Quantum Entanglement, Symmetric Nonnegative Quadratic Polynomials and Moment Problems

Abstract

Quantum states are represented by positive semidefinite Hermitian operators with unit trace, known as density matrices. An important subset of quantum states is that of separable states, the complement of which is the subset of \textit{entangled} states. We show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis. Only a small number of such moments are accessible experimentally, and so in practice the question of quantum entanglement of a many-body system can be reduced to a truncated moment problem. By considering quantum entanglement of $n$ identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of $n$ copies of unit balls in $\mathbb{R}^d$. We work with moments up to degree $2$. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality of size at most $2d+2$, which is independent of $n$. The linear matrix inequalities can be converted into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large $n$. The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares relaxation. We show that the sum-of-squares relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques (J. Pure Appl. Algebra 192, no. 1-3, 95, arXiv:1205.3102), it can be written as a small linear matrix inequality of size at most $2d+2$, which is independent of $n$. For the cone of symmetric nonnegative polynomials with the relevant support we also prove an analogue of the half-degree principle for globally nonnegative symmetric polynomials (J. Math. Anal. Appl. 284, no. 1, 174, J. Pure Appl. Algebra 216, no. 4, 850).