# Entangled edge states of corank one with positive partial transposes

## TL;DR

This paper constructs explicit families of low-rank entangled states with positive partial transpose in high-dimensional systems, revealing extreme violations of separability criteria and providing the first known examples for dimensions four and above.

## Contribution

It introduces the first explicit construction of $n ext{-}n$ PPT entangled edge states of corank one for all $n
eq 2$, with minimal coranks, expanding understanding of entanglement structure.

## Key findings

- Constructed parameterized families of PPT states of corank one for all $n
geq 3$.
- Demonstrated these states violate the range criterion in the most extreme way.
- Provided the first explicit examples of such states for $n extgreater 3$.

## Abstract

We construct a parameterized family of $n\otimes n$ PPT (positive partial transpose) states of corank one for each $n\ge 3$. With a suitable choice of parameters, we show that they are $n\otimes n$ PPT entangled edge states of corank one for $3\le n\le 1000$. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by $2n-3$, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first explicit examples of such states for $n\ge 4$.