Applications of the generalized Pauli group in quantum information

          @misc{ pirsa_PIRSA:07100013,
            doi = {10.48660/07100013},
            url = {https://pirsa.org/07100013},
            author = {Durt, Thomas},
            keywords = {Quantum Information},
            language = {en},
            title = {Applications of the generalized Pauli group in quantum information},
            publisher = {Perimeter Institute},
            year = {2007},
            month = {oct},
            note = {PIRSA:07100013 see, \url{https://pirsa.org}}
          }
          

Abstract

It is known that finite fields with d elements exist only when d is a prime or a prime power. When the dimension d of a finite dimensional Hilbert space is a prime power, we can associate to each basis state of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group. Its elements can be expressed, in terms of the elements of a Galois field. This group presents numerous applications in Quantum Information Science e.g. tomography, dense coding, teleportation, error correction and so on. The aim of our talk is to give a general survey of these properties and to present recently obtained results in connection with three problems: -the so-called ''Mean King's problem'' in prime power dimension, -discrete Wigner distributions, -and quantum tomography . Finally we shall discuss a limitation of the possible dimensions in which the so-called epistemic interpretation can be consistently formulated, in relation with the existence of finite affine planes, Euler's conjecture and the 36 officers problem.