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Physical Review A

Certifying separability and entanglement classes of not-completely-permutation-symmetric qudit states

Artur Barasiński, Mateusz Nowotarski

Published Dec 15, 2016• 4 citations
Publisher

Abstract

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 $k$-positive map witness, optimal Schmidt-number witnesses, and entanglement measures.

DOI
10.1103/physreva.94.062319

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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.
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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.
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 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.
We introduce an operational entanglement classification of symmetric mixed states for an arbitrary number of qubits based on stochastic local operations assisted with classical communication (SLOCC operations). We define families of SLOCC entanglement classes successively embedded into each other, we prove that they are of nonzero measure, and we construct witness operators to distinguish them. Moreover, we discuss how arbitrary symmetric mixed states can be realized in the laboratory via a one-to-one correspondence between well-defined sets of controllable parameters and the corresponding entanglement families.
We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion is provided to easily identify whether given symmetric states are maximally entangled in that respect or not. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin of the system vanishes, as well as in corollary the dipole moment of the Husimi function. We establish the link between this kind of maximal entanglement, the anticoherence properties of spin states, and the degree of polarization of light fields. We analyze the relationship between the MES states and the classes of states equivalent through stochastic local operations with classical communication (SLOCC). We provide a nonexistence criterion of MES states within SLOCC classes of qubit states and show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. The 4-qubit system is analyzed exhaustively and all MES states of this system are identified and characterized. Finally the entanglement content of MES states is analyzed with respect to the geometric and barycentric measures of entanglement, as well as to the generalized NNN-tangle. We show that the geometric entanglement of MES states is ensured to be larger than or equal to 1/21/21/2, but also that MES states are not in general the symmetric states that maximize the investigated entanglement measures.