Dualities and positivity in the study of quantum entanglement

TL;DR

This paper surveys mathematical frameworks for quantum entanglement, focusing on convex cone duality, positivity conditions, and the relationship between separable state length and Schmidt rank, providing explicit characterizations and limitations.

Contribution

It introduces a generalized duality framework for entanglement witnesses, solves polynomial positivity problems for specific operator families, and explores the connection between separable state length and Schmidt rank.

Findings

Explicit description of entanglement witnesses in a three-parameter family.

Proved that certain block positivity tests are inherently insufficient.

Established that separable states with length ≤ 3 have Schmidt rank equal to their length.

Abstract

We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank. In particular, we prove that separable states of length lower of equal 3 have Schmidt ranks equal to their lengths. We also give an example of a state which has length 4 and Schmidt rank 3.