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arXiv:quant-ph/0504172 · Physical Review A

Classical statistics inherent in a quantum density matrix

A. K. Rajagopal, R. W. Rendell

Published Aug 18, 2005• 21 citations
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Abstract

A density-matrix formulation of classical bipartite correlations is constructed. This leads to an understanding of the appearance of classical statistical correlations intertwined with the quantum correlations as well as a physical underpinning of these correlations. As a by-product of this analysis, a physical basis of the classical statistical correlations leading to additive entropy in a bipartite system discussed recently by Tsallis emerges as inherent classical spin fluctuations. It is found that in this example, the quantum correlations shrink the region of additivity in phase space.

DOI
10.1103/physreva.72.022322

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A density-matrix formulation of classical bipartite correlations is constructed. This leads to an understanding of the appearance of classical statistical correlations intertwined with the quantum correlations as well as a physical underpinning of these correlations. As a by-product of this analysis, a physical basis of the classical statistical correlations leading to additive entropy in a bipartite system discussed recently by Tsallis emerges as inherent classical spin fluctuations. It is found that in this example, the quantum correlations shrink the region of additivity in phase space.
The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
The characterization of quantum correlations, being stronger than classical, yet weaker than those appearing in non-signaling models, still poses many riddles. In this work, we show that the extent of binary correlations in a general class of nonlocal theories can be characterized by the existence of a certain covariance matrix. The set of quantum realizable two-point correlators in the bipartite case then arises from a subtle restriction on the structure of this general covariance matrix. We also identify a class of theories whose covariance has neither a quantum nor an "almost quantum" origin, but which nevertheless produce the accessible two-point quantum mechanical correlators. Our approach leads to richer Bell-type inequalities in which the extent of nonlocality is intimately related to a non-additive entropic measure. In particular, it suggests that the Tsallis entropy with parameter q=1/2 is a natural operational measure of non-classicality. Moreover, when generalizing this covariance matrix, we find novel characterizations of the quantum mechanical set of correlators in multipartite scenarios. All these predictions might be experimentally validated when adding weak measurements to the conventional Bell test (without adding postselection).
We provide a characterization of energy in the form of exchanged heat and work between two interacting constituents of a closed, bipartite, correlated quantum system. By defining a binding energy we derive a consistent quantum formulation of the first law of thermodynamics, in which the role of correlations becomes evident, and this formulation reduces to the standard classical picture in relevant systems. We next discuss the emergence of the second law of thermodynamics under certain-but fairly general-conditions such as the Markovian assumption. We illustrate the role of correlations and interactions in thermodynamics through two examples.
The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space-a common meeting point to both classical and quantum density statistical descriptors. Here, this issue is tackled by investigating the behavior of classical analogs arising upon the removal of all interference traits displayed by the Wigner distribution functions associated with a given pure quantum state. Accordingly, the dynamical evolution of the linear and von Neumann entropies is numerically computed for a continuous-variable bipartite system, and compared with the corresponding classical counterparts, in the case of two quartic oscillators nonlinearly coupled under regular and chaos conditions. Three quantum states for the full system are considered: a Gaussian state, a cat state, and a Bell-type state. By comparing the quantum and classical entropy values, and particularly their trends, it is shown that, instead of entanglement production, such entropies rather provide us with information on the system (either quantum or classical) delocalization. This gradual loss of information translates into an increase in both the quantum and the classical realms, directly connected to the increase in the correlations between both parties' degrees of freedom which, in the quantum case, is commonly related to the production of entanglement.
We establish a fluctuation theorem for an open quantum bipartite system that explicitly manifests the role played by quantum correlation. Generally quantum correlations may substantially modify the universality of classical thermodynamic relations in composite systems. Our fluctuation theorem finds a non-equilibrium parameter of genuinely quantum nature that sheds light on the emerging quantum information thermodynamics. Specifically we show that the statistics of quantum correlation fluctuation obtained in a time-reversed process can provide a useful insight into addressing work and heat in the resulting thermodynamic evolution. We illustrate these quantum thermodynamic relations by two examples of quantum correlated systems.