A partial ordering of CS m defined by the generalized matrix functions

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number , positive semi-definite matrices , , and generalized matrix functions such as the determinant and permanent, etc., we have A general scheme is introduced to prove more general inequalities involving m positive semi-definite matrices for that extend the results of other authors.

Inequalities which compare elements of the convex cone of positive semidefinite hermitian matrices with products of roots of elements are proved. They yield inequalities for Schur functions (generalized matrix functions) which, when specialized to the determinant, give a result of R. Bellman and L. Mirsky.

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

For normalised generalized matrix functions f and g, we say that f dominates g if f(A)⩾g(A) for every M-matrix A. We first demonstrate a finite set of test matrices for any such inequality. Then, using results from group representation theory, all comparisons among immanants in certain classes are determined. This work parallels ongoing research into gmf inequalities on positive semidefinite matrices, for which no finite set of test matrices is available. However, the inequalities for the two classes are quite different, and the test matrices permit more rapid progress in the M-matrix case. Just as in the positive semidefinite case, the gmf inequalities we prove may be used to verify previously unknown determinantal inequalities for M-matrices, such as the symmetrized Fischer inequalities recently proved in the positive semidefinite case.

Given a partition a = {ax , a2, ... , as} , ax > a2 > > as, of n we let X denote the derived irreducible character of S" , and we associate with a a derived partition a = {a, -1, a2 -1, .. , ott -1 , at+x , ... , as , 1 } where t denotes the smallest positive integer such that at > a(+1 (aJ+r = 0).We show that if y is a decomposable C-valued n-linear function on Cm x Cm x x Cm (-copies) then (Xa Y, Y) > (Xa, Y, Y) .Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions dx and dx ( , namely that (Xa(e))-xdx(B)>(Xa,(e))-ldxAB) for each n x n positive semidefinite Hermitian matrix B .This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.

The cone of positive generalized matrix functions

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

Inequalities and identities for generalized matrix functions

Sparse matrix decompositions and graph characterizations

We study the reduction from a bipartite Hermitian matrix H to two single-qubit reduced Hermitian matrices HA and HB and the inverse problem. The inverse problem extends the so-called marginal problem from quantum information theory. We explicitly obtain the expressions of the matrices mentioned above and then extend them to positive semidefinite matrices by showing several inequalities involving a few parameters from the matrices. We also extend our results to multi-quit matrices. Our study constructs the intimate connection between the generalized marginal problems in terms of Hermitian matrices and the traditional marginal problem in terms of positive semidefinite matrices.

The states of the qubit, the basic unit of quantum information, are 2 × 2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.

On vector configurations that can be realized in the cone of positive matrices

Generalized matrix functions and Hermitian matrices