Toward Exact 2 x 2 Hilbert-Schmidt Determinantal Probability Distributions via Mellin Transforms and Other Approaches

TL;DR

This paper aims to derive exact probability distributions for 2x2 quantum systems' Hilbert-Schmidt moments using Mellin transforms and other methods, providing insights into quantum entanglement boundaries.

Contribution

It introduces a novel approach combining Mellin transforms with existing techniques to construct exact univariate distributions for 2x2 quantum states.

Findings

Derived approximate separability/entanglement boundaries.

Connected moments to Dyson-index-like parameters.

Enhanced understanding of quantum state distributions.

Abstract

We attempt to construct the exact univariate probability distributions for 2 x 2 quantum systems that yield the (balanced) univariate Hilbert-Schmidt determinantal moments <(|rho| |rho^{PT}|)^n>, obtained by Slater and Dunkl (J. Phys. A, 45, 095305 [2012]). To begin, we follow--to the extent possible--the Mellin transform-based approach of Penson and Zyczkowski in their study of Fuss-Catalan and Raney distributions (Phys. Rev. E, 83, 061118 [2011]). Further, we approximate the y-intercepts (separability/entanglement boundaries)--at which |rho^{PT}|=0-- of the associated probability distributions based on the (balanced) moments, as well as the previously reported unbalanced determinantal moments <|rho^{PT}|^n>, as a function of the seventy-two values of the Dyson-index-like parameter alpha = 1/2 (rebits), 1 (qubits),...,2 (quaterbits),...35.