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Physical Review A

Quantifying entanglement properties of qudit mixed states with incomplete permutation symmetry

Artur Barasiński, Mateusz Nowotarski

Published Apr 21, 2017• 7 citations
Publisher

Abstract

The characterization of entanglement properties in mixed states is important from both a theoretical and a practical point of view. While the estimation of entanglement of bipartite pure states is well established, for mixed states it is a considerably much harder task. The key elements of the mixed-state entanglement theory are given by the exact solutions which sometimes are possible for special states of high symmetry problems. In this paper, we present the exact investigation on the entanglement properties for a five-parameter family of highly symmetric two-qudit mixed states with equal but arbitrary finite local Hilbert space dimension. We achieve this by extensive analysis of various conditions of separability and the entanglement classification with respect to stochastic local operations and classical communication. Furthermore, our results can be used for an arbitrary state by proper application of the proposed twirling operator.

DOI
10.1103/physreva.95.042333

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The characterization of entanglement properties in mixed states is important from both a theoretical and a practical point of view. While the estimation of entanglement of bipartite pure states is well established, for mixed states it is a considerably much harder task. The key elements of the mixed-state entanglement theory are given by the exact solutions which sometimes are possible for special states of high symmetry problems. In this paper, we present the exact investigation on the entanglement properties for a five-parameter family of highly symmetric two-qudit mixed states with equal but arbitrary finite local Hilbert space dimension. We achieve this by extensive analysis of various conditions of separability and the entanglement classification with respect to stochastic local operations and classical communication. Furthermore, our results can be used for an arbitrary state by proper application of the proposed twirling operator.
We analyze the entanglement properties for the not-completely-permutation-symmetric states of quantum systems composed of two subsystems with an equal but arbitrary finite local Hilbert space dimension. We investigate both pure and mixed states with such a symmetry obtained by relaxing the symmetry requirement of the axisymmetric states. For such states we discuss the entanglement classification with respect to stochastic local operations and classical communication and establish the entanglement quantitatively by means of concurrence and negativity. In particular, we determine the separability criterion in the frame of various methods, including the kkk-positive map witness, optimal Schmidt-number witnesses, and entanglement measures.
The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that generic nnn-qubit symmetric states are also isolated, we consider particular families for which we can determine to be, on the contrary, endowed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. In the course of the study of LOCC transformations, we also characterize the local symmetries of symmetric states.
In this paper, we consider the local unitary classification of the class of qudit bipartite mixed states for which no information can be obtained locally. These states are represented by symmetrical density matrices in which both tracial states are maximally mixed. Interestingly, this symmetry facilitates the local unitary classification of two-qubit states. However, the same formalism fails in the case of systems of higher dimensions. We consider a broader set of states by introducing a family of qudit bipartite mixed states with maximally mixed marginals. For this family of states, we determine several constants which are in variant under local unitary transformations so can be used for entanglement classification. Finally, we consider the two-qutrit case and in particular, a two-parameter family of states for which the local unitary classification is complete. We relate this classification to known entanglement measures such as purity and negativity.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
Quantum entanglement is a key property of quantum information theory, that is at the heart of numerous promising applications in fields such as quantum cryptography, quantum computing or quantum sensing. In the past decades, the advent of such innovative technologies has reinforced the need for a better understanding of entanglement. The aim of this thesis is to contribute to this effort through the development of new tools targeting the characterization of several features of entanglement. Concerning the issue of entanglement detection, we present an optimization of the approach that exploits the concept of generalized concurrences to solve the separability problem for pure states. We then reformulate the separability question of mixed states into a matrix analysis problem, from which we obtain general separability criteria for multipartite states of ranks two and three. We also briefly discuss some properties of separable states. In particular, we characterize optimal separable decompositions of symmetric (i.e. permutation invariant) states of two and three qubits with maximal rank properties. Regarding the quantification of entanglement, we propose a function to quantify the entanglement of symmetric multiqubit states within classes of entangled states gathering states that are stochastically equivalent through local operations assisted with classical communication. This function establishes a link between the amount of entanglement of a symmetric state and the distribution of its Majorana points on the Bloch sphere. We finally investigate the robustness of entanglement with respect to particle loss and provide a full description of all multiqubit states that are fragile for the loss of one of their qubits. For symmetric states, the fragility for the loss of one qubit is shown to be related to a particular symmetry of the Majorana points.
We analyze entanglement classes for permutation-symmetric states for nnn qudits (i.e., ddd-level systems), with respect to local unitary operations (LU equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operators is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the Greenberger-Horne-Zeilinger and WWW states for the many-particle qudit case. In the latter case, we introduce the excitation state as a natural generalization of the WWW state for d>2d>2d>2.