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Journal of Physics A Mathematical and General

Joint quasiprobability distribution for N spin-1/2 systems

Ravinder Rupchand Puri

Published Nov 7, 1995• 3 citations
Publisher

Abstract

A quantum system is described by a density matrix which determines the joint probabilities for simultaneous eigenstates of its observables. Since there do not exist any simultaneous eigenstates of non-commuting observables, the density matrix cannot provide any information about the joint probability of their eigenstates. The quasiprobability distributions (QPDs) constructed by using the density matrix provide a means of determining these joint probabilities. Here the QPDs for a system of N spin-1/2 are constructed and the joint probabilities for their components determined.

DOI
10.1088/0305-4470/28/21/025

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A quantum system is described by a density matrix which determines the joint probabilities for simultaneous eigenstates of its observables. Since there do not exist any simultaneous eigenstates of non-commuting observables, the density matrix cannot provide any information about the joint probability of their eigenstates. The quasiprobability distributions (QPDs) constructed by using the density matrix provide a means of determining these joint probabilities. Here the QPDs for a system of N spin-1/2 are constructed and the joint probabilities for their components determined.
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.
The (Fisher) information matrix (I2p) for a p×p density matrix (ρ) is taken to be the (2p2+p)-dimensional information matrix of a 2p-variate real normal distribution with an a priori zero mean vector and a covariance matrix (Σ2p), the two diagonal blocks of which are formed from the symmetric (real) component of ρ, and the two off-diagonal blocks from the skew-symmetric (imaginary) part of ρ. The inverse square root ‖nI2p‖−1/2 (where n is the number of observations) serves as a unitary invariant prior distribution over (the precision matrices) Σ2p−1 and ‖nI2p‖1/2 over Σ2p—and consequently over the p×p (mixed and pure) density matrices. The availability of these priors should facilitate the process of state determination. An uncertainty principle for the estimation of ρ is formulated. The determinant of Σ2p proves for integral p≥1 to equal (2−p‖ρ‖)2.
It is known that non-commuting observables in quantum mechanics do not have joint probability. This statement refers to the precise (additive) probability model. I show that the joint distribution of any non-commuting pair of variables can be quantified via upper and lower probabilities, i.e. the joint probability is described by an interval instead of a number (imprecise probability). I propose transparent axioms from which the upper and lower probability operators follow. The imprecise probability depend on the non-commuting observables, is linear over the state (density matrix) and reverts to the usual expression for commuting observables.
The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator. While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently. In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates. In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system. Therefore, our method presents a complementary approach to prominent quasiprobability representations. We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light. In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement.
We introduce a general method for the construction of quasiprobability representations for arbitrary notions of quantum coherence. Our technique yields a non-negative probability distribution for the decomposition of any classical state. Conversely, quantum phenomena are certified in terms of signed distributions, i.e., quasiprobabilities, and a residual component unaccessible via classical states. Our unifying method combines well-established concepts, such as phase-space distributions in quantum optics, with resources of quantumness relevant for quantum technologies. We apply our approach to analyze various forms of quantum coherence in different physical systems. Moreover, our framework renders it possible to uncover complex quantum correlations between systems, for example, via quasiprobability representations of multipartite entanglement.