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arXiv:0803.1430 · Journal of Physics A Mathematical and Theoretical

The thermostatistical aspect of Werner-type states and quantum entanglement

Sumiyoshi Abe, A. R. Usha Devi, A. K. Rajagopal

Published Dec 23, 2009• 5 citations
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Abstract

A general Werner-type state is studied from two viewpoints: (i) an application of the dynamical interaction of the objective system with its environment, represented by a unital positive operator-valued measure (POVM), which ensures an increase of entropy, making the system evolve from an initial pure state to a mixed state of the Werner type, and (ii) the space of the objective system is constrained to have a given value of the fidelity. Then, the maximum entropy principle is shown to yield the Werner-type state as a canonical ensemble with a projector Hamiltonian. This novel observation is illustrated by examples of bipartite systems, the separability criteria of which are given in terms of the values of temperature. The present viewpoint may cast light on the relevance of thermostatistics to the physics of quantum entanglement. In addition, the POVM scheme presented here offers a way of experimentally generating the Werner-type thermal states.

DOI
10.1088/1751-8113/43/4/045303

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A general Werner-type state is studied from two viewpoints: (i) an application of the dynamical interaction of the objective system with its environment, represented by a unital positive operator-valued measure (POVM), which ensures an increase of entropy, making the system evolve from an initial pure state to a mixed state of the Werner type, and (ii) the space of the objective system is constrained to have a given value of the fidelity. Then, the maximum entropy principle is shown to yield the Werner-type state as a canonical ensemble with a projector Hamiltonian. This novel observation is illustrated by examples of bipartite systems, the separability criteria of which are given in terms of the values of temperature. The present viewpoint may cast light on the relevance of thermostatistics to the physics of quantum entanglement. In addition, the POVM scheme presented here offers a way of experimentally generating the Werner-type thermal states.
Dynamical AAA and BBB maps have been employed extensively by Sudarshan and co-workers to investigate open-system evolution of quantum systems. A canonical structure of the AAA map is introduced here. It is shown that this canonical AAA map enables us to investigate whether the dynamics is completely positive (CP) or not completely positive (NCP) in an elegant way and, hence, it subsumes the basic results on open-system dynamics. Identifying memory effects in open-system evolution is gaining increasing importance recently and, here, a criterion of non-Markovianity, based on the relative entropy of the dynamical state is proposed. The relative entropy difference of the dynamical system serves as a complementary characterization---though not related directly---to the fidelity difference criterion proposed recently. Three typical examples of open-system evolution of a qubit, prepared initially in a correlated state with another qubit (environment), and evolving jointly under a specific unitary dynamics---which corresponds to a NCP dynamical map---are investigated by employing both the relative entropy difference and fidelity difference tests of non-Markovianity. The two-qubit initial states are chosen to be (i) a pure entangled state, (ii) the Werner state, which exemplifies both entangled and separable states of qubits, depending on a real parameter, and (iii) a separable mixed state. Both the relative entropy and fidelity criteria offer a nice display of how non-Markovianity manifests itself in all three examples.
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We describe the evolution of the quantum entanglement in a system composed of two interacting bosonic modes immersed in a thermal reservoir, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. The evolution of entanglement is described in terms of the covariance matrix for Gaussian initial states. We calculate the logarithmic negativity and show that for separable initial squeezed thermal states entanglement generation may take place, for definite values of squeezing parameter, average photon numbers, temperature of the thermal bath, dissipation constant and the strength of interaction between the two modes. After its generation one can observe temporary suppressions and revivals of the entanglement. For entangled initial squeezed thermal states, entanglement suppression takes place, for all temperatures of the reservoir, and temporary revivals and suppressions of entanglement can be observed too. In the limit of infinite time the system evolves asymptotically to an equilibrium state which may be entangled or separable.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal environment. Using the Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a repeated collapse and revival of entanglement takes place. We determine the asymptotic Gaussian maximally entangled mixed states (GMEMS) and their corresponding asymptotic maximal logarithmic negativity.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we present a description of continuous-variable entanglement for a system consisting of two non-interacting modes embedded in a thermal environment. By using the Peres–Simon necessary and sufficient criterion for the separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place for non-zero temperatures of the environment. Only for zero temperature of the thermal bath, the initial entangled state remains entangled for finite times. We also show that, independent of its type, namely separable or entangled, the initial state evolves asymptotically to an equilibrium state that is always separable.
This dissertation describes the application of symmetry to multipartite quantum systems from the viewpoint of quantum information theory. It is written in the framework of abstract quantum information theory, i.e. without distinguishing the physical nature of the d-level systems considered. The results are mathematically rigorous and aim at investigating entanglement properties as well as operational properties of states of multipartite systems. The first chapter gathers the mathematical tools and concepts needed in the following chapters and can be omitted by the experienced reader. In chapter two we apply the concept of symmetry to multipartite systems and show how to construct families of symmetric states that can be described by few parameters only regardless of the size of single systems. Chapter three is devoted to the states of tripartite systems having Werner symmetry. We characterize their separability properties and analyze the strength of the known separability criteria. Furthermore we investigate their entanglement properties in terms of the relative entropy of entanglement, the entanglement monotone induced by the trace norm distance and the maximal violation of Bell inequalities. As a third aspect we analyze the possibility of embedding bipartite Werner states and explore the inner geometry of the manifold given by the statistical distance of two neighbouring tripartite Werner states. In the fourth chapter we introduce multipartite quantum data hiding as an application of multipartite Werner states. We prove the security of this protocol for any coalition and give various examples. The last chapter is concerned with the shared fidelity, its frustration and how it can be maximized in certain multipartite systems.