Entangled subspaces and generic local state discrimination with pre-shared entanglement

Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [42]. In this work, we determine this number in a more general setting in which the local parties have access to pre-shared entanglement in the form of a resource state. We find that, for an arbitrary pure resource state, this number is equal to the Krull dimension of (the closure of) the set of pure states obtainable from the resource state by SLOCC. Surprisingly, a generic resource state maximizes this number. Local state discrimination is closely related to the topic of entangled subspaces, which we study in its own right. We introduce r-entangled subspaces, which naturally generalize previously studied spaces to higher multipartite entanglement. We use algebraic-geometric methods to determine the maximum dimension of an r-entangled subspace, and present novel explicit constructions of such spaces. We obtain similar results for symmetric and antisymmetric rentangled subspaces, which correspond to entangled subspaces of bosonic and fermionic systems, respectively.

Automated summary

This paper extends the theory of discriminating quantum states in the context of local measurements and shared entangled resources. The maximum distinguishability is found to be determined by the properties of the resource state. Moreover, the paper explores the topic of entangled subspaces and presents novel constructions.