Permutation Symmetry Determines the Discrete Wigner Function

Abstract

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2-design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.

5+2 pages, connection with unitary 2-designs added, accepted by Phys. Rev. Lett. as Editors' Suggestion

References in corpus (7)

• Contextuality supplies the magic for quantum computation
• Evenly distributed unitaries: on the structure of unitary designs
• Multiqubit Clifford groups are unitary 3-designs
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Super-symmetric informationally complete measurements
• Sharply covariant mutually unbiased bases

Cited by in corpus (32)

• Multiqubit Clifford groups are unitary 3-designs
• Contextuality and Wigner function negativity in qubit quantum computation
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• Equivalence between contextuality and negativity of the Wigner function for qudits
• Simulation of Qubit Quantum Circuits via Pauli Propagation
• Heisenberg-Weyl Observables: Bloch vectors in phase space
• The Clifford group fails gracefully to be a unitary 4-design
• Uniqueness of noncontextual models for stabilizer subtheories
• Quasiprobability representations of quantum mechanics with minimal negativity
• A hidden variable model for universal quantum computation with magic states on qubits
• Super-symmetric informationally complete measurements
• Discrete Wigner Formalism for Qubits and Non-Contextuality of Clifford Gates on Qubit Stabilizer States
• Probability representation of quantum dynamics using pseudostochastic maps
• Mutually unbiased bases as minimal Clifford covariant 2-designs
• Discrete Wigner Functions from Informationally Complete Quantum Measurements
• The role of cohomology in quantum computation with magic states
• The magic of universal quantum computing with permutations
• Quantum speed limits in arbitrary phase spaces
• Limitations of Classically-Simulable Measurements for Quantum State Discrimination
• Classical codes in quantum state space
• Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?
• Hidden variable model for quantum computation with magic states on qudits of any dimension
• Covariant mutually unbiased bases
• Nonexistence of sharply covariant mutually unbiased bases in odd prime dimensions
• Generalized reduction formula for Discrete Wigner functions of multiqubit systems
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space
• Symmetries and Wigner representations of operational theories
• Duality theory for Clifford tensor powers
• Twisted Fourier analysis and pseudo-probability distributions
• Simulating Quantum Circuits by Shuffling Paulis
• Maximally symmetric stabilizer MUBs in even prime-power dimensions