Joint Hilbert-Schmidt Determinantal Moments of Product Form for the Two-Rebit and Two-Qubit and Higher-Dimensional Quantum Systems

2011/04/01 by Paul B. Slater, Slater, Paul B. · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math-ph #math.MP #quant-ph

paper · pdf · doi:10.48550/arxiv.1104.0217

23 pages, 5 figures; Greatly expanded set of computational results, largely stemming from a suggestion of C. Dunkl regarding the use of the Cholesky-decomposition to parameterize density matrices. A two-page appendix of Dunkl is also included

openalex publication_date 2011/04/01 · arxiv created 2011/05/24 · arxiv updated 2011/05/26 · openalex created_date 2022/09/13 · openalex updated_date 2026/07/28

Abstract

We report formulas for the joint moments of the determinantal products (detrho)^k (detrho^PT)^K (k=0, 1, 2,...,N; K = 0, 1, 2, 3, 4) of Hilbert-Schmidt (HS) probability distributions over the two-rebit and (K = 0, 1) two-qubit density matrices rho. Here PT denotes the partial transposition operation of quantum-information-theoretic importance. Each formula is the product of the expression for the moments of (detrho)^k, k=0,1,2,...,N--already known from work of Cappellini, Sommers and Zyczkowski (Phys. Rev. A 74, 062322 (1996))--and an adjustment factor. The factor is a biproper rational function, with its numerators and denominators both being 3 K-degree polynomials in k. We infer the structure that the denominators follow for arbitrary K in both the two-rebit and two-qubit cases, and the six leading-order coefficients of k of the numerators in the two-rebit scenario.. We also commence an analogous investigation of generic rebit-retrit and qubit-qutrit systems. This research was motivated, in part, by the objective of using the computed moments to well reconstruct the associated HS probabilities (for which we present plots) over the two determinants, and to ascertain--at least to high accuracy--the associated (separability) probabilities of "philosophical, practical and physical" interest that (detrho^PT)>0.

Cited by

• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities

Related

• Moments of the Hilbert-Schmidt probability distributions over determinants of real two-qubit density matrices and of their partial transposes
• Hilbert-Schmidt Orthogonality of det(rho) and det(rho^PT) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses
• Toward Exact 2 x 2 Hilbert-Schmidt Determinantal Probability Distributions via Mellin Transforms and Other Approaches
• A Balanced ₁₀F₉ Hypergeometric Hybrid Hilbert-Schmidt/Bures Two-Qubit Function and Related Constructions
• Symmetric Halves of the 8/33-Probability that the Joint State of Two Quantum Bits is Disentangled