Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information

TL;DR

This paper develops a Gelfand-Tsetlin basis for the algebra of partially transposed permutation matrices, enabling significant simplifications in quantum information tasks like channel optimization and quantum teleportation.

Contribution

It provides an explicit formula for the algebra's generators in the Gelfand-Tsetlin basis and applies this to improve quantum protocols and algorithms.

Findings

Simplifies semidefinite programs in quantum information

Derives an efficient quantum circuit for port-based teleportation

Provides a classical algorithm for the mixed quantum Schur transform

Abstract

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.