Entanglement topology of two qubits

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.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.

arXiv:1007.1749International Journal of Modern Physics BMar 20, 201214 citations

Entanglement sudden-death time: a geometric quantity

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

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.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.

arXiv:1309.3555Sep 13, 20130 citations

Geometrical meaning of two-qubit entanglement and its symmetries

Helena Braga, Simone Souza, Salomon Sylvain Mizrahi

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- $7$ 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.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-$7$ 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.

Physical Review AApr 16, 20107 citations

Asymptotic entanglement dynamics and geometry of quantum states

Raphael C. Drumond, Marcelo Terra Cunha

A given dynamics for a composite quantum system can exhibit several distinct properties for the asymptotic entanglement behavior, such as entanglement sudden death, asymptotic death of entanglement, sudden birth of entanglement, etc. A classification of the possible situations was given by Terra Cunha (2007 New J. Phys. 9 237) but for some classes there were no known examples. In this work, we give a better classification for the possible relaxing dynamics in light of the geometry of their set of asymptotic states and give explicit examples for all the classes. Although the classification is completely general, in the search for examples it is sufficient to use two qubits with dynamics given by differential equations in the Lindblad form (some of them are non-autonomous). We also investigate, in each case, the probabilities of finding each possible behavior for random initial states.A given dynamics for a composite quantum system can exhibit several distinct properties for the asymptotic entanglement behavior, such as entanglement sudden death, asymptotic death of entanglement, sudden birth of entanglement, etc. A classification of the possible situations was given by Terra Cunha (2007 New J. Phys. 9 237) but for some classes there were no known examples. In this work, we give a better classification for the possible relaxing dynamics in light of the geometry of their set of asymptotic states and give explicit examples for all the classes. Although the classification is completely general, in the search for examples it is sufficient to use two qubits with dynamics given by differential equations in the Lindblad form (some of them are non-autonomous). We also investigate, in each case, the probabilities of finding each possible behavior for random initial states.

arXiv:0809.4445Journal of Physics A Mathematical and TheoreticalJun 24, 200918 citations

Dynamics of two qubits in common environment

E. N. Bratus’, Leonid Andreevich Pastur

We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment. We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso- and even nanosize. We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition. We then work out an analog of the Born–Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord. We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits. In particular, we find the vanishing of the entanglement (Entanglement Sudden Death) at finite moments and its revivals (Entanglement Sudden Birth). Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment. We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso- and even nanosize. We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition. We then work out an analog of the Born–Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord. We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits. In particular, we find the vanishing of the entanglement (Entanglement Sudden Death) at finite moments and its revivals (Entanglement Sudden Birth). Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.

Reviews in Mathematical PhysicsApr 28, 20203 citations

Coherence and entanglement in a two-qubit system

Miguel Orszag, Maritza Hernandez

Entanglement is a fundamental concept in quantum mechanics. In this review, we study various aspects of coherence and entanglement, illustrated by several examples. We relate the concepts of loss of coherence and disentanglement, via a model of two two-level atoms in different types of reservoir, including cases of both independent and common baths. Finally, we relate decoherence and disentanglement, by focusing on the sudden death of the entanglement and the dependence of the death time with the distance of our initial condition from the decoherence-free subspace. In particular, we study the sudden death of the entanglement in a two-atom system with a common reservoir.Entanglement is a fundamental concept in quantum mechanics. In this review, we study various aspects of coherence and entanglement, illustrated by several examples. We relate the concepts of loss of coherence and disentanglement, via a model of two two-level atoms in different types of reservoir, including cases of both independent and common baths. Finally, we relate decoherence and disentanglement, by focusing on the sudden death of the entanglement and the dependence of the death time with the distance of our initial condition from the decoherence-free subspace. In particular, we study the sudden death of the entanglement in a two-atom system with a common reservoir.

Advances in Optics and PhotonicsMar 17, 201036 citations

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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.