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Physical Review A

Geometrical meaning of two-qubit entanglement and its symmetries

Helena Braga, Simone Souza, Salomon Sylvain Mizrahi

Published Apr 16, 2010• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.81.042310

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Journal of Physics Conference Series•Jul 8, 2011•0 citations
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×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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.
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres-Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.
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.
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.
Entanglement is a key resource in quantum information. It can be destroyed or sometimes created by interactions with a reservoir. In recent years, much attention has been devoted to the phenomena of entanglement sudden death and sudden birth, i.e., the sudden disappearance or revival of entanglement at finite times resulting from a coupling of the quantum system to its environment [1, 2, 3]. We investigate the evolution of the entanglement of noninteracting qubits coupled to reservoirs under monitoring of the reservoirs by means of continuous measurements. Because of these measurements, the qubits remain at all times in a pure state, which evolves randomly. To each measurement result (or "realization") corresponds a quantum trajectory in the Hilbert space of the qubits. We show that for two qubits coupled to independent baths subjected to local measurements, the average of the qubits' concurrence over all quantum trajectories is either constant or decays exponentially. The corresponding decay rate depends on the measurement scheme only. This result contrasts with the entanglement sudden death phenomenon exhibited by the qubits' density matrix in the absence of measurements. Our analysis applies to arbitrary quantum jump dynamics (photon counting) as well as to quantum state diffusion (homodyne or heterodyne detections) in the Markov limit. We discuss the best measurement schemes to protect the entanglement of the qubits. We also analyze the case of two qubits coupled to a common bath. Then, the average concurrence can vanish at discrete times and may coincide with the concurrence of the density matrix. The results explained in this article have been presented during the "Fifth International Workshop DICE2010" by the first author and have been the subject of a prior publication [4].