Related papers: State polynomials: positivity, optimization and no…
In this work we study the problem of writing a Hermitian polynomial as a Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea of Putinar-Scheiderer to obtain necessary matrix positivity conditions for Hermitian…
We consider a hierarchy of upper approximations for the minimization of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$ proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward…
This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This…
We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles…
In this paper, we present a method to solve the quantum marginal problem for symmetric $d$-level systems. The method is built upon an efficient semi-definite program that determines the compatibility conditions of an $m$-body reduced…
The connection between quantum optical nonclassicality and the violation of Bell's inequalities is explored. Bell type inequalities for the electromagnetic field are formulated for general states of quantised radiation and their violation…
We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the…
Greenberger-Horne-Zeilinger states are intuitively known to be the most non-classical ones. They lead to the most radically nonclassical behavior of three or more entangled quantum subsystems. However, in case of two-dimensional systems, it…
The violation of Mermin's inequalities is analyzed by making use of two different Bell setups built with pseudospin operators. Employing entangled states defined by means of squeezed and coherent states, the expectation value of Mermin's…
Proposals for Bell inequality tests on systems restricted by superselection rules often require operations that are difficult to implement in practice. In this paper, we derive a new Bell inequality, where pairs of states are used to…
In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as…
We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
Using the convex optimization method and Helstrom family of ensembles introduced in Ref. [1], we have discussed optimal ambiguous discrimination in qubit systems. We have analyzed the problem of the optimal discrimination of N known quantum…
Bell's test, initially devised to distinguish quantum theory from local hidden variable models through {violations of local bounds}, is also a common tool for detecting entanglement. For this purpose, one can assume the quantum description…
Finding all Bell inequalities for a given number of parties, measurement settings, and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can…
This paper studies the complexity of matrix Putinar's Positivstellens{\"a}tz on the semialgebraic set that is given by the polynomial matrix inequality. \rev{When the quadratic module generated by the constrained polynomial matrix is…
We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method…
This paper studies stochastic optimization problems with polynomials. We propose an optimization model with sample averages and perturbations. The Lasserre type Moment-SOS relaxations are used to solve the sample average optimization.…
A necessary and sufficient entanglement criterion based on variances of Mermin-Klyshko's Bell operators is proved for multiqubit pure states. Contrary to Bell's inequalities, entangled pure states strictly satisfy a quadratic inequality but…