We revisit the basic problem of quantum state certification: given copies of
unknown mixed state $\rho\in\mathbb{C}^{d\times d}$ and the description of a
mixed state $\sigma$, decide whether $\sigma = \rho$ or $\|\sigma -
\rho\|_{\mathsf{tr}} \ge \epsilon$. When $\sigma$ is maximally mixed, this is
mixedness testing, and it is known that $\Omega(d^{\Theta(1)}/\epsilon^2)$
copies are necessary, where the exact exponent depends on the type of
measurements the learner can make [OW15, BCL20], and in many of these settings
there is a matching upper bound [OW15, BOW19, BCL20].
Can one avoid this $d^{\Theta(1)}$ dependence for certain kinds of mixed
states $\sigma$, e.g. ones which are approximately low rank? More ambitiously,
does there exist a simple functional $f:\mathbb{C}^{d\times
d}\to\mathbb{R}_{\ge 0}$ for which one can show that
$\Theta(f(\sigma)/\epsilon^2)$ copies are necessary and sufficient for state
certification with respect to any $\sigma$? Such instance-optimal bounds are
known in the context of classical distribution testing, e.g. [VV17].
Here we give the first bounds of this nature for the quantum setting, showing
(up to log factors) that the copy complexity for state certification using
nonadaptive incoherent measurements is essentially given by the copy complexity
for mixedness testing times the fidelity between $\sigma$ and the maximally
mixed state. Surprisingly, our bound differs substantially from instance
optimal bounds for the classical problem, demonstrating a qualitative
difference between the two settings.

本文主要研究关于量子态证明的基础问题，展示了使用非自适应的非相干测量量子态证明所需的拷贝数目与混合度测量所需的拷贝数目之间存在关联，实现了一个实例优化的上限。