Measures of quantum state purity and classical degree of polarization

There is a well-known mathematical similarity between two-dimensional classical polarization optics and two-level quantum systems, where the Poincaré 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 $N=3$ 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.

Figure 1

Classes of cross section of the eight-dimensional space in which the generalized Bloch vectors exist, based on a diagram by Kimura [22]. In each diagram, the shaded region represents the allowable states, while the outer circle is the enclosing hypersphere. The pure states are where the shaded region touches the outer circle. Points $A$, $B$, $C$, and $D$ are specific states we examine.

Figure 2

Values of purity measures for various eigenvalues of a three-dimensional density matrix. Each graph has fixed ${\lambda }_1$, with ${\lambda }_2$ against the horizontal axis, and ${\lambda }_3\equiv 1-{\lambda }_1-{\lambda }_2$. In each graph the top solid curve in black is Barakat's last measure ${\Pi }_B$, the short-dashed curve is the SSKF purity ${\Pi }_{\text{SSKF}}$, the long-dashed curve is the von Neumann purity ${\Pi }_v$, and the thick solid curve is the KW purity ${\Pi }_{\text{KW}}$.