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

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.

32 pages

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Cited by in corpus (19)

• Open quantum systems with local and collective incoherent processes: Efficient numerical simulation using permutational invariance
• Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality
• Asymptotic performance of port-based teleportation
• Efficient multi port-based teleportation schemes
• Dimension-free entanglement detection in multipartite Werner states
• Multiport based teleportation -- transmission of a large amount of quantum information
• Optimality of the pretty good measurement for port-based teleportation
• Optimal Multi-port-based Teleportation Schemes
• RepLAB: a computational/numerical approach to representation theory
• Minimal Port-based Teleportation
• Linear programming with unitary-equivariant constraints
• The Asymmetric Quantum Cloning Region
• Universal adjointation of isometry operations using conversion of quantum supermaps
• Teleportation of Post-Selected Quantum States
• From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications
• Square-root measurements and degradation of the resource state in port-based teleportation scheme
• Iterative construction of ${\mathfrak{S}}_p\times {\mathfrak{S}}_p$ group-adapted irreducible matrix units for the walled Brauer algebra
• New constructive counterexamples to additivity of minimum output Rényi p-entropy of quantum channels
• A resource theory of asynchronous quantum information processing