Group-Adapted Irreducible Matrix Units for the Walled Brauer Algebra
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918322239438848 |
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| author | Studziński, Michał Młynik, Tomasz Mozrzymas, Marek Horodecki, Michał Grinko, Dmitry |
| author_facet | Studziński, Michał Młynik, Tomasz Mozrzymas, Marek Horodecki, Michał Grinko, Dmitry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Group-Adapted Irreducible Matrix Units for the Walled Brauer Algebra Studziński, Michał Młynik, Tomasz Mozrzymas, Marek Horodecki, Michał Grinko, Dmitry Quantum Physics Mathematical Physics 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. |
| title | Group-Adapted Irreducible Matrix Units for the Walled Brauer Algebra |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2501.13067 |