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arXiv:1309.3555

Entanglement sudden-death time: a geometric quantity

Ramsés J. Sánchez, Esteban Isasi, Douglas F. Mundarain

Published Sep 13, 2013• 0 citations
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Abstract

We study the entanglement evolution of the set of Bell diagonal states for a two-qubit system coupled to two independent vacuum noise sources. This set can be represented geometrically as the set of points inside a tetrahedron in a three-dimensional Euclidean space and contains the maximally entangled states for bipartite systems. We show that the set of entangled Bell diagonal states can be divided into two bounded subsets in this representation: states that evolve into separable states in a finite time and states that lose their entanglement asymptotically. Additionally, we find that the finite time in which the Bell diagonal states lose their entanglement depends only on the distances from their position in the three-dimensional representation to the boundaries of both, the set of separable states and the set of states that remains always entangled.

DOI
10.48550/arxiv.1309.3555

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We study the entanglement evolution of the set of Bell diagonal states for a two-qubit system coupled to two independent vacuum noise sources. This set can be represented geometrically as the set of points inside a tetrahedron in a three-dimensional Euclidean space and contains the maximally entangled states for bipartite systems. We show that the set of entangled Bell diagonal states can be divided into two bounded subsets in this representation: states that evolve into separable states in a finite time and states that lose their entanglement asymptotically. Additionally, we find that the finite time in which the Bell diagonal states lose their entanglement depends only on the distances from their position in the three-dimensional representation to the boundaries of both, the set of separable states and the set of states that remains always entangled.
In this work we present an explicit geometric classification of the set Bell-diagonal in terms of the asymptotic behaviour of a system of two qubits evolving in the presence of a pair of independent thermal baths at zero temperature. We show that the set of initial Bell-diagonal states is separated in three bounded and simply-connected sets exhibiting: asymptotic entanglement dead, sudden entanglement dead and separability. The notion of entanglement dead time is related to the distance in the three dimensional geometric representation.
The dynamics of a two-qubit system is considered with the aim of a general categorization of the different ways in which entanglement can disappear in the course of the evolution, e.g., entanglement sudden death. The dynamics is described by the function ~n(t), where ~n is the 15-dimensional polarization vector. This representation is particularly useful because the components of ~n are direct physical observables, there is a meaningful notion of orthogonality, and the concurrence C can be computed for any point in the space. We analyze the topology of the space S of separable states (those having C = 0), and the often lower-dimensional linear dynamical subspace D that is characteristic of a specific physical model. This allows us to give a rigorous characterization of the four possible kinds of entanglement evolution. Which evolution is realized depends on the dimensionality of D and of D \cap S, the position of the asymptotic point of the evolution, and whether or not the evolution is distance-Markovian, a notion we define. We give several examples to illustrate the general principles, and to give a method to compute critical points. We construct a model that shows all four behaviors.
The dynamics of a two-qubit system is considered with the aim of a general categorization of the different ways in which entanglement can disappear in the course of the evolution, e.g., entanglement sudden death. The dynamics is described by the function n(t), where n is the 15-dimensional polarization vector. This representation is particularly useful because the components of n are direct physical observables, there is a meaningful notion of orthogonality, and the concurrence C can be computed for any point in the space. We analyze the topology of the space S of separable states (those having C = 0), and the often lower-dimensional linear dynamical subspace D that is characteristic of a specific physical model. This allows us to give a rigorous characterization of the four possible kinds of entanglement evolution. Which evolution is realized depends on the dimensionality of D and of D∩S, the position of the asymptotic point of the evolution, and whether or not the evolution is "distance-Markovian", a notion we define. We give several examples to illustrate the general principles, and to give a method to compute critical points. We construct a model that shows all four behaviors.
For pure and mixed two-qubit states we present an analysis based on symmetries of vectors and matrices associated to the density operator. These symmetries are revealed by doing reflection operations on the vectors and the matrices by Cartesian planes. Then we consider states whose $4\ifmmode\times\else\texttimes\fi{}4$ matrices belong to the D-777 manifold class and introduce a new set of parameters. This procedure allows us to establish the Peres-Horodecki separability criterion in terms of a squared distance having a Minkowski metric. Thereafter, defining a phase space in terms of those parameters, we identify two regions, a cone and elsewhere. Separable states are represented by points located within the cone or on its surface (the border), while elsewhere the points stand for the entangled states. For a system evolving in time, if its state is initially entangled, then depending on the values of its parameters and on the nature of the interactions with the environment, the trajectory in the phase space may cross the border at a finite time, going from the entangledlike to the separablelike region and vice-versa. As a particular feature, crossings in one direction stand for the phenomenon of sudden death of entanglement, while crossings in the opposite direction hint for the sudden birth of entanglement. Our method dispenses with operations on the density matrix; it is sufficient to write the expressions directly in terms of matrix entries. Some illustrative examples are worked out.
This paper will analyze the impact of “stochastic noises” on open quantum systems, also known as systems exposed to their environment. In the real world, entangled systems are often subject to stochastic environmental noises that result in decoherence and corruption of their quantum properties. This poses a significant challenge for the implementation of quantum entanglement in fields such as quantum computing and quantum informatics. We will describe the dynamics and evolution of these entangled systems as they decay from a maximally entangled state to a separated state. These dynamics can be highly sensitive to the type and strength of noise, leading to a wide variety of decay behaviors. A special phenomenon we encounter is Entanglement Sudden Death (ESD), where entanglement dies inexplicably within a short period of time. We will also analyze the time-dependent distance from the separability boundary, denoted Λ, within these evolving systems.
We investigate the phenomenon of sudden death of entanglement in a high-dimensional bipartite system subjected to dissipative environments with an arbitrary initial pure entangled state between two fields in the cavities. We find that in a vacuum reservoir, the presence of the state where one or more than one (two) photons in each cavity are present is a necessary condition for the sudden death of entanglement. Otherwise entanglement remains for infinite time and decays asymptotically with the decay of individual qubits. For pure two-qubit entangled states in a thermal environment, we observe that sudden death of entanglement always occurs. The sudden death time of the entangled states is related to the number of photons in the cavities, the temperature of the reservoir and the initial preparation of the entangled states.