2017/04/30 by T. Hashimoto, A. Hayashi, M. Horibe · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Mathematical analysis #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Quantum state #Statistical physics #Uncertainty principle #quant-ph
paper · pdf · doi:10.1103/physreva.99.022126
published in Physical Review A 99(2) (American Physical Society) · 12 pages, 3 figures; improved presentation, final version
openalex publication_date 2019/02/25 · arxiv created 2019/02/26 · arxiv updated 2019/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We construct minimum-uncertainty states and a non-negative quasiprobability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relation of the two unitary operators related by the discrete Fourier transformation. It is shown that some assumptions in their proof can be justified by the use of the Perron-Frobenius theorem. The minimum-uncertainty states are the ones that saturate this uncertainty inequality. The continuum limit is closely analyzed by introducing a scale factor in the limiting scheme. Using the minimum-uncertainty states, we construct a non-negative quasiprobability distribution. Its marginal distributions are smeared out. However, we show that this quasiprobability is optimal in the sense that there does not exist a non-negative quasiprobability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasiprobability is complete, i.e., it contains full information of the state. We show that the obtained quasiprobability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.