Resolving the Edge of a Quantum Pyramid

Abstract:Standing on the shoulders of giants, we resolve the quantum pyramids conjecture, confirming the globally information-optimal measurement for an ensemble of equiangular equiprobable pure states, as conjectured by Englert and Řeháček (arXiv:0905.0510). We do so by proving the remaining entropy inequalities of Holevo and Utkin (arXiv:2506.06700), which certify optimality for obtuse and flat pyramids. For obtuse pyramids, our key contribution is a rigorous proof that local minimizers of the corresponding entropy inequality cannot have three distinct coordinate values. We show that eliminating this family can be reduced to a neat algebraic reciprocal inequality relating branches of the Lambert $W$ function, which may be of independent interest. For flat pyramids, we prove a tight $\ell^p$ inequality for zero-sum vectors that was recently conjectured, proved analytically in dimension $d=3$, and computationally verified for $d\leq 200$ by Holevo and Utkin (arXiv:2603.24017). We prove this bound for all $d\geq 2$ via a technique in symmetric inequalities known as the equal variables method.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Functional Analysis (math.FA)

arXiv:2606.14698 [quant-ph]

(or arXiv:2606.14698v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.14698 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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