On bipartite operators defined by sets of completely different permutations

Abstract

We introduce a class of bipartite operators acting on H⊗H (H being an n-dimensional Hilbert space) defined by a set of n Completely Different Permutations (CDPs). 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 the so-called PPT (positive under partial transposition) states which are of great importance for quantum information. Using well-known relation between bipartite operators and linear maps, one can use also construct linear maps related to CDPs.