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

TL;DR

This paper explores how pre-shared entanglement enhances local quantum state discrimination, linking it to algebraic geometry and entangled subspaces, and introduces new concepts like $r$-entangled subspaces with explicit constructions.

Contribution

It generalizes local state discrimination results to include pre-shared entanglement, relates it to the Krull dimension of SLOCC classes, and introduces $r$-entangled subspaces with algebraic-geometric analysis.

Findings

Maximum number of states is the Krull dimension of SLOCC orbit closure.

Generic resource states maximize the discriminability number.

Explicit constructions of $r$-entangled subspaces and their symmetric/antisymmetric variants.

Abstract

Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]. 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 $r$-entangled subspaces, which correspond to entangled subspaces of bosonic and fermionic systems, respectively.