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Area of Science:

  • Quantum mechanics
  • Quantum information theory
  • Mathematical physics

Background:

  • The Wigner function is a quasiprobability representation of quantum mechanics with broad physics applications.
  • Its properties are linked to the symmetry of phase point operators, allowing transformations between any pair.
  • Discrete Wigner functions are crucial for quantum computation and information processing.

Purpose of the Study:

  • To investigate the conditions for high symmetry in discrete Wigner functions.
  • To establish the equivalence between permutation symmetry and unitary 2-designs in discrete quantum systems.
  • To identify unique discrete Wigner functions based on symmetry properties.

Main Methods:

  • Analysis of symmetry groups in discrete quantum systems.
  • Proof of equivalence between permutation symmetry and unitary 2-designs.
  • Investigation of Clifford covariance in different dimensional spaces.

Main Results:

  • High permutation symmetry in discrete Wigner functions is equivalent to the symmetry group being a unitary 2-design.
  • Such symmetry is restricted to odd prime power dimensions, plus dimensions 2 and 8.
  • A unique discrete Wigner function is singled out by this symmetry.
  • Clifford covariance uniquely determines this discrete Wigner function, but it does not exist in even prime power dimensions.

Conclusions:

  • The study identifies unique discrete Wigner functions based on symmetry principles.
  • The findings restrict the existence of highly symmetric Wigner functions to specific dimensions.
  • The results highlight the importance of symmetry and Clifford covariance in quantum representations.