Simplified formalism of the algebra of partially transposed permutation operators with applications

TL;DR

This paper advances the representation theory of permutation operator algebras with partial transposition, simplifying key expressions, deriving new irreducible representations, and applying these to analyze port-based teleportation schemes.

Contribution

It introduces partially reduced irreducible representations, simplifies algebraic expressions, and applies these to port-based teleportation, providing new insights and proofs.

Findings

Simplified algebraic expressions for permutation operators with partial transposition.

Derived new irreducible representations and matrix forms for algebra generators.

Connected algebraic properties to teleportation fidelity and scheme characterization.

Abstract

Hereunder we continue the study of the representation theory of the algebra of permutation operators acting on the $n$-fold tensor product space, partially transposed on the last subsystem. We develop the concept of partially reduced irreducible representations, which allows to simplify significantly previously proved theorems and what is the most important derive new results for irreducible representations of the mentioned algebra. In our analysis we are able to reduce complexity of the central expressions by getting rid of sums over all permutations from symmetric group obtaining equations which are much more handy in practical applications. We also find relatively simple matrix representations for the generators of underlying algebra. Obtained simplifications and developments are applied to derive characteristic of the deterministic port-based teleportation scheme written purely in terms of irreducible representations of the studied algebra. We solve an eigenproblem for generators of algebra which is the first step towards to hybrid port-based teleportation scheme and gives us new proofs of asymptotic behaviour of teleportation fidelity. We also show connection between density operator characterising port-based teleportation and particular matrix composed of irreducible representation of the symmetric group which encodes properties of the investigated algebra.