Construction of three-qubit positive-partial-transpose entangled states of rank four

TL;DR

This paper characterizes three-qubit PPT entangled states of rank four, classifies them based on Lorentz invariants, and provides explicit constructions and SLOCC equivalence analysis.

Contribution

It introduces a classification of rank-four three-qubit PPT entangled states based on Lorentz invariants and offers explicit constructions for each type.

Findings

States with nonzero invariants are constructed via UPB.

States with zero invariant are represented with one complex parameter.

The paper analyzes SLOCC equivalence and Lorentz invariants for these states.

Abstract

Multiqubit positive-partial-transpose (PPT) entangled states play an important role in quantum information theory. We characterize such states of minimum rank in three-qubit system, namely rank four. Depending on whether the Lorentz invariant is zero, we classify such states into two types. The PPT entangled states constructed by unextendible product bases (UPB) have nonzero invariants, which belong to type I. We provide a method to effectively determine whether a state can be constructed from UPB. For states with zero invariant, which belong to type II, we provide an explicit expression up to equivalence of stochastic local operations and classical communications (SLOCC). It turns out that we can represent them with only one complex parameter. We further study SLOCC-equivalence relation within the expression. We also investigate the Lorentz invariants of multiqubit states with rank less than three and analyze their range.