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We study representation theory of the partially transposed permutation matrixalgebra, a matrix representation of the diagrammatic walled Brauer algebra.This algebra plays a prominent role in mixed Schur-Weyl duality that appears invarious contexts in quantum information. Our main technical result is anexplicit formula for the action of the walled Brauer algebra generators in theGelfand-Tsetlin basis. It generalizes the well-known Gelfand-Tsetlin basis forthe symmetric group (also known as Young's orthogonal form or Young-Yamanouchibasis).  We provide two applications of our result to quantum information. First, weshow how to simplify semidefinite optimization problems overunitary-equivariant quantum channels by performing a symmetry reduction.Second, we derive an efficient quantum circuit for implementing the optimalport-based quantum teleportation protocol, exponentially improving the knowntrivial construction. As a consequence, this also exponentially improves theknown lower bound for the amount of entanglement needed to implement unitariesnon-locally.  Both applications require a generalization of quantum Schur transform totensors of mixed unitary symmetry. We develop an efficient quantum circuit forthis mixed quantum Schur transform and provide a matrix product staterepresentation of its basis vectors. For constant local dimension, this yieldsan efficient classical algorithm for computing any entry of the mixed quantumSchur transform unitary.

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