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arXiv:2112.12738 · Journal of Physics A Mathematical and Theoretical

Positive maps from the walled Brauer algebra

Maria Balanzó-Juandó, Michał Studziński, Felix Huber

Published Feb 21, 2024• 4 citations
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Abstract

Abstract We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.

DOI
10.1088/1751-8121/ad2b86

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Abstract We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley–Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.
Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras ℬ( ) and ℬ( ), where and are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on ℬ( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.
This paper investigates the representation theory of the algebra of partially transposed permutation operators, Ap,pd\mathcal{A}^d_{p,p}Ap,pd​, 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 C[Sp]×C[Sp]\mathbb{C}[S_p]\times \mathbb{C}[S_p]C[Sp​]×C[Sp​] subalgebra, where SpS_pSp​ 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.