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

Abstract

Walgate and Scott have determined the maximum number of generic pure quantum states in multipartite space that can be unambiguously discriminated by an LOCC measurement [WS08]. 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. This dimension is known for several resource states, for example the GHZ state.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.

New techniques for bounding stabilizer rank

“…Since dim(S n (C 2 )) = n + 1, it follows from Proposition 5.3 that χ n ≥ n + 1. We note that a similar proof as below can be used to show that a generic product state of the form ψ 1 ⊗ • • • ⊗ ψ n , for qubit states ψ i ∈ S(C 2 ), has stabilizer rank 2 n , where in this context we define generic in the same algebraic-geometric sense as in [20]. In Proposition 3.7 we have presented an explicit sequence of product states of stabilizer rank 2 n , which is maximal.…”

Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses

“…However, these numbers just give lower bounds on how many negative eigenvalues P ∨ (vv * ) Γ P ∨ can have. The general ( d 2 ) = d(d − 1)/2 upper bound of [LJ20] applies in this setting, and Figure 1 compares these various lower and upper bounds.…”