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arXiv:quant-ph/9806011 · Acta Physica Slovaca Reviews and Tutorials

Entanglement and pseudomixtures

Maciej Kuna

Published Jun 3, 1998• 1 citation
arXivHTMLPDF

Abstract

In a recent paper Sanpera et al. have shown, that for the simplest binary composite systems any density matrix can be described in terms of only product vectors. The purpose of this note is to show that posibillity of decomposing any state as pseudomixtures does not depend on dimension of the subsystems.
DOI
10.48550/arxiv.quant-ph/9806011

Related papers

ENTANGLEMENT AND PSEUDOMIXTURES 1

Maciej Kuna

In a recent paper Sanpera et al. have shown [1], that for the simplest binary composite systems any density matrix can be described in terms of only product vectors. The purpose of this note is to show that posibillity of decomposing any state as pseudomixtures does not depend on dimension of the subsystems. In a composite quantum system one can find states that involve different degrees of correlation between subsystems. From the physical point of view the most important characterization of such states is related to the nature of these correlations. Separable states involve correlations which can be explained by classical models [2]. The notion of inseparability or entanglement is related to specific relations existing only in pure quantum systems. Recently a general characterization of separable states was given in the language of vectors from natural cone of the Tomita-Takesaki theory, and quantum origin of this notion was explained [3]. Quantum character of entaglement plays a crucial role in quantum communication [4], cryptography [5], and quantum computation [6]. A state of a composite quantum system is inseparable if it cannot by represented as a convex mixture of tensor products

•Jan 1, 1998•0 citations

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arXiv:quant-ph/0304019•Journal of Physics A Mathematical and General•Feb 11, 2004•10 citations
In a recent paper Sanpera et al. have shown [1], that for the simplest binary composite systems any density matrix can be described in terms of only product vectors. The purpose of this note is to show that posibillity of decomposing any state as pseudomixtures does not depend on dimension of the subsystems. In a composite quantum system one can find states that involve different degrees of correlation between subsystems. From the physical point of view the most important characterization of such states is related to the nature of these correlations. Separable states involve correlations which can be explained by classical models [2]. The notion of inseparability or entanglement is related to specific relations existing only in pure quantum systems. Recently a general characterization of separable states was given in the language of vectors from natural cone of the Tomita-Takesaki theory, and quantum origin of this notion was explained [3]. Quantum character of entaglement plays a crucial role in quantum communication [4], cryptography [5], and quantum computation [6]. A state of a composite quantum system is inseparable if it cannot by represented as a convex mixture of tensor products
In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B , is associated with a tensor product space HA ? HB . Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems. As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement , which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing. Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al . [80] and Plenio et al . [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.
Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.
We point out that density matrices can only be used to describe quantum states, so the entanglement contained in a density matrix is just quantum entanglement. This means a bipartite state described by a density matrix contains quantum entanglement, unless the density matrix has the form ρAB=ρA⊗ρBρ_{AB}=ρ_A\otimes ρ_BρAB​=ρA​⊗ρB​.
It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of a product set) on the separable–entangled boundary, the optimal Lewenstein–Sanpera (LS) decomposition (with respect to an arbitrary separable set) can be obtained via a direct optimization procedure for a generic entangled density matrix. On the basis of this, we obtain the optimal LS decomposition for some bipartite systems such as 2 ⊗ 2 and 2 ⊗ 3 Bell decomposable (BD) states, a generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one-parameter and three-parameter local operations and classical communications (LOCC), d ⊗ d Werner and isotropic states and a one-parameter 3 ⊗ 3 state. We also obtain the optimal decomposition for multi-partite isotropic states. It is shown that in all 2 ⊗ 2 systems considered here the average concurrence of the decomposition is equal to the concurrence. We also show that for some 2 ⊗ 3 Bell decomposable states, the average concurrence of the decomposition is equal to the lower bound of the concurrence of the state presented recently in Lozinski et al (2003 Preprint quant-ph/0302144), so an exact expression for concurrence of these states is obtained. It is also shown that for a d ⊗ d isotropic state where decomposition leads to a separable and an entangled pure state, the average I-concurrence of the decomposition is equal to the I-concurrence of the state.