Mixed Schur–Weyl duality in quantum information

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.