bibkey: rotundo2023entropic authors: Antonio F. Rotundo; René Schwonnek year: 2023 title: “An entropic uncertainty principle for mixed states” doi: 10.1103/PhysRevResearch.6.033043 url: https://arxiv.org/abs/2303.11382v1 claim: “Conjecture 1 (Extended MUB regime): a doubly stochastic matrix has mixed operator norm d^(1-lambda-mu) whenever ((1-mu)/mu)((1-lambda)/lambda) >= sigma_2^2.” strata_touched:

• D5/S3/Quantum/Information/ExtendedMubNormRefutation license: citation-only triage: anchor

An entropic uncertainty principle for mixed states

Antonio F. Rotundo and René Schwonnek, arXiv:2303.11382v1; Phys. Rev. Research 6, 033043 (2024).

Conjecture 1 on p. 2 of the arXiv v1 paper states:

(Extended MUB regime). Let $C^{(2)}$ be a doubly stochastic matrix. Its norm is equal to that of $C_{\mathrm{MUB}}^{(2)}$, i.e. it is given by eq. (7), as long as $\frac{1-\mu }{\mu }\frac{1-\lambda }{\lambda }\ge {\sigma }_2^2$, where ${\sigma }_2$ is the second largest singular value of $C^{(2)}$.

Equation (7) is $\log|C^{(2)}{\mathrm{MUB}}|{1/\mu\to1/(1-\lambda)} =(1-\lambda-\mu)\log d$. Equation (5) defines the operator norm using the usual vector p-norms, with complex vectors. The formalized claim uses real vectors and real doubly stochastic matrices, with $d\ge 2$ and $0<\mu ,\lambda <1$. The real vector producing the strict lower bound is also an admissible complex vector in equation (5).

The journal version restates Conjecture 1 as an entropic inequality. The norm statement in arXiv v1 is the statement refuted by D5/S3/Quantum/Information/ExtendedMubNormRefutation.result; the journal’s entropic reformulation is outside this theorem.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevResearch.6.033043
• URL: https://arxiv.org/abs/2303.11382v1 — p. 2, Conjecture 1 and equations (5), (7), (9).