Quantum

Tight Quasiprobabilistic Entropic Uncertainty Relations for Discrete Quantum Systems

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Briefings

A 2026 arXiv preprint (arXiv:2611.04009v1) proves that for discrete quantum systems, the minimum entropy of a quasiprobability distribution is achieved by pure states, contradicting prior conjectures for Wigner and Kirkwood-Dirac distributions. This finding establishes that pure states play the same minimal-uncertainty role as coherent states in continuous-variable systems, with large nonpositivity in these distributions leading to increased uncertainty. The study focuses on mutually unbiased bases and Gross-Wigner distributions, confirming that the conjecture holds only for specific discrete cases, not universally for all positive distributions.',

Key points

• The 2026 arXiv preprint arXiv:2611.04009v1 proves that for discrete quantum systems, the Shannon differential entropy of the Kirkwood-Dirac distribution is minimized by pure states with positive quasiprobability distributions.
• Prior to this work, it was conjectured that the Wigner function's differential entropy of states with positive Wigner function is bounded below by the entropy of pure Gaussian states, a claim disproven by the new proof.

Context

This research addresses the long-standing question of whether pure states universally minimize entropy across different quasiprobability distributions in discrete quantum systems, extending the continuous-variable result to discrete cases.

Limitations

The proof applies only to mutually unbiased bases and Gross-Wigner distributions, not all possible discrete systems. The study does not cover all quasiprobability distributions, and the results are based on a preprint with potential future revisions.

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