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arXiv:1208.6494

Positive definite matrices with Hermitian blocks and their partial traces

Jean-Christophe Bourin, Eunyoung Lee, Minghua Lin

Published Aug 31, 2012• 1 citation
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Abstract

Let $H$ be a positive semi-definite matrix partitioned in $β\times β$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le β$. 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.

DOI
10.48550/arxiv.1208.6494

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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.
Let Sm\mathcal{S}_{m}Sm​ be the set of all m×mm\times mm×m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈Sm\rho\in\mathcal{S}_{m}ρ∈Sm​ based on outcomes of nnn measurements of observables X1,…,Xn∈HmX_{1},\dots,X_{n}\in\mathbb{H}_{m}X1​,…,Xn​∈Hm​ (Hm\mathbb{H}_{m}Hm​ being the space of m×mm\times mm×m Hermitian matrices) for a quantum system identically prepared nnn times in state ρ.\rho.ρ. Outcomes Y1,…,YnY_{1},\dots,Y_{n}Y1​,…,Yn​ of such measurements could be described by a trace regression model in which Eρ(Yj∣Xj)=tr(ρXj),j=1,…,n.\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.Eρ​(Yj​∣Xj​)=tr(ρXj​),j=1,…,n. The design variables X1,…,XnX_{1},\dots,X_{n}X1​,…,Xn​ are often sampled at random from the uniform distribution in an orthonormal basis {E1,…,Em2}\{E_{1},\dots,E_{m^{2}}\}{E1​,…,Em2​} of Hm\mathbb{H}_{m}Hm​ (such as Pauli basis). The goal is to estimate the unknown density matrix ρ\rhoρ based on the data (X1,Y1),…,(Xn,Yn).(X_{1},Y_{1}),\dots,(X_{n},Y_{n}).(X1​,Y1​),…,(Xn​,Yn​). Let Z^:=m2n∑j=1nYjXj\hat{Z}:=\frac{m^{2}}{n}\sum_{j=1}^{n}Y_{j}X_{j}Z^:=nm2​∑j=1n​Yj​Xj​ and let ρˇ\check{\rho}ρˇ​ be the projection of Z^\hat{Z}Z^ onto the convex set Sm\mathcal{S}_{m}Sm​ of density matrices. It is shown that for estimator ρˇ\check{\rho}ρˇ​ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten ppp-norm distances, p∈[1,∞]p\in[1,\infty]p∈[1,∞] and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator ρˇ\check{\rho}ρˇ​ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.
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).