Irreducible matrix representations for the walled Brauer algebra

TL;DR

This work develops a matrix-model framework for the walled Brauer algebra via the algebra of partially transposed permutation operators $\mathcal{A}^d_{p,p}$, central to mixed Schur–Weyl duality in quantum information. The authors construct irreducible matrix units in the second-highest ideal $\mathcal{M}^{(p-1)}$ that are covariant under the $\mathbb{C}[\mathcal{S}_p]\times\mathbb{C}[\mathcal{S}_p]$ action, and present a recursive approach to generate irreducible bases for the remaining ideals. They introduce spanning operator families $F$ and $H$ to build explicit bases for $\mathcal{M}^{(p)}$ and $\mathcal{M}^{(p-1)}$, including a diagonalization scheme for the lower-ideal construction via matrices $B^{\mu\nu}$ and unitary diagonalisers $U^{\mu\nu}$. The paper then derives twirl/trace identities and computes the spectra of the twirled operators $\rho(p)$ and $\rho(p-1)$ in these irreducible bases, with a detailed analytic example at $p=3$ and $d=3$ illustrating block-diagonal structure and eigenoperators. These results have practical implications for port-based teleportation and related quantum-information tasks that exploit walled Brauer symmetry. The methods offer a pathway to efficient symmetry-adapted computations in quantum information and multi-partite quantum processes.

Abstract

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 $\mathbb{C}[\mathcal{S}_p]\times \mathbb{C}[\mathcal{S}_p]$ subalgebra, where $\mathcal{S}_p$ 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. 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.

Figures (6)

• Figure 1: Graphic presents five possible Young frames for $p=4$, which also corresponds to all possible abstract irreducible representations of $\mathcal{S}_4$. Considering representation space $(\mathbb{C}^d)^{\otimes 4}$, there appear only irreps for which the height of corresponding Young frames is no larger than $d$. For example, if $d=2$, irreps $(2,1,1), (1,1,1,1)$ do not appear since $\operatorname{ht}((2,1,1)) = 3$, and $\operatorname{ht}((1,1,1,1))=4$ respectively.
• Figure 2: Graphic presents possible Young diagrams $\alpha\vdash 2$, which can be obtained from a diagram $\mu=(2,1)$ by removing a single box, depicted here in red. In this particular case, by writing $\alpha =\mu-\Box$ only possible $\alpha$ are $(2), (1,1)$. In the same manner, we define adding a box into a Young diagram.
• Figure 3: The Bratteli diagram with four consecutive layers labeled by permutation group from $\mathcal{S}_1$ to $\mathcal{S}_4$. By the red arrow, we present a possible path from irrep $\mu=(1)$ of $\mathcal{S}_1$ to irrep $\mu'=(2,1,1)$ of $\mathcal{S}_4$.
• Figure 4: A graphical depiction of the composition of two diagrams $\sigma,\pi \in \mathcal{B}_{3,2}^\delta$. Identifying a closed loop (in red) results in multiplying the diagram by a scalar $\delta \in \mathbb{C}$, showing that the composition $\sigma\pi$ remains within $\mathcal{B}_{3,2}^\delta$.
• Figure 5: Top: Graphic illustration on the diagram level of the operator $V^{(p)}$ from equation \ref{['eq:Vp']}. Bottom: Graphic illustration on the diagram level of the operator $V^{(p-1)}$ from equation \ref{['eq:Vp-1']}. On the abstract level, these objects are special cases of elements from the walled Brauer algebra $\mathcal{B}_{p,p}^\delta$ with $\delta=d$. On the representation space $(\mathbb{C}^d)^{\otimes 2p}$ operators $V^{(p)},V^{(p-1)}$ are elements of the algebra of the partially transposed permutation operators $\mathcal{A}_{p,p}^{d}$.

Theorems & Definitions (58)

• Lemma 1: Lemma 2 in Ebler_2023
• Lemma 2: Lemma 7 in StudzinskiIEEE22
• proof
• Remark 4
• proof
• Lemma 7
• proof
• Proposition 8
• proof
• Theorem 9