Broader applications of the moment-matrix bound

Investigate whether the moment-matrix bound \(\|M^\rho-M^\omega\|_{\mathrm{op}}\le 2\), where \(M^\rho\) is the quartic moment matrix of an arbitrary state and \(M^\omega\) is that of a mixture of Gaussian states, has broader applications.

Background

A central technical result shows that the quartic moment matrix of any fermionic state can be approximated in operator norm by the moment matrix of a mixture of Gaussian states. The paper uses this result to derive the main eigenvalue bound, but does not determine whether the bound is useful in other settings or problems.