Separability, entanglement, and full families of commuting normal matrices (original) (raw)

On Entanglement and Separability

We propose two necessary sufficient (NS) criteria to decide the separability of quantum states. They follow from two independent ideas: i) the Bloch-sphere-like-representation of states and ii) the proportionality of lines (rows, columns etc.) of certain multimatrix [1] associated with states. The second criterion proposes a natural way to determine the possible partial (or total, when possible) factorization of given multipartite state and in a sense can be used to determine the structure of the entanglement. We also introduce three entanglement measures based on the proposed new characterizations of entanglement. At last we discuss the second criterion mentioned above in the language of density matrix which is an inevitable language especially for mixed states.

An analytic approach to the problem of separability of quantum states based upon the theory of cones

Quantum Information Processing, 2011

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)C ρ + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice C ρ = M 1 ⊗ M 2 we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ 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. We argue that this approach is not exhausted with the first simple choices included herein.

A Cone Approach to the Quantum Separability Problem

Arxiv preprint arXiv: …, 2010

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)C ρ + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice C ρ = M 1 ⊗ M 2 we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ 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.

Separability and Entanglement of Composite Quantum Systems

Physical Review Letters, 1998

We show that all density operators of 2×N-dimensional quantum systems that remain invariant after partial transposition with respect to the first system are separable. Based on this criterion, we derive a sufficient separability condition for general density operators in such quantum systems. We also give a simple proof of the separability criterion in 2 × 2dimensional systems [A.

Separable and entangled states of composite quantum systems; Rigorous description

arXiv (Cornell University), 1997

We present a general description of separable states in Quantum Mechanics. In particular, our result gives an easy proof that inseparabitity (or entanglement) is a pure quantum (noncommutative) notion. This implies that distinction between separability and inseparabitity has sense only for composite systems consisting of pure quantum subsystems. Moreover, we provide the unified characterization of pure-state entanglement and mixed-state entanglement.

Mixed State Entanglement Measures for Intermediate Separability

Arxiv preprint arXiv: …, 2009

To determine whether a given multipartite quantum state is separable with respect to some partition we construct a family of entanglement measures R_m. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Further, we provide an analytically computable lower bound for R_m via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for each intermediate separability.

SEPARABILITY OF PURE N-QUBIT STATES: TWO CHARACTERIZATIONS

International Journal of Foundations of Computer Science, 2003

Given a pure state ψ N N ∈H of a quantum system composed of n qubits, where H N is the Hilbert space of dimension N n = 2 , this paper answers two questions: what conditions should the amplitudes in ψ N satisfy for this state to be separable (i) into a tensor product of n qubit states ψ ψ ψ 2 0 2 1 2 1 ⊗ ⊗ ⊗ − ... n , and (ii), into a tensor product of 2 subsystems states ψ ψ P Q ⊗ with P p = 2 and Q q = 2 such that p q n + = ? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. These conditions bear some relation with entanglement measures, and a number of more refined questions about separability in n qubit systems can be studied on the basis of these results.

There exist infinitely many kinds of partial separability/entanglement

Journal of Mathematical Physics, 2022

In tri-partite systems, there are three basic biseparability, A- BC, B- CA, and C- AB, according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three-qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members to see that there are infinitely many kinds of three-qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three-qubit Greenberger–Horne–Zeilinger diagonal states distinguishing those convex sets in the chain.

Entanglement or separability: the choice of how to factorize the algebra of a density matrix

The European Physical Journal D, 2011

Quantum entanglement has become a resource for the fascinating developments in quantum information and quantum communication during the last decades. It quantifies a certain nonclassical correlation property of a density matrix representing the quantum state of a composite system. We discuss the concept of how entanglement changes with respect to different factorizations of the algebra which describes the total quantum system. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this free choice of factorization, which for an experimentalist is often naturally fixed. For theorists it offers an extension of the interpretations and is adequate to generalizations, as we point out in the examples of quantum teleportation and entanglement swapping.