Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers for instance the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel.Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the d th order cumulant can be presented in the form of an d-dimensional tensor, the algorithm is presented using tensor operations. The algorithm provided in the paper takes advantage of supersymmetry of cumulant and moment tensors. We show that the proposed algorithm considerably reduces the computational complexity and the computational memory requirement of cumulant calculation as compared with existing algorithms. For the sizes of interest, the reduction is of the order of d! compared to the naïve algorithm.
Abstract. The totality of normalised density matrices of order N forms a convex set Q N in R N 2 −1 . Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N . For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W (A), induced by the unitarily invariant FubiniStudy measure on the complex projective manifold CP N −1 . We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.