Purity indices yield more complete data than entropy

Many physical systems are characterized by means of density matrices or covariance matrices whereby the Von Neumann entropy (i.e., a measure of the number of independent ways in which a whole polarization state may be arranged) provides an overall measure of statistical purity. Nevertheless, this approach does not provide complete information. A particular example of this problem, with important applications in several fields, is characterizing the polarimetric purity of 3D polarized light: a light beam whose direction of propagation fluctuates.¹

The polarimetric purity of 2D polarized light, whose direction of propagation is well defined, can be measured either by the degree of polarization, P (i.e., the ratio of the intensity of the totally polarized part to the intensity of the whole light beam), or by the Von Neumann entropy S. P is widely preferred and currently used as the representative quantity owing to its simple and meaningful definition. However, in the case of 3D polarized light, characterizing its purity structure requires a pair of physical quantities instead of an overall measure. We have found that a better method is the 3D 'indices of purity,' P₁ and P₂, which provide a complete measure of polarimetric purity.¹

Figure 1. The degree of directionality, P₂, of a 3D light beam is a measure of the angular aperture φof the direction of propagation and sets an upper limit for the 2D degree of polarization, P₁.

The values of these quantity pairs are nested in the form 0≤P₁≤P₂≤1, so that P₂ is a measure of the stability of the direction of propagation (see Figure 1, where the angular aperture φ, given by cos φ = P₂, represents the mean aperture angle of the direction of propagation) and sets an upper limit for P₁, the measure of the degree of polarization.

Figure 2. The 4D purity space is determined by the region limited by the edges OA, OB, OC, AB, AC, and BC. The 3D purity space, P₃ = 1, is determined by the area limited by the edges AB, AC, and BC.

Figure 3. The n–1 indices of purity of an n xn covariance matrix are numbered P₁, P₂, P₃, . . . ,P_{n - 2}, P_{n - 1} and are ordered in a simple sequence in which the value of a given index, P_i, sets a limit for the value of the previous one, P_{i - 1}.

When the dimensionality increases from 2 to 3 in response to the fluctuations of the direction of propagation, it becomes necessary to add an additional parameter, P₂, to the conventional degree of polarization, P₁. The overall degree of purity, P_Δ3, is derived from the first two indices of purity through a weighted quadratic average.¹

The indicated approach can be extended to characterize the polarimetric purity of material samples^{1, 3,4} in such a manner that the corresponding 4D purity space (see Figure 2) is built from a set of three indices of purity, P₁, P₂, P₃, thereby satisfying 0≤P₁≤P₂≤P₃≤1, which provide the complete required information and open non-destructive polarimetric techniques to a wide range of applications including medicine, remote sensing, and industry. These results have allowed us to establish a generalized polarization algebra^{2, 3} as a common mathematical framework for polarized light (2D and 3D) and for the description of the polarimetric properties of material samples. In such cases, the indices of purity arise as relevant measurable quantities that we expect will allow experimentalists to substantially improve several analytical techniques. Examples include the identification of terrestrial or marine targets through synthetic aperture radar (SAR) imaging polarimetry or early diagnosis of certain kinds of cancer.

These are just a few of the applications for and significance of the indices of purity, which can be easily defined for covariance or density matrices of any dimensionality.⁴ Our work has enabled us to show that, given a system characterized by n stochastic variables, the statistical purity of the system is determined by a set of n–1 nested indices P_i (i = 1,2, . . . , n–1)—see Figure 3—that reflect, in a surprisingly simple and natural way, the purity structure and provide much more detailed information than just a single value of entropy. From a weighted quadratic average of these indices, we have defined an overall degree of purity, P_Δn, which contains information akin to that provided by the Von Neumann entropy. Consequently, the indices constitute a powerful tool for addressing both theoretical and experimental problems (such as polarization, n–level systems, remote sensing, and non-destructive diagnosis) in which the system is represented by means of covariance or density matrices.

Our ongoing efforts are mainly focused on applications of the indices of purity both to characterize and improve electromagnetic reverberation chambers, which are widely used for studies and tests of electromagnetic compatibility. We also hope to exploit imaging polarimeters for the study of biological tissues or remote sensing technologies based on SAR polarimeters.