Exponential de Finetti Theorems for Fermionic Gaussian States

Abstract

We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dependence of our bound on the local Hilbert space dimension is polylogarithmic, giving an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these results to the broader class of Gaussian-invariant states, which contains, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas and always admit a purification into a larger Gaussian-symmetric state. As an application, we derive explicit convergence bounds for a hierarchy of convex relaxations of polynomial optimisation over fermionic Gaussian states. For variational energy minimisation, the hierarchy gives lower bounds converging to the global Gaussian minimum. Combined with upper bounds from Gaussian trial states, these bounds quantify how far a variational calculation can be from the best energy attainable within the Gaussian family.

arXiv comment7+23 pages, 1 figure; v2 contains applications to optimisation over Fermionic Gaussian states