Entanglement detection by Bragg scattering

Abstract

We show how to measure the structural witnesses proposed in [P. Krammer et al., Phys. Rev. Lett. 103, 100502 (2009)] for detecting entanglement in a spin chain using photon scattering. The procedure, moreover, allows one to measure the two-point correlation function of the spin array. This proposal could be performed in existing experimental platforms realizing ion chains in Paul traps or atomic arrays in optical lattices.Multipartite entanglement plays a crucial role in various tasks of quantum information, such as in multiparty quantum secret sharing [1] and in the computational speed-up in quantum algorithms [2,3]. This motivates the great interest in detecting entanglement in many-body systems. An experimentally feasible method to achieve this task for spin systems was recently proposed in Ref.[4] and later tested in a quantum optical experiment [5]. Such a method is based on the construction of entanglement witnesses related to structure factors operators and it is suitable to detect entanglement for various families of multipartite entangled states, such as the Dicke states, without the need of any a priori knowledge on the Hamiltonian describing the physical system under consideration. In this paper we discuss specifically how this method can be implemented in ion chains or atomic arrays in optical lattices. In addition, we propose how to measure all elements of the structure factor matrix, thereby allowing one to reconstruct the two-body density matrix of the spin system.We first briefly review the structural witnesses method and define the structure factor matrix. Consider a chain of N spin 1/2 particles which are ordered in a onedimensional array at the positions r i (i = 1, . . . N ). The structural witness method is based on the measurement of the 3 × 3 matrix S(q), which is a function of the transfered wave vector q and whose elements are given by the static structure factorsIn the above expression S α i is the α component of the spin operator of particle i at position r i (α = x, y, z) and the average is taken over the initial state of the N spins, whose entanglement is to be detected. As shown in Ref. [4], for different values of q and different linear combinations of S αβ (q) it is possible to detect different families of multipartite entangled states. For example, with the choice q = 0 one can construct the witness operator W Dwhere 1 denotes the identity operator for the N -qubit system andŜ αβ (q) = i<j e iq·(r i −r j ) S α i S β j are the structure factor operators whose expectation values are given in Eq. (1). Whenever operatorŴ D gives a negative expectation value on a density matrix of the spin array ρ 0 , Tr{Ŵ D ρ 0 } < 0, we are guaranteed that the N -spin state under examination is entangled. The witness in Eq. (2) is suitable for detecting symmetric Dicke states with any number of excitations, namely its average value for symmetric Dicke states is always negative.More generally, the structural witness operators take the formŴwhere c α are real coefficients with The underlying detect...

Environment mediated multipartite and multidimensional entanglement

“…Our results can be tested experimentally, as scattering experiments have already proved to be useful in quantifying multipartite entanglement 28 . Structure factors are macroscopic quantities measurable in various periodic systems, e.g., Bragg scattering on ultracold atoms in optical lattices 29 or from ion chains 30 and neutron scattering from solid state systems. Furthermore, the structure factor, particularly the dynamic structure factor, is a central observable quantity in determining single-particle and collective electronic excitations in many-body systems 31,32 .…”

Quantifying entanglement with scattering experiments

“…Examples include optical Bragg scattering from ultracold atoms in optical lattices [34] or from ion chains [35] and neutron scattering from magnetic materials [36]. These observables may be written aŝ…”

“…We consider observables that arise in scattering experiments from N spins arranged on a lattice. Examples include optical Bragg scattering from ultracold atoms in optical lattices [34] or from ion chains [35] and neutron scattering from magnetic materials [36]. These observables may be written as…”

Witnessing entanglement in hybrid systems

“…A method to detect multipartite entanglement by means of structure factors was recently proposed [6] and successfully tested in a quantum optical experiment [7]. The method was later applied to detect entanglement in a spin chain using photon scattering [8]. Related work in condensed matter systems was reported in [9].…”

“…In the first scenario we focus, as in Refs. [6][7][8][9], on spin entanglement witnesses and we ask ourselves if and how the presence of other quantum degrees of freedom (the vibrational quanta in the trapped ion case) affects the detection of entanglement between spins. In the second scenario we identify some cases for which the generalised witnesses here introduced allow to detect hybrid entanglement.…”