A triality pattern in entanglement theory

TL;DR

This paper uncovers a unifying pattern linking three classes of quantum states, revealing bounds on their spectral radii and ranks, and demonstrating their separability under certain conditions, advancing understanding in entanglement theory.

Contribution

It establishes new mathematical bounds and relationships among positive partial transpose, symmetric, and realignment-invariant quantum states, highlighting a triality pattern in entanglement theory.

Findings

Unified upper bound for spectral radii of the three state types

Existence of a lower bound for their ranks

States are separable when bounds are attained

Abstract

In this work, we present new connections between three types of quantum states: positive under partial transpose states, symmetric with positive coefficients states and invariant under realignment states. First, we obtain a common upper bound for their spectral radii and a result on their filter normal forms. Then we prove the existence of a lower bound for their ranks and the fact that whenever this bound is attained the states are separable. These connections add new evidence to the pattern that for every proven result for one of these types, there are counterparts for the other two, which is a potential source of information for entanglement theory.