Related papers: Decomposition and partial trace of positive matric…
In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive…
Under the name prime decomposition (pd), a unique decomposition of an arbitrary $N$-dimensional density matrix $\rho$ into a sum of seperable density matrices with dimensions given by the coprime factors of $N$ is introduced. For a class of…
This is a concise review of the complex, real and quaternion real Ginibre random matrix ensembles and their elliptic deformations. Eigenvalue correlations are exactly reduced to two-point kernels and discussed in the strongly and weakly…
A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace…
If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are…
A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum…
We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in…
Considering the problem of sampling from the output photon-counting probability distribution of a linear-optical network for input Gaussian states, we obtain results that are of interest from both quantum theory and the computational…
A theorem of Blackwell about comparison between information structures in classical statistics is given an analogue in the quantum probabilistic setup. The theorem provides an operational interpretation for trace-preserving completely…
We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…
We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66…
Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…
We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the…
Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p…
We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum…
Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of…