On Bipartite Operators Defined by Completely Different Permutations

TL;DR

This paper introduces a new class of bipartite operators based on Completely Different Permutations (CDP), providing a method to construct PPT states and related linear maps for quantum information applications.

Contribution

It presents a novel construction of bipartite operators using CDPs, revealing their structure and properties, especially when CDPs form an abelian group, and links to PPT states.

Findings

Construction of bipartite operators from CDPs.

Partial transpose of these operators corresponds to a different set of CDPs.

Potential to generate new PPT states for quantum information.

Abstract

We introduce a class of bipartite operators acting on $\mathcal{H} \otimes \mathcal{H}$ ($\mathcal{H}$ being an $n$-dimensional Hilbert space) defined by a set of $n$ Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.