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arXiv:quant-ph/0601044 · Physical Review A

Another convex combination of product states for the separable Werner state

Hiroo Azuma, Masashi Ban

Published Mar 15, 2006• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.73.032315

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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.
In general, the calculation of robustness of entanglement for the mixed entangled quantum states is rather difficult to handle analytically. Using the convex semi-definite programming method, we claculate exactly the robustness of entanglement of some mixed entangled quantum states such as the generic two-qubit state in the Wootters basis, 2 ⊗ 2 Bell decomposable (BD) states, iso-concurrence decomposable states, 2 ⊗ 3 Bell decomposable states, d ⊗ d Werner and isotropic states, a one parameter 3 ⊗ 3 state and a multi-partite isotropic state. The results are in agreement with those of 2 ⊗ 2 density matrices and have already been calculated in Refs. 1 and 2. Also, an analytic expression is given for separable states that wipe out all the entanglements. It is further shown that they are on the boundary of separable states, as pointed out in Ref. 3.
Abstract In the study of optical quantum information, quantum teleportation can be realized by preparing multiphoton entangled states. Entanglement is a special quantum correlation, and there is also a quantum correlation in separable quantum States, which comes from the measurement of quantum states. Convex combinations of arbitrary quantum states are still quantum states. From the spectral decomposition of a quantum state, an ensemble formed by a complete eigenvector system can be determined, which corresponds to an orthogonal projection operator family with rank 1, and the convex combination of the operator family will correspond to a quantum state. The characterization of quantum correlation is an important issue in the study of quantum information theory. A Quantum state that remains unchanged under local orthogonal projection is a classical correlation state, which is not affected by decoherence. Based on the standard orthogonal basis, this paper considers the correlation of the quantum states formed by the convex combination of the operator family. According to the separability and entanglement of the ground state and the mutual dissimilarity of the corresponding non-zero eigenvalues, it is concluded that the convex combination of the operator family formed by the separable standard orthogonal basis in the composite system must be a classical correlation state. When the entangled pure state is an eigenvector, it is a quantum correlation state if the corresponding eigenvalue is a non-zero single value. In the case of multiple roots, we can illustrate that both are possible. For the correlation of unitary evolution of quantum states under different standard orthogonal bases, the evolution law is revealed by considering important quantum gates.
The 'Werner gap' is the range of relevant parameters characterizing a quantum state for which it is both entangled and admits a local hidden variable model. Werner showed that the gap becomes maximal for entanglement mixed with white noise if subsystems have infinitely many levels. Here we study pure entangled states mixed with simple coloured noise modelled as a single pure product state. We provide an explicit local hidden variable model for quantum correlations of some states of this family and provide hints that there is probably a model for all quantum predictions. This demonstrates essentially a maximal Werner gap already for just two qubits. Additionally to its fundamental interest, the study has implications for quantum computation and communication. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
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.
Abstract We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of ‘polygon diagram’ Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.