Global Minimax Readout of a Qubit Direction

• URL: http://arxiv.org/abs/2608.16840v1
• Date: Mon, 17 Aug 2026 17:26:32 GMT
• Title: Global Minimax Readout of a Qubit Direction
• Abstract summary: We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length $$.<n>Every fixed-local architecture, including recorded classical randomization, reduces exactly to a zero-barycenter probability measure on the Bloch ball.<n>For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid.
• Score: 5.246986428523558
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length $η$. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For $N$ copies, inverse-Fisher $A$ loss has the exact value $2/[Nf(η)]$, where $ f(η)= \frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely $A$- and $D$-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have $A$-loss coefficient $4/η^2$, whereas collective measurements attain $2(1+η)/η^2$. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.

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