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Reliable Quantum State Estimation from Quantum Scoring Rules

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Inferring a quantum system\'s state, from repeated measurements, is critical for verifying theories and designing quantum hardware. It\'s also surprisingly easy to do wrong, as illustrated by maximum likelihood estimation (MLE), the current state of the art. I\'ll explain why MLE yields unreliable and rank-deficient estimates, why you shouldn\'t be a quantum frequentist, and why we need a different approach. I\'ll show how operational divergences -- well-motivated metrics designed to evaluate estimates -- follow from quantum strictly proper scoring rules. This motivates Bayesian Mean Estimation (BME), and I\'ll show how it fixes most of the problems with MLE. I\'ll conclude with a couple of speculations about the future of quantum state and process estimatio