arXiv:1305.6183

Commutant structuture of Ux...xUxU* transformations

Michał Studziński, Michał Horodecki, Marek Mozrzymas

Published May 27, 2013• 4 citations

Abstract

In this paper we have found irreducible representations (irreps) of the algebra of partially transposed permutation operators on last subsystem. We give here direct method of irreps construction for an arbitrary number of subsystems n and local dimension d. Our method is inspired by representation theory of symmetric group S(n), theory of Brauer Algebras and Walled Brauer Algebras.

DOI: 10.48550/arxiv.1305.6183

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arXiv:1305.6183

Commutant structuture of Ux...xUxU* transformations

Michał Studziński, Michał Horodecki, Marek Mozrzymas

Published May 27, 2013• 4 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1305.6183

Related papers

Using partial transpose and realignment to generate local unitary invariants

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This paper investigates the representation theory of the algebra of partially transposed permutation operators,

\mathcal{A}^d_{p,p}

, which provides a matrix representation for the abstract walled Brauer algebra. This algebra has recently gained significant attention due to its relevance in quantum information theory, particularly in the efficient quantum circuit implementation of the mixed Schur-Weyl transform. In contrast to previous Gelfand-Tsetlin type approaches, our main technical contribution is the explicit construction of irreducible matrix units in the second-highest ideal that are group-adapted to the action of

\mathbb{C}[S_p]\times \mathbb{C}[S_p]

subalgebra, where

S_p

is the symmetric group. This approach suggests a recursive method for constructing irreducible matrix units in the remaining ideals of the algebra. The framework is general and applies to systems with arbitrary numbers of components and local dimensions. In addition, we present a complementary construction method based on tensor networks of Clebsch-Gordan coefficients of the unitary group. This approach enables the construction of all group-adapted irreducible matrix units, but requires knowledge of certain Littlewood-Richardson coefficients. This method can be successfully applied for a reasonably small number of particles with the support of dedicated software. The obtained results are applied to a special class of operators motivated by the mathematical formalism appearing in all variants of the port-based teleportation protocols through the mixed Schur-Weyl duality. We demonstrate that the given irreducible matrix units are, in fact, eigenoperators for the considered class.