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Frontiers in Optics

Measures of Quantum State Purity and Classical Degree of Polarization

Omar Gamel, Daniel F. V. James

Published Jan 1, 2012• 4 citations
Publisher

Abstract

Classical degree of polarization and quantum purity are in fact the same quantity. We analyze rival measures of polarization in N>2 dimensions. Interesting relationships, advantages of each measure and application to entanglement measures will be discussed.
DOI
10.1364/fio.2012.ftu4f.6

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There is a well-known mathematical similarity between two-dimensional classical polarization optics and two-level quantum systems, where the Poincar\'e and Bloch spheres are identical mathematical structures. This analogy implies that the classical degree of polarization and quantum purity are in fact the same quantity. We make extensive use of this analogy to analyze various measures of polarization for higher dimensions proposed in the literature, in particular the N=3N=3N=3 case, illustrating interesting relationships that emerge as well as the advantages of each measure. We also propose a different class of measures of entanglement based on the purity of subsystems.
There has been much discussion in the literature about rival measures of classical polarization in three dimensions. We gather and compare the various proposed measures of polarization, creating a geometric representation of the polarization state space in the process. We use majorization, previously used in quantum information, as a criterion to establish a partial ordering on the polarization state space. Using this criterion and other considerations, the most useful polarization measure in three dimensions is found to be one dependent on the Bloch vector decomposition of the polarization matrix.
Several approaches and descriptors have been proposed to characterize the purity of coherency or density matrices describing physical states, including the polarimetric purity of 2D and 3D partially polarized waves. This work introduces two interpretations of the degree of purity: one derived from statistics and another from algebra. In the first one, the degree purity is expressed in terms of the mean and standard deviation of the eigenvalue spectrum of the density or coherency matrix of the corresponding state. The second one expresses the purity in terms of two specific measures obtained by decomposing the coherency matrix as a sum of traceless symmetric, antisymmetric, and scalar matrices. We believe these two approaches offer better insights into the purity measure. Furthermore, interesting relations with existing quantities in polarization optics also are described.
The classical analogues of quantum correlations have been the subject of considerable interest in the past few years; however, all of these investigations consider the classical analogue of only pure quantum mechanical states. In this work, we study mixed classical light states and derive relations between the polarization coherence of the field and the (classical) entanglement between its degrees of freedom. We show, for a specific model of mixed states, where the purity is determined by a single parameter, how the coherence shared between polarization and entanglement shrinks as the level of purity decreases. The sum of the square of polarization and entanglement remains constant, a behavior consistent with previous results [Phys. Rev. Lett.117, 153901 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.153901] of pure states, even though the constant for the mixed-state case is now smaller in value.