## 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 [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. 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.

**Authors:**

- Nathaniel Johnston
- Benjamin Lovitz

**Download:**

- Preprint from arXiv:2010.02876 [quant-ph]
- Local preprint [pdf]

**Cite as:**

- B. Lovitz and N. Johnston.
*Entangled subspaces and generic local state discrimination with pre-shared entanglement*. E-print: arXiv:2010.02876 [quant-ph], 2020.

**Suplemenentary Material:**

- Entangled-subspaced-code repo – MATLAB and Macaulay2 code for checking whether or not a multipartite subspace is r-entangled

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