The Non-m-Positive Dimension of a Positive Linear Map

Abstract

We introduce a property of a matrix-valued linear map Φ that we call its "non-m-positive dimension" (or "non-mP dimension" for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of I m ⊗ Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map I m ⊗ Φ * can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive nontrivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer-Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

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

“…. We construct a subspace that attains this bound in a somewhat similar manner to the non-positive partial transpose subspaces constructed in [Joh13,JLP19]. Let…”

“…It is well-known, at least in the bipartite case, that entangled subspaces can be used to construct entanglement witnesses with the maximum number of negative eigenvalues (see [Joh13,JLP19], for example). We now show that the same is true in the multipartite case, and even for not-Z witnesses in general, as long as Z is a Euclidean closed cone.…”

“…It is known that, under various notions of entanglement, the maximum dimension of an entangled subspace is precisely the maximum number of negative eigenvalues of an entanglement witness [ATL11,Joh13,JLP19]. The number of negative eigenvalues quantifies "how good" the witness is at detecting entanglement.…”

“…There are other types of entangled subspaces that have been studied in previous works: non-positive partial transpose subspaces, for which every mixed state supported on that subspace has non-positive partial transpose [Joh13,JLP19]; genuinely entangled subspaces, for which every element is non-product with respect to every bipartition [CMW08,DA18]; and subspaces of bipartite space with high entropy of entanglement [HLW06]. Entangled subspaces are connected to unextendible product bases, and have found applications, for example, in quantum error correction [GW07,Sco04] and quantum tomography [HMW13].…”

Positive maps and trace polynomials from the symmetric group

“…Linear maps from the positive cone to itself are known as positive maps. These have applications in the study of quantum dynamics and entanglement [9][10][11][12][13][14][15][16], and can be understood as matrix inequalities for the positive cone.…”