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arXiv:1210.2922 · International Journal of Mathematics

DECOMPOSITION AND PARTIAL TRACE OF POSITIVE MATRICES WITH HERMITIAN BLOCKS

Jean-Christophe Bourin, Eunyoung Lee

Published Jan 1, 2013• 17 citations
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Abstract

Let H = [As, t] be a positive definite matrix written in β × β Hermitian blocks and let Δ = A1, 1 + ⋯ + Aβ, β be its partial trace. Assume that β = 2p for some p ∈ ℕ. Then, up to a direct sum operation, H is the average of β matrices isometrically congruent to Δ. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen–Kempe separability criterion.

DOI
10.1142/s0129167x13500109

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Let H = [As, t] be a positive definite matrix written in β × β Hermitian blocks and let Δ = A1, 1 + ⋯ + Aβ, β be its partial trace. Assume that β = 2p for some p ∈ ℕ. Then, up to a direct sum operation, H is the average of β matrices isometrically congruent to Δ. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen–Kempe separability criterion.
Let HHH be a positive semi-definite matrix partitioned in β×ββ\times ββ×β Hermitian blocks, H=[As,t]H=[A_{s,t}]H=[As,t​], 1≤s,t,≤β1\le s,t,\le β1≤s,t,≤β. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.
A block matrix [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] is positive partial transpose (PPT) if both [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] and [AX∗XB]\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right][AX​X∗B​] are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC (((PPT))) matrix can be decomposed into a sum of s+1s+1s+1 SPC (((PPT))) matrices of which the first sss are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC (((PPT))) matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC (((PPT))) matrices to the case of weakly irreducible SPC (((PPT))) matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).