Constraints on the uncertainties of entangled symmetric qubits

Abstract

We derive necessary and sufficient inseparability conditions imposed on the variance matrix of symmetric qubits. These constraints are identified by examining a structural parallelism between continuous variable states and two-qubit states. Pairwise entangled symmetric multiqubit states are shown here to obey these constraints. We also bring out an elegant local invariant structure exhibited by our constraints. © 2006 Elsevier B.V. All rights reserved. Entanglement is a central property of multipartite quantum systems as it forms the corner stone of all aspects of quantum information, communication, and computation [1]. The first task is to find a criterion if a given state is entangled or not. Peres-Horodecki inseparability criterion [2] viz., positivity under partial transpose (PPT) has been extremely fruitful in addressing this question and provides necessary and sufficient conditions for (2 × 2)-and (2 × 3)-dimensional systems. It is found that the PPT criterion is significant in the case of infinite-dimensional bipartite continuous variable (CV) states too. An important advance came about through an identification of how Peres-Horodecki criterion gets translated elegantly into the properties of the second moments (uncertainties) of CV states [3]. This results in restrictions [3,4] on the covariance matrix of an entangled bipartite CV state. In the special case of two-mode Gaussian states, where the basic entanglement properties are imbibed in the structure of its covariance matrix, the restrictions on the covariance matrix are found [3,4] to be necessary and sufficient for inseparability. Investigations on the structure of variance matrix have proved to be crucial in understanding the issue of entanglement in CV states [5][6][7] and * Corresponding author.E-mail address: arutth@rediffmail.com (A.R. Usha Devi).a great deal of interest has been catching up in experimentally accessible, simple conditions of inseparability involving higher order moments [8][9][10]. In a parallel direction, growing importance is being evinced towards quantum correlated macroscopic atomic ensembles [11][12][13][14][15]. In the last few years experimental generation of entangled multiqubit states in trapped-ion systems [16][17][18][19], where individual particles can be manipulated, has been accomplished, giving new hopes for scalable quantum information processing.In the present Letter, we explore a structural parallelism in CV states and two-qubit systems by constructing covariance matrix of the latter. We show that pairwise entanglement between any two-qubits of a symmetric N -qubit state is completely characterized by the off-diagonal block of the two-qubit covariance matrix. We establish the inseparability constraints satisfied by the covariance matrix and these are identified to be equivalent to the generalized spin squeezing inequalities [20] for two-qubit entanglement. The interplay between two basic principles viz., the uncertainty principle and the nonseparability gets highlighted through the restriction on the ...

Local sum uncertainty relations for angular momentum operators of bipartite permutation symmetric systems

“…We show that the angular momentum LSUR, which places lower bound on the set of all bipartite separable states, gets violated if and only if the covariance matrix of the two-qubit reduced system of the N-qubit permutation symmetric state is not positive semidefinite. Since it has been shown [47][48][49] that the covariance matrix negativity serves as a necessary and sufficient condition for entanglement in a two-qubit symmetric system, our result establishes a one-to-one connection between violation of the LSUR and pairwise entanglement.…”

“…Permutation symmetric N-qubit states draw attention due to their experimental feasibility and for the mathematical simplicity offered by them. [45][46][47][48] This class of states belongs to the d = 2 j + 1 = N + 1 dimensional subspace of the 2 N dimensional Hilbert space, which corresponds to the maximum value j = N/2 of angular momentum of N-qubit system. Invariance under exchange of particles labeled by α, β in a multiparty state ρ sym is exhibited by the property…”

Dynamics of Pair of Entangled Spin-1/2 Particles and Quantification of the Dynamics in terms of Correlations

“…) is the polarization of the frst spin and (Y 1 , Y 2 , Y 3 ) is the polarization of the second spin. Te coefcients X a Y b can be interpreted as correlations between the constituent qubits [10]. It should be noted that the indices 1, 2 in the above equation indicate that the constituents are polarized along diferent directions.…”

“…Terefore, the second-order solution is exact [4,5]. Assuming the initial state ρ(0) as the pure state projection operator (10). Comparing the stochastic diferential equations (SDEs) on both sides of equation ( 25), we get…”

Relation between nonlocality and contextuality for a biphoton

“…This state is fully described by eight parameters: three corresponding to local Bloch vector and five corresponding to the symmetric correlation tensor T [9]. Three out of five parameters of T correspond to off-diagonal terms.…”