Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

TL;DR

This paper investigates the orthogonality of determinants related to two-rebit quantum states under Hilbert-Schmidt measure, revealing a unique orthogonality property in nine dimensions and exploring its implications for quantum entanglement analysis.

Contribution

It demonstrates the orthogonality of the determinants |rho| and |rho^{PT}| in two-rebit systems under Hilbert-Schmidt measure and examines the failure of this property in higher dimensions and other measures.

Findings

Determinants form orthogonal polynomials under HS measure in 2-rebit systems.

Orthogonality does not extend to two-qubit systems or other measures like Bures.

The results relate to the probability of quantum state separability.

Abstract

A complete description of the multitudinous ways in which quantum particles can be entangled requires the use of high-dimensional abstract mathematical spaces. We report here a particularly interesting feature of the nine-dimensional convex set-endowed with Hilbert-Schmidt (Euclidean/flat) measure-composed of two-rebit (4 x 4) density matrices (rho). To each rho is assigned the product of its (nonnegative) determinant |rho| and the determinant of its partial transpose |rho^{PT}| -negative values of which, by the results of Peres and Horodecki, signify the entanglement of rho. Integrating this product, |rho| |rho^{PT}| =|rho rho^{PT}|, over the nine-dimensional space, using the indicated (HS) measure, we obtain the result zero. The two determinants, thus, form a pair of multivariate orthogonal polynomials with respect to HS measure. It is hypothesized that the analogous two determinants for the standard fifteen-dimensional convex set of two-qubit density matrices are similarly orthogonal. However, orthogonality does not hold, we find, if the symmetry of the nine-dimensional two-rebit scenario is broken slightly, nor with the use of non-flat measures, such as the prominent Bures (minimal monotone) measure-nor in the full HS extension to the 15-dimensional convex set of two-qubit density matrices. We discuss relations-involving the HS moments of |\rho^{PT}|-to the long-standing problem of determining the probability that a generic pair of rebits/qubits is separable.