Understand
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.