Toward Instance-Optimal State Certification With Incoherent Measurements

We revisit the basic problem of quantum state certification: given copies of unknown mixed state ρ ∈ Cd×d and the description of a mixed state σ, decide whether σ = ρ or ‖σ − ρ‖tr ≥ ε. When σ is maximally mixed, this is mixedness testing, and it is known that Ω(d/ε) 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 dependence for certain kinds of mixed states σ, e.g. ones which are approximately low rank? More ambitiously, does there exist a simple functional f : Cd×d → R≥0 for which one can show that Θ(f(σ)/ε) copies are necessary and sufficient for state certification with respect to any σ? 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 σ 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. This work was supported in part by NSF Award 2103300, NSF CAREER Award CCF-1453261, NSF Large CCF-1565235, and Ankur Moitra’s ONR Young Investigator Award. Some of this work was done while the author was working at Microsoft Quantum. Supported by NSF grant FET1909310 and ARO grant W911NF2110001. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF). ar X iv :2 10 2. 13 09 8v 2 [ qu an tph ] 1 0 N ov 2 02 1