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International Journal of Quantum Information

SEPARABILITY: A NEW APPROACH FROM THE CONIC STRUCTURE OF POSITIVE OPERATORS

David Cienfuegos Salgado, J. L. Sánchez‐Gómez, Miguel Ferrero

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We exploit the cone structure of unnormalized quantum states to reformulate the separability problem. Firstly a convex combination of every quantum state ρ in terms of a state Cρ with the same rank and another one Eρ with lower rank is perfomed, with weights 1 − λρ and λρ, respectively. Secondly a scalar [Formula: see text] is computed. Then ρ is separable if, and only if, [Formula: see text]. The computation of [Formula: see text] has been undergone under the simplest choice for Cρ as a product matrix and Eρ being a pure state, valid for any bipartite and multipartite system in arbitrary dimensions. A necessary condition is also formulated when Eρ is not pure in the bipartite case.

DOI
10.1142/s021974991100737x

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We exploit the cone structure of unnormalized quantum states to reformulate the separability problem. Firstly a convex combination of every quantum state ρ in terms of a state Cρ with the same rank and another one Eρ with lower rank is perfomed, with weights 1 − λρ and λρ, respectively. Secondly a scalar [Formula: see text] is computed. Then ρ is separable if, and only if, [Formula: see text]. The computation of [Formula: see text] has been undergone under the simplest choice for Cρ as a product matrix and Eρ being a pure state, valid for any bipartite and multipartite system in arbitrary dimensions. A necessary condition is also formulated when Eρ is not pure in the bipartite case.
Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as ρ=(1−λ)Cρ+λEρρ=(1-λ)C_ρ+λE_ρρ=(1−λ)Cρ​+λEρ​, where CρC_ρCρ​ is a separable matrix whose rank equals that of ρρρ and the rank of EρE_ρEρ​ is strictly lower than that of ρρρ. With the simple choice Cρ=M1⊗M2C_ρ=M_{1}\otimes M_{2}Cρ​=M1​⊗M2​ we have a necessary condition of separability in terms of λλλ, which is also sufficient if the rank of EρE_ρEρ​ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication (SLOCC). We argue that this approach is not exhausted with the first simple choices included herein.
We show how the separability problem is dual to that of decomposing any given matrix into a conic combination of rank-one partial isometries, thus offering a duality approach different to the positive maps characterization problem. Several inmediate consequences are analyzed: (i) a sufficient criterion for separability for bipartite quantum systems, (ii) a complete solution to the separability problem for pure states also of bipartite systems independent of the classical Schmidt decomposition method and (iii) a natural generalization of these results to multipartite systems.
We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in C d ⊗ C d (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for d ≤ 4 . Furthermore, for d ≥ 5 , we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on N -partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.
It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.
Let H 1 , H 2 , …, H n , K ( n ⩾ 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state ρ in H 1 ⊗ H 2 ⊗⋅⋅⋅⊗ H n ⊗ K is fully separable if and only if (Λ⊗Id)ρ ⩾ 0 for all (finite-rank) LPP elementary operators Λ from into . Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.