Minimax Quantum State Tomography in a Known Trace-Distance Ball

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $\rho$. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write $\mathrm{\Gamma }=d(\mathrm{T}\mathrm{r}\sqrt{\rho }{)}^2+d^{2}R$. For $d\ge 2$, $0<R\le 1$, and $n\ge 1$ copies, the expected risk is $\mathrm{\Theta }(\min \{R,\sqrt{\mathrm{\Gamma }\mathrm{/}n}\})$. At failure probability $0<\delta \le 1\mathrm{/}3$, the minimax confidence radius is $$\mathrm{\Theta }\text{\hspace{0.17em}⁣}(\min \{R,\sqrt{\frac{\mathrm{\Gamma }+\log (1\mathrm{/}\delta )}{n}}\})\mathrm{.}$$ For $\epsilon \le c_{0}R$ with a universal $c_0>0$, the copy complexity is $\mathrm{\Theta }((\mathrm{\Gamma }+\log (1\mathrm{/}\delta ))\mathrm{/}{\epsilon }^2)$. For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.