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.