Global Minimax Readout of a Qubit Direction

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 $\eta$. 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 $2\mathrm{/}[Nf(\eta )]$, where $f(\eta )=\frac{2\eta -(1-{\eta }^2)\log [(1+\eta )\mathrm{/}(1-\eta )]}{4\eta }$. 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 $4\mathrm{/}{\eta }^2$, whereas collective measurements attain $2(1+\eta )\mathrm{/}{\eta }^2$. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.