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arXiv:quant-ph/0604147 · Communications in Theoretical Physics

Necessary and Sufficient Conditions of Separability for Multipartite Pure States

Dafa Li, Xiangrong Li, Huang Hongtao, Xinxin Li

Published May 1, 2008• 7 citations
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Abstract

In this paper we present a necessary and sufficient condition of separability for multipartite pure states and variants of it. These conditions are very simple and calculable, and they do not require Schmidt decomposition (for two subsystems) or tracing out operations. We also give a necessary condition for a local unitary equivalence class for a bipartite system in terms of the determinant of the matrix of amplitudes.

DOI
10.1088/0253-6102/49/5/29

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In this paper we present a necessary and sufficient condition of separability for multipartite pure states and variants of it. These conditions are very simple and calculable, and they do not require Schmidt decomposition (for two subsystems) or tracing out operations. We also give a necessary condition for a local unitary equivalence class for a bipartite system in terms of the determinant of the matrix of amplitudes.
In this report, we discuss a necessary and sufficient condition for the existence of Schmidt decomposition of a pure state in a multipartite Hilbert space. Schmidt decomposition exists for any pure state in a bipartite Hilbert space. It is extremely useful since it reduces the number of terms needed to represent a pure state using orthonormal bases of the respective parts. However, for 3 or more partite Hilbert spaces, Schmidt decomposition does not exist in general for an arbitrary pure state. In this thesis, we derive a condition that states whether a state in an n-partite Hilbert space is Schmidt-decomposable or not, i.e., whether we can represent it through the orthonormal bases of the respective parts using less number of terms that is required in general.
In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let MMM be the matrix of the amplitudes of ∣ψ⟩\ketψ∣ψ⟩, we prove MMM is a compact operator. We also prove ∣ψ⟩\ketψ∣ψ⟩ is separable if and only if MMM is a bounded linear operator with rank 1, that is the image of MMM is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.
We show that pure states of multipartite quantum systems are multiseparable (i.e., give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite-bound entangled state having a positive partial transpose is tri-inseparable, i.e., has none of its three bipartite partial traces separable.
In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of(D-1) pure states, whereDis the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.