The Quantum Schur Transform: Theory and Implementations

The quantum Schur transform is a unitary change of basis from the computational product basis to the so-called Schur basis, a basis labeled by the irreducible representations of the symmetric and unitary groups, based on the Schur-Weyl duality in the group representation theory. The latter basis is called the Schur basis. The quantum Schur transform is utilized in several quantum information protocols for optimal purification and spectrum estimation, universal entanglement concentration and source coding, decoherence-free subspaces, sample efficient state tomography, quantum communications without shared reference frame, etc.