We present a description of finite-dimensional quantum entanglement based on a study of the space of all convex decompositions of a given density matrix. On this space we construct a system of real polynomial equations describing separable states. We further study this system using methods of statistical mechanics. As an example, we finally apply our techniques to Werner states of two qubits and obtain a sufficient criterion for separability.
We present a description of finite-dimensional quantum entanglement based on a study of the space of all convex decompositions of a given density matrix. On this space we construct a system of real polynomial equations describing separable states. We further study this system using methods of statistical mechanics. As an example, we finally apply our techniques to Werner states of two qubits and obtain a sufficient criterion for separability.
Abstract This chapter gives a detailed overview of entanglement for pure and mixed states. We begin with the Schmidt-decomposition theorem, which allows us to associate the subsystem mixedness of pure state with entanglement. We discuss the purification of mixed quantum states, the convexity of the set of separable states, and entanglement as a form of correlation. As an example we examine the Bloch decomposition for two qubits. For the detection of entanglement the Peres-Horodecki criterion is introduced, before we analyse the relation between entanglement and non-locality via the CHSH-Operator Criterion and the family of Werner states. We further consider Tsirelson’s bound and hidden non-locality. We then discuss important mathematical tools for entanglement detection, including the positive-map theorem, the Størmer-Woronowicz theorem, and the reduction criterion. In this context we examine the family of isotropic states. Finally, the Weyl states provide an example for which we study the geometry of two-qubit entanglement
In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study related issues such as classification of pure and mixed states, Von Neumann entropy, separability of multipartite quantum states and quantum operations in terms of the graphs associated with quantum states. In order to address the separability and entanglement questions using graphs, we introduce a modified tensor product of weighted graphs, and establish its algebraic properties. In particular, we show that Werner's definition (Werner 1989 Phys. Rev. A 40 4277) of a separable state can be written in terms of graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to Braunstein et al (2006 Phys. Rev. A 73 012320) to a class of weighted graphs with real weights. We have given some criteria for the Laplacian associated with a weighted graph to be positive semidefinite.
A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral
Quantum entanglement is a unique correlation phenomenon in quantum mechanics, and the measurement of quantum entanglement plays an important role in quantum computing and quantum communication. Many mainstream entanglement criteria and measurement methods currently known have shortcomings in certain aspects, such as not being sufficient or necessary conditions for entanglement, or only being effective in simple cases such as 2-qubits or pure states. In this work, we will propose a scheme for measuring quantum entanglement, which starts with treating the set of quantum separable states as a convex hull of quantum separable pure states, and analyzes the properties of the convex hull to obtain a new form of entanglement measurement. Although a large amount of data is required in the measurement process, this method is not only applicable to 2-qubit quantum states, but also a entanglement measurement method that can be applied to any dimension and any fragment. We will provide several examples to compare their results with other entanglement metrics and entanglement determination methods to verify their feasibility.
We consider low-rank density operators $\ensuremath{\varrho}$ supported on a $M\ifmmode\times\else\texttimes\fi{}N$ Hilbert space for arbitrary M and N (M<N), and with a positive partial transpose (PPT) ${\ensuremath{\varrho}}^{{T}_{A}}>~0.$ For rank $r(\ensuremath{\varrho})<~N$ we prove that having a PPT is necessary and sufficient for $\ensuremath{\varrho}$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\ensuremath{\varrho}$ and ${\ensuremath{\varrho}}^{{T}_{A}}$ satisfies $r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})<~2MN\ensuremath{-}M\ensuremath{-}N+2.$ This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.