2013/11/18 by Paul B. Slater, Slater, Paul B.
Chemistry · Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1311.4447
20 pages, one figure, new section IV reporting computation for the Bures n = 2, alpha = 1/2 moment formula
openalex publication_date 2013/11/18 · arxiv created 2014/03/07 · arxiv updated 2014/03/10 · openalex created_date 2017/06/05 · openalex updated_date 2026/07/28
We seek to develop a Bures (minimal monotone/statistical distinguishability) metric-based series of formulas for the moments of probability distributions over the determinants |ρ| and |ρPT| of 4 × 4 density matrices, ρ, for generalized (rebit, quater[nionic]bit,…) two-qubit systems, analogous to a series that has been obtained for the Hilbert-Schmidt (HS) metric. In particular, we desire--using moment-inversion procedures--to be able to closely test the previously-developed conjecture (J. Geom. Phys., 53, 74 [2005]) that the Bures separability probability over the (standard, fifteen-dimensional convex set of) two-qubit states is (1680 (√(2)-1))/(π8) ≈ 0.0733389--while, in the HS context, strong evidence has been adduced, along the indicated analytical lines, that the counterpart of this value is (8)/(33) (J. Phys. A, 45, 095305 [2012]). Working within the "utility function" framework of Dunkl employed in that latter study, we obtain an interesting 10F9 balanced hypergeometric function based on a "hybridization" of known Bures and HS terms. This exercise appears to provide an upper bound on the Bures two-qubit separability probability of 0.0798218. We also examine the yet unresolved HS qubit-qutrit scenario. Mathematica calculations indicate that if the same form of hypergeometric paradigm as has been established for the generalized two-qubit HS moments is followed in either the HS qubit-qutrit or Bures two-qubit cases, then a balanced hypergeometric function pFp-1 with p>9 would be required.