Minimax risk and confidence bounds for quantum state tomography near a known state
Minimax Quantum State Tomography in a Known Trace-Distance Ball
Information Theory
Summary
Quantum state tomography is a way to figure out the exact state of a quantum system by making measurements. This paper studies this task when the unknown state is close to a known reference state within a certain distance. The authors find the best possible accuracy limits you can achieve with any measurement method, depending on how close the states are and how many copies you measure. They also provide mathematical formulas showing how many copies are needed to reach a desired accuracy with high confidence.
What this means in practice
- •For quantum computing engineers: Design measurement processes that optimally estimate quantum states close to known references with minimal errors and confidence guarantees.
- •For quantum hardware testers: Evaluate the number of measurements needed to confidently verify hardware states near expected reference states using improved accuracy bounds.
A theory result. No direct application yet.
Authors
Yonglong Li
Abstract
We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $ρ$. 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 $Γ=d(\mathrm{Tr}\sqrtρ)^2+d^2R$. For $d\ge2$, $0<R\le1$, and $n\ge1$ copies, the expected risk is $Θ(\min\{R,\sqrt{Γ/n}\})$. At failure probability $0<δ\le1/3$, the minimax confidence radius is \[ Θ\!\left(\min\left\{R,\sqrt{\frac{Γ+\log(1/δ)}{n}}\right\}\right). \] For $\varepsilon\le c_0R$ with a universal $c_0>0$, the copy complexity is $Θ((Γ+\log(1/δ))/\varepsilon^2)$. 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.