Resolving the Edge of a Quantum Pyramid

• URL: http://arxiv.org/abs/2606.14698v1
• Date: Fri, 12 Jun 2026 17:59:33 GMT
• Title: Resolving the Edge of a Quantum Pyramid
• Abstract summary: We prove the entropy inequalities of Holevo and Utkin, which certify optimality for obtuse and flat pyramids.<n>For obtuse pyramids, our key contribution is a rigorous proof that local minimizers of the corresponding entropy inequality cannot have three distinct coordinate values.<n>For flat pyramids, we prove a tight $ellp$ inequality for zero-sum vectors that was recently conjectured.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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