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.

翻译：我们重新审视量子状态认证的基本问题 : 当 $\\ grama\ in\ mathbb{C\\\ d\ f time d} 提供未知混合状态 $\ rho\ prho\ $\ rgma} 或 $\ rho$\ gramafsf\ tr\\\ ge\ \ eepsilon$ 最大混合时, 这是混合度测试, 并且已知 $( d\\\ theta(1)} 位数差异测试 ) 需要 $( lapha) 和 $\\\ c\\\\\ time d} 混合度的描述类型类型, 在两个混合状态中可以避免这种 $( lagreg$), 例如, 更雄心的是, 存在一个简单的功能 $\\\\\\\\\\\\\ lideal deal decregial\ drass report 。