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arXiv:0905.0933 · Physical Review A

“Classical” quantum states

Marek Kuś, Ingemar Bengtsson

Published Aug 13, 2009• 23 citations
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Abstract

We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, ``most classical'' can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons, as well as mixed spin states decomposable into probabilistic mixtures of pure coherent states. The latter was the subject of the recent paper by Giraud et al. [Phys. Rev. A 78, 042112 (2008)], who showed that in the lowest-dimensional nontrivial case of spin 1, each such state can be decomposed into a mixture of eight pure states. Using our method, we prove that in fact four pure states always suffice.

DOI
10.1103/physreva.80.022319

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We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, ``most classical'' can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons, as well as mixed spin states decomposable into probabilistic mixtures of pure coherent states. The latter was the subject of the recent paper by Giraud et al. [Phys. Rev. A 78, 042112 (2008)], who showed that in the lowest-dimensional nontrivial case of spin 1, each such state can be decomposed into a mixture of eight pure states. Using our method, we prove that in fact four pure states always suffice.
Pure coherent states are known as the most classical states in quantum mechanics. Similarly, pure spin coherent states based on the rotation group also exist in finite dimensions. We call these states pure classical states, and the mixed classical states are the statistical mixtures of these states. The set of all classical states is a subset of the set of all pure & mixed states. We present an essentially complete description of the geometry of this set for 3-dimensional Hilbert space. While studying its geometry we find a remarkable similarity between the geometry of this set and the geometry of the set of all classical probability distributions. This study also improves a recent work on this set done by Giraud at el [4] and relates it to an earlier work by Wootters [14].
The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.
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.
The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of "bit symmetry", the violation of "no simultaneous encoding" (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.
A sufficient condition for a quantum state of a system of spin-1/2 particles to admit a local hidden variable description, i.e., to be classical, is the separability of the density matrix characterizing its state, but not all classical states are separable. This leads one to infer that separability and classicality are two different concepts. These concepts are examined here in the framework of a criterion for identifying the classicality of a system of spin-1/2 particles based on the concept of joint quasiprobability (JQP) for the eigenvalues of spin components [Puri, J. Phys. A 29, 5719 (1996)]. The said criterion identifies a state as classical if a suitably defined JQP of the eigenvalues of spin components in three suitably chosen orthogonal directions is non-negative. In agreement with other approaches, the JQP-based criterion also leads to the result that all nonfactorizable pure states of two spin-1/2 particles are nonclassical. In this paper it is shown that the application of the said criterion to mixed states suggests that the states it identifies as classical are also separable and that there exist states which, identified as classical by other methods, may not be identified as classical by the criterion as it stands. However, results in agreement with the known ones are obtained if the criterion is modified to identify as classical also those states for which the JQP of the eigenvalues of the spin components in two of the three prescribed orthogonal directions is non-negative. The validity of the modified criterion is confirmed by comparing its predictions with those arrived at by other methods when applied to several mixed states of two spin-1/2 particles and the Werner-like state of three spin-1/2 particles [T\`oth and Ac\`{\i}n, Phys. Rev. A 74, 030306(R) (2006)]. The JQP-based approach, formulated as it is along the lines of the PPP-function approach for identifying classical states of the electromagnetic field, offers a unified approach for systems of an arbitrary number of spin-1/2 particles and the possibility of linking classicality with the nature of the measurement process.