Abstract
The information contained in an n-dimensional (nD) density matrix ρ is parametrized and interpreted in terms of its asymmetry properties through the introduction of a family of components of purity that are invariant with respect to arbitrary rotations of the nD Cartesian reference frame and that are composed of two categories of meaningful parameters of different physical nature: the indices of population asymmetry and the intrinsic coherences. It is found that the components of purity coincide, up to respective simple coefficients, with the intrinsic Stokes parameters, which are also introduced in this work, and that determine two complementary sources of purity, namely the population asymmetry and the correlation asymmetry, whose weighted square average equals the overall degree of purity of ρ. A discriminating decomposition of ρ as a convex sum of three density matrices, viz. the pure, the fully random (maximally mixed) and the discriminating component, is introduced, which allows for the definition of the degree of nonregularity of ρ as the distance from ρ to a density matrix of a system composed of a pure component and a set of 2D, 3D,… and nD maximally mixed components. The chiral properties of a state ρ are analyzed and characterized from its intimate link to the degree of correlation asymmetry. The results presented constitute a generalization to nD systems of those established and exploited for polarization density matrices in a series of previous works.
Highlights
Density matrices play a key role in both quantum mechanics and classical treatment of mixed states [1], as for instance in the characterization of the second-order polarization properties of electromagnetic waves [2,3]
Since Pc is invariant under orthogonal similarity transformations of ρdO it follows that Pc = Pc and we deduce that the degree of correlation asymmetry of the discriminating component is limited by 0 ≤ Pc ≤ 1/2, where the minimum Pc = 0 corresponds to regular states, and otherwise, ρ is a nonregular state, with a maximum Pc = 1/2 for states with maximal nonregularity, hereafter called perfect nonregular states
The definition of an intrinsic coordinate system for each given n-dimensional density matrix ρ is exploited in order to define a set of quantities that provide complete information on the rotational invariant properties associated with ρ in a hierarchical and meaningful manner
Summary
Density matrices play a key role in both quantum mechanics and classical treatment of mixed states [1], as for instance in the characterization of the second-order polarization properties of electromagnetic waves [2,3]. The concepts of discriminating decomposition, intrinsic density matrix, sources of purity (population and correlation asymmetry), intrinsic Stokes parameters, degree of randomness and nonregularity are introduced and analyzed in terms of certain types of asymmetry exhibited by density matrices representing n-dimensional systems. The fact that these notions have proven to be very fruitful for the study and interpretation of polarization density matrices (three-dimensional systems), supports their generalization to n-dimensional (nD) density matrices. We have followed the criterion of Byrd and Khaneja [9] for the definition of parameters sk , in such a manner that the absolute value |s| of the Bloch vector equals 1 for pure states and, as shown in Section 4, in general coincides with the degree of purity of the state represented by ρ
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