A Singular Value Inequality for Heinz Means

We prove a matrix inequality for matrix monotone functions, and apply it to prove a singular value inequality for Heinz means recently conjectured by X. Zhan.Comment: 4 pages; Mistake in proof of Theorem 1 correcte

On a block matrix inequality quantifying the monogamy of the negativity of entanglement

We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a very special case. Given n matrices Ai, i = 1,..., n, of the same size, let Z1 and Z2 be the block matrices Z1 := ( A j A* i) n i, j= 1 and Z2 := ( A* j Ai) n i, j= 1, respectively. Then, the conjectured inequality is (|| Z1|| 1 - Tr Z1) 2 + (|| Z2|| 1 - Tr Z2) 2 =.. ( i ) = j || Ai || 2|| A j || 2.. 2, where || . || 1 and || . || 2 denote the trace norm and the Hilbert- Schmidt norm, respectively. We prove this inequality for the already challenging case n = 2 with A1 = I

A sharp version of Kahan's theorem on clustered eigenvalues

AbstractLet the n × n Hermitian matrix A have eigenvalues λ1, λ2, …, λn, let the k × k Hermitian matrix H have eigenvalues μ1 ≤ μ2 … ≤ μk, and let Q be an n × k matrix having full column rank, so 1 ≤ k ≤ n. It is proved that there exist k eigenvalues λi1 ≤ λi2 ≤ … ≤ λik of A such thatmax1≤j≤k|μj − λij| ≤ Cσmin(Q) ‖AQ − QH‖2always holds with c = 1, where σmin (Q) is the smallest singular value of Q, and ‖; · ‖;2 denotes the biggest singular value of a matrix. The inequality was proved for c ≤ √2 in 1967 by Kahan, who also conjectured that it should be true for c = 1

Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems

Extensive numerical integration results lead us to conjecture that the silver mean, that is, s = \sqrt{2}-1 = .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of separable states consisting generically of rank-four 4 x 4 density matrices has volume (``hyperarea'') 55s/39 and that part composed of rank-three density matrices, 43s/39, so the total boundary hyperarea would be 98s/39. While the Bures probability of separability (0.07334) dominates that (0.050339) based on the Wigner-Yanase metric (and all other monotone metrics) for rank-four states, the Wigner-Yanase (0.18228) strongly dominates the Bures (0.03982) for the rank-three states.Comment: 30 pages, 6 tables, 17 figures; nine new figures and one new table in new section VII.B pertaining to 14-dimensional hyperareas associated with various monotone metric