Partitions of the polytope of Doubly Substochastic Matrices

In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given

Conditions for the generalized numerical range to be real

AbstractNecessary and sufficient conditions are given for the C-numerical range of a matrix A to be a subset of the real axis. In particular, it is shown that both A and C must be translates of hermitian matrices

Rearrangement and extremal results for Hermitian matrices

AbstractThis paper contains extreme value results for concave and convex symmetric functions of the eigenvalues of B + P∗AP as functions of the partial isometry P. The matrices A and B are Hermitian

Extending Kolmogorov's axioms for a generalized probability theory on collections of contexts

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Auff\'eves and Grangier, which in turn are inspired by Gleason's theorem.Comment: 18 pages, 3 figures, greatly revise

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{\gamma+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices