Sinkhorn-Knopp Theorem for PPT states

TL;DR

This paper presents an algorithm to determine if a PPT quantum state can be transformed into a filter normal form, aiding the study of quantum entanglement, by analyzing associated positive maps and eigenvectors.

Contribution

It introduces a novel algorithm for checking the equivalence of positive maps to doubly stochastic maps for PPT states, extending to mixed dimensions.

Findings

Algorithm effectively tests for filter normal form conversion.

Extension of the method to PPT states in higher dimensions.

Provides a practical tool for entanglement analysis.

Abstract

Given a PPT state $A=\sum_{i=1}^nA_i\otimes B_i \in M_k\otimes M_k$ and a vector $v\in\Im(A)\subset\mathbb{C}^k\otimes\mathbb{C}^k$ with tensor rank $k$, we provide an algorithm that checks whether the positive map $G_A:M_k\rightarrow M_k$, $G_A(X)=\sum_{i=1}^n tr(A_iX)B_i$, is equivalent to a doubly stochastic map. This procedure is based on the search for Perron eigenvectors of completely positive maps and unique solutions of, at most, $k$ unconstrained quadratic minimization problems. As a corollary, we can check whether this state can be put in the filter normal form. This normal form is an important tool for studying quantum entanglement. An extension of this procedure to PPT states in $M_k\otimes M_m$ is also presented.