Minimax Quantum State Tomography in a Known Trace-Distance Ball

Abstract: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 $\Gamma=d(\mathrm{Tr}\sqrt\rho)^2+d^2R$. For $d\ge2$, $0<R\le1$, and $n\ge1$ copies, the expected risk is $\Theta(\min\{R,\sqrt{\Gamma/n}\})$. At failure probability $0<\delta\le1/3$, the minimax confidence radius is \[ \Theta\!\left(\min\left\{R,\sqrt{\frac{\Gamma+\log(1/\delta)}{n}}\right\}\right). \] For $\varepsilon\le c_0R$ with a universal $c_0>0$, the copy complexity is $\Theta((\Gamma+\log(1/\delta))/\varepsilon^2)$. 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.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2610.05374 [quant-ph]

(or arXiv:2610.05374v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.05374 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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