Instance-Optimal Quantum State Certification with Entangled Measurements
This addresses an open question in quantum information theory by providing instance-optimal bounds for certification with entangled measurements, which is incremental but important for optimizing quantum testing protocols.
The paper tackles the problem of determining the optimal copy complexity for quantum state certification on an instance-by-instance basis when testers can use fully entangled measurements, showing it equals the worst-case complexity multiplied by the fidelity between the hypothesis state and the maximally mixed state.
We consider the task of quantum state certification: given a description of a hypothesis state $σ$ and multiple copies of an unknown state $ρ$, a tester aims to determine whether the two states are equal or $ε$-far in trace distance. It is known that $Θ(d/ε^2)$ copies of $ρ$ are necessary and sufficient for this task, assuming the tester can make entangled measurements over all copies [CHW07,OW15,BOW19]. However, these bounds are for a worst-case $σ$, and it is not known what the optimal copy complexity is for this problem on an instance-by-instance basis. While such instance-optimal bounds have previously been shown for quantum state certification when the tester is limited to measurements unentangled across copies [CLO22,CLHL22], they remained open when testers are unrestricted in the kind of measurements they can perform. We address this open question by proving nearly instance-optimal bounds for quantum state certification when the tester can perform fully entangled measurements. Analogously to the unentangled setting, we show that the optimal copy complexity for certifying $σ$ is given by the worst-case complexity times the fidelity between $σ$ and the maximally mixed state. We prove our lower bounds using a novel quantum analogue of the Ingster-Suslina method, which is likely to be of independent interest. This method also allows us to recover the $Ω(d/ε^2)$ lower bound for mixedness testing [OW15], i.e., certification of the maximally mixed state, with a surprisingly simple proof.