Beyond Shor's Conjecture: Optimal Quantum Measurements at Fixed Error

Abstract

We prove Shor's conjecture: an orthogonal measurement maximizes the information obtainable from any two quantum states in finite dimension. We then show that an error constraint can require three outcomes. For a class of full-rank qubit pairs, the best orthogonal basis switches abruptly as the error penalty increases. Near the switch, three outcomes yield more information at the same error than any orthogonal measurement or random mixture of them. The extra outcome distinguishes confidence levels in one guess; every outcome is used.

arXiv comment33 pages, 6 figures, 4 tables. Substantially expanded to include information-error trade-offs, a three-outcome measurement advantage, and generalized losses. Title and author list updated. The original version was publicly available on Zenodo on 13 August 2026: https://doi.org/10.5281/zenodo.21911604