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

Benjamin Lovitz1 and Nathaniel Johnston2,3

1Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, 200 University Ave W, Waterloo, ON, Canada
2Department of Mathematics & Computer Science, Mount Allison University, Canada
3Department of Mathematics & Statistics, University of Guelph, Canada

### 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.

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### Cited by

[1] Nathaniel Johnston, Benjamin Lovitz, and Aravindan Vijayaraghavan, "Complete hierarchy of linear systems for certifying quantum entanglement of subspaces", Physical Review A 106 6, 062443 (2022).

[2] Somshubhro Bandyopadhyay and Vincent Russo, "Entanglement cost of discriminating noisy Bell states by local operations and classical communication", Physical Review A 104 3, 032429 (2021).

[3] Gabriel Champagne, Nathaniel Johnston, Mitchell MacDonald, and Logan Pipes, "Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses", arXiv:2108.10405, (2021).

[4] Benjamin Lovitz and Vincent Steffan, "New techniques for bounding stabilizer rank", Quantum 6, 692 (2022).

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