We study representation theory of the partially transposed permutation matrix algebra, a matrix representation of the diagrammatic walled Brauer algebra. This algebra plays a prominent role in mixed Schur-Weyl duality that appears in various contexts in quantum information. Our main technical result is an explicit formula for the action of the walled Brauer algebra generators in the Gelfand-Tsetlin basis. It generalizes the well-known Gelfand-Tsetlin basis for the symmetric group (also known as Young's orthogonal form or Young-Yamanouchi basis). We provide two applications of our result to quantum information. First, we show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels by performing a symmetry reduction. Second, we derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol, exponentially improving the known trivial construction. As a consequence, this also exponentially improves the known lower bound for the amount of entanglement needed to implement unitaries non-locally. Both applications require a generalization of quantum Schur transform to tensors of mixed unitary symmetry. We develop an efficient quantum circuit for this mixed quantum Schur transform and provide a matrix product state representation of its basis vectors. For constant local dimension, this yields an efficient classical algorithm for computing any entry of the mixed quantum Schur transform unitary.
We study representation theory of the partially transposed permutation matrix algebra, a matrix representation of the diagrammatic walled Brauer algebra. This algebra plays a prominent role in mixed Schur-Weyl duality that appears in various contexts in quantum information. Our main technical result is an explicit formula for the action of the walled Brauer algebra generators in the Gelfand-Tsetlin basis. It generalizes the well-known Gelfand-Tsetlin basis for the symmetric group (also known as Young's orthogonal form or Young-Yamanouchi basis). We provide two applications of our result to quantum information. First, we show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels by performing a symmetry reduction. Second, we derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol, exponentially improving the known trivial construction. As a consequence, this also exponentially improves the known lower bound for the amount of entanglement needed to implement unitaries non-locally. Both applications require a generalization of quantum Schur transform to tensors of mixed unitary symmetry. We develop an efficient quantum circuit for this mixed quantum Schur transform and provide a matrix product state representation of its basis vectors. For constant local dimension, this yields an efficient classical algorithm for computing any entry of the mixed quantum Schur transform unitary.
This thesis explores the interplay between representation theory and quantum information. Specifically, we focus on mixed Schur–Weyl duality, which considers the action of the unitary group on mixed tensors. This setting naturally arises in quantum information tasks involving unitary-equivariant channels, such as port-based teleportation, quantum majority vote, and universal transposition of unitary operators. A key contribution of this thesis is an explicit derivation of the action of the generators of the partially transposed permutation matrix algebra—the commutant of the mixed unitary action—in the Gelfand–Tsetlin basis. As another key result of this thesis, we develop efficient quantum circuits for the mixed quantum Schur transform, a novel primitive in quantum information. The key ingredient of our construction is new efficient circuits for the dual Clebsch–Gordan transform of the unitary group. A significant application of our findings is the construction of efficient quantum algorithms for port-based teleportation, a variant of quantum teleportation that eliminates the need for corrective operations. Another application is a symmetry reduction of semidefinite optimisation problems with unitary equivariance symmetry. Finally, we study the extendibility of quantum states possessing unitary, mixed unitary, or orthogonal symmetry on the complete graph. We obtain analytically the exact maximum values for projections onto the maximally entangled state and the antisymmetric state for each of the three symmetry classes. This thesis demonstrates the usefulness of mixed Schur–Weyl duality in quantum information and computing. We expect that our tools will help address other problems in other areas of quantum information processing, such as communication, cryptography, and simulation.
This paper investigates the representation theory of the algebra of partially transposed permutation operators, 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] subalgebra, where Sp 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.
Port-based teleportation (PBT) is a variant of quantum teleportation that, unlike the canonical protocol by Bennett et al., does not require a correction operation on the teleported state. Since its introduction by Ishizaka and Hiroshima in 2008, no efficient implementation of PBT was known. We close this long-standing gap by building on our recent results on representations of partially transposed permutation matrix algebras and mixed quantum Schur transform. We construct efficient quantum algorithms for probabilistic and deterministic PBT protocols on n ports of arbitrary local dimension, both for EPR and optimized resource states. We describe two constructions based on different encodings of the Gelfand-Tsetlin basis for n qudits: a standard encoding that achieves O(n) time and O(nlog(n)) space complexity, and a Yamanouchi encoding that achieves O(n2) time and O(log(n)) space complexity, both for constant local dimension and target error. We also describe efficient circuits for preparing the optimal resource states.
The Schur transform, which block-diagonalizes the tensor representation U⊗n of the unitary group Ud on n qudits, is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) to the case of mixed tensor U⊗n⊗Uˉ⊗m, where Uˉ is the dual representation. This representation is the symmetry of unitary-equivariant channels, which find various applications in quantum majority vote, multiport-based teleportation, asymmetric state cloning, black-box unitary transformations, etc. The "mixed" Schur transform contains several natural extensions of the representation theory used in the Schur transform, in which the main ingredient is a duality between the mixed tensor representations and the walled Brauer algebra. Another element is an efficient implementation of a "dual" Clebsch-Gordan transform for Uˉ. The overall circuit has complexity O((n+m)d4). Finally, we show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings and discuss other potential applications, including an extension of permutational quantum computing that includes partial transposes.
Exploiting permutation invariance to reduce the exponential scaling of semidefinite programs in quantum information has emerged as a powerful computational technique. In this work, we develop a systematic framework for using this reduction via Schur-Weyl duality for optimization problems, and establish methods that allow one to work fully inside the permutation invariant subspace while performing operations such as (partially) applying channels and taking (partial) traces, or computing expressions like the quantum relative entropy. We then apply our techniques to the problem of computing efficient lower bounds on the channel fidelity over n parallel uses of a quantum channel. The algorithm, which we call symmetric seesaw method, exploits permutation-invariant codes to yield improved lower bounds on the channel fidelity over n uses of the depolarizing and amplitude-damping channel in the regime of tens of channel uses, and was used in [arxiv:2604.27042] to demonstrate non-asymptotic superactivation of quantum capacity for n=17. An implementation of our methods, aimed at being suitable for various quantum information theoretic optimization problems, is also available as an open-source Python package.
The walled Brauer algebras BN(m,n) govern Schur--Weyl duality for unitary groups U(N) acting on mixed tensor spaces VN⊗m⊗VN⊗n and play an important role in applications ranging from AdS/CFT to quantum information theory. In the stable regime N≥m+n the algebra is semisimple and its representation theory is well understood. For N<m+n, however, BN(m,n) becomes non-semisimple. The representation of the algebra on tensor space has a non-trivial kernel and the corresponding quotient algebra is semisimple, with representation dimensions differing from those in the stable regime.
We introduce \emph{restricted Bratteli diagrams}, obtained by modifying the standard Bratteli diagrams for BN(m,n). This construction provides a systematic way to use representation-theoretic data from the stable regime to compute the dimension modifications arising in the non-semisimple regime. In the regime N=m+n−l, with l small compared to m,n, we show that the restricted diagrams exhibit a stability property and enable an efficient counting of the paths responsible for these dimension corrections.
Remarkably, the resulting generating functions are governed by the partition function of an infinite tower of simple harmonic oscillators. We briefly discuss implications for the construction of orthogonal bases of matrix invariants in gauge theory and related applications in quantum information theory.