H. Weinfurter, PRL 2009 ]

29 An experiment: Dicke state with photons II 29 / 42 A photo of a real experiment (six-photon Dicke state, Weinfurter group, 2009):

30 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 30 / 42

31 Experiment: W-state with ions 31 / 42 Experimental observation of an 8-qubit W-state with trapped ions. H. Haeffner, W. Haensel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Koerber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, R. Blatt, Nature 438, (2005).

32 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 32 / 42

33 Singlet states Another interesting state is the permutationally invariant singlet ϱ s j = 0, j z = 0, α j = 0, j z = 0, α. α Entanglement condition for singlets. For separable states we have ( J x ) 2 + ( J y ) 2 + ( J z ) 2 Nj. [GT PRA 2004; GT and M.W. Mitchell, NJP 2010.] [Singlet of a two-mode:two-mode system: T.Sh. Iskhakov, I.N. Agafonov, M.V. Chekhova, and G. Leuchs, arxiv: ] [Singlet with fermions: J. Meineke, J.-P. Brantut, D. Stadler, T. Müller, H. Moritz & T. Esslinger, Nature Phys. (2012)]

34 Singlet states II For j = 1/2, there is a unique permutationally invariant singlet. [I. Urizar-Lanz, P. Hillus, I. Egusquiza, GT arxiv 2012.] For a given even number of particles N, the permutationally invariant singlet state can be expressed as and ϱ s = lim T 0 Tr e Jx 2+J2 y +J2 z T ( e J2 x +J2 y +J2 z T ϱ s Π k ( Ψ Ψ Ψ Ψ ) Π k, k where the two-particle singlet is Ψ = 1 2 ( ). )

35 Singlet states III Equal mixture of all chains of two-particle singlets:

36 Outline 1 Motivation Why Dicke states are important? 2 Dicke states and genuine multipartite entanglement Genuine multipartite entanglement Entanglement detection close to Dicke states 3 Spin squeezing inequalities for Dicke states Spin squeezing inequalities for particles with j=1/2 Spin squeezing inequalities for particles with j>1/2 4 Spin squeezing inequalities for Dicke states for higher spins 5 Metrology with Dicke states Basic task of metrology Entanglement conditions with the quantum Fisher information 6 Photonic experiments with Dicke states Dicke states with photons W-state with ions 7 Singlet states Properties of singlets Experiments with singlets 36 / 42

37 Experiments for creating many-body singlets with cold atomic ensembles Picture from M.W. Mitchell, ICFO, Barcelona.

38 Experiments for creating many-body singlets Singlets can be created by squeezing the white noise [GT and M.W. Mitchell, NJP 2010.] We squeezed first J x, then J y, then J z and obtain a singlet 2 squeezing J y squeezing J z 1.5 ξ s squeezing J x

39 Using singlets in metrology Singlet are invariant under the effect of homogenous magnetic fields. But are sensitive to field gradients. Thus, they can be used for differential magnetometry. [I. Urizar-Lanz, P. Hyllus, I. Egusquiza, GT, arxiv] [ N. Behbood, B. Dubost, M. Napolitano, M. Koschorreck, R. Sewell, G. Tóth, and M.W. Mitchell, CEWQO 2011] [ongoing experiment in the Mitchell group at ICFO, Barcelona]

40 Symemtric Dicke states vs. singlets Comparing the basic properties of the two states we obtain the following. J l J 2 x + J2 y J2 z H D (N) N/2 0 max. 0 (J2 x + J2 y ) singlet (Jx 2 + J2 y + J2 z ) F Q [ϱ, J l ] l=x,y,z F Q [ϱ, J l ] D (N) N/2 almost max. max. singlet 0 0

41 Group Philipp Hyllus Iñigo Urizar-Lanz Giuseppe Vitagliano Iagoba Appellaniz Research Fellow 2nd year Ph.D. Student 2nd year Ph.D. Student M.Sc. Student (advisor P. Hyllus) Topics Multipartite entanglement and its detection Metrology, cold gases Collaborating on experiments: Funding: Weinfurter group, Munich, (photons) Mitchell group, Barcelona, (cold gases) European Research Council starting grant , 5-6 people for 5 years CHIST-ERA QUASAR collaborative EU project Grants of the Spanish Government and the Basque Government

42 Summary We discussed the interesting properties of Dicke states. We considered the symmetric Dicke state with J z = 0 is one of the most interesting states. We also considered singlet states. THANK YOU FOR YOUR ATTENTION! OTKA