  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.
