A sufficient condition of violating the SPA conjecture

TL;DR

This paper identifies a sufficient condition under which the structural physical approximation (SPA) conjecture is violated, providing geometric insights into the relationship between entanglement witnesses and quantum state sets.

Contribution

It introduces a new sufficient condition for violating the SPA conjecture and explores the geometric relationship between entanglement witnesses and quantum states.

Findings

A specific condition for SPA conjecture violation is established.

Geometric illustrations clarify the relationship between witnesses and state sets.

Discussion of SPA conjecture in the context of decomposable entanglement witnesses.

Abstract

Based on the general form of entanglement witnesses constructed from separable states, we first show a sufficient condition of violating the structural physical approximation (SPA) conjecture [Phys. Rev. A 78, 062105 (2008)]. Then we discuss the SPA conjecture for decomposable entanglement witnesses. Moreover, we make geometric illustrations of the connection between entanglement witnesses and the sets of quantum states, separable states, and entangled states comparing with planes and vectors in Euclidean space.