Toward Instance-Optimal State Certification With Incoherent Measurements

TL;DR

This paper establishes quantum state certification bounds that depend on the mixedness and fidelity of the state, revealing a fundamental difference from classical distribution testing.

Contribution

It introduces the first instance-optimal bounds for quantum state certification with incoherent measurements, linking copy complexity to mixedness testing and fidelity.

Findings

Copy complexity is proportional to mixedness testing complexity times fidelity.

Bounds differ significantly from classical distribution testing, highlighting quantum-specific phenomena.

Provides nearly tight bounds for nonadaptive incoherent measurement-based certification.

Abstract

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.