Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0703240 · Journal of Physics A Mathematical and Theoretical

A decomposition of separable Werner states

Razmik G. Unanyan, Hermann Kampermann, Dagmar Bruß

Published May 30, 2007• 9 citations
arXivHTMLPDF

Abstract

We derive an integral convex combination of product states for a range of separable Werner states. Our method consists of expanding the sought-after local density operators in terms of Wigner operators. For dimension d = 2, our decomposition holds for the whole separable range of Werner states, while for d > 2 it is valid for a subset of separable Werner states. We illustrate the general method with the explicit examples d = 2 and d = 3.

DOI
10.1088/1751-8113/40/24/f07

Related papers

Another convex combination of product states for the separable Werner state

Hiroo Azuma, Masashi Ban

In this paper, we write down the separable Werner state in a two-qubit system explicitly as a convex combination of product states, which is different from the convex combination obtained by Wootters' method. The Werner state in a two-qubit system has a single real parameter and varies from inseparable to separable according to the value of its parameter. We derive a hidden variable model that is induced by our decomposed form for the separable Werner state. From our explicit form of the convex combination of product states, we understand the following: The critical point of the parameter for separability of the Werner state comes from positivity of local density operators of the qubits.

arXiv:quant-ph/0601044•Physical Review A•Mar 15, 2006•12 citations

Another convex combination of product states for the separable Werner state (5 pages)

Hiroo Azuma

Physical Review A•Jan 1, 2006•0 citations

On the notion of entanglement in Hilbert spaces

Александр Семенович Холево, Максим Евгеньевич Широков, Reinhard F. Werner

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

Russian Mathematical Surveys•Apr 30, 2005•47 citations

Smallest state spaces for which bipartite entangled quantum states are separable

Hussain Anwar, Sania Jevtic, Oliver Rudolph +1 more

According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators.If we relax the requirement that the local operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators.This form of separability can be used to construct classical models and simulation methods when only a restricted set of measurements is available.With these motivations in mind, we ask what are the smallest sets of local operators such that a pure bipartite entangled quantum state becomes separable?We find that in the case of maximally entangled states there are many inequivalent solutions, including for example the sets of phase point operators that arise in the study of discrete Wigner functions.We therefore provide a new way of interpreting these operators, and more generally, provide an alternative method for constructing local hidden variable models for entangled quantum states under subsets of quantum measurements.

arXiv:1411.5907•New Journal of Physics•Sep 25, 2015•3 citations

A statistical-mechanical description of quantum entanglement

Jarosław K. Korbicz, F. Hulpke, Andreas Osterloh +1 more

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.

arXiv:0704.3357•Journal of Physics A Mathematical and Theoretical•Aug 11, 2008•4 citations

Note on Separability of the Werner states in arbitrary dimensions

Pittenger, A O, Rubin, M H

Recently, a great deal of progress has been made in proving sufficient conditions for Werner density matrices on the direct product space of n d-level systems (qudits). In this brief note we complete the process by proving the sufficient condition for general n and d.

•Feb 1, 2000•35 citations
We derive an integral convex combination of product states for a range of separable Werner states. Our method consists of expanding the sought-after local density operators in terms of Wigner operators. For dimension d = 2, our decomposition holds for the whole separable range of Werner states, while for d > 2 it is valid for a subset of separable Werner states. We illustrate the general method with the explicit examples d = 2 and d = 3.
In this paper, we write down the separable Werner state in a two-qubit system explicitly as a convex combination of product states, which is different from the convex combination obtained by Wootters' method. The Werner state in a two-qubit system has a single real parameter and varies from inseparable to separable according to the value of its parameter. We derive a hidden variable model that is induced by our decomposed form for the separable Werner state. From our explicit form of the convex combination of product states, we understand the following: The critical point of the parameter for separability of the Werner state comes from positivity of local density operators of the qubits.
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
According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators.If we relax the requirement that the local operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators.This form of separability can be used to construct classical models and simulation methods when only a restricted set of measurements is available.With these motivations in mind, we ask what are the smallest sets of local operators such that a pure bipartite entangled quantum state becomes separable?We find that in the case of maximally entangled states there are many inequivalent solutions, including for example the sets of phase point operators that arise in the study of discrete Wigner functions.We therefore provide a new way of interpreting these operators, and more generally, provide an alternative method for constructing local hidden variable models for entangled quantum states under subsets of quantum measurements.