Quantum systems as embarrassed colleagues: what do tax evasion and state tomography have in common?

Quantum state estimation (a.k.a. ``tomography'') plays a key role in designing quantum information processors. As a problem, it resembles probability estimation -- e.g. for classical coins or dice -- but with some subtle and important discrepancies. We demonstrate an improved classical analogue that captures many of these differences: the ``noisy coin.'' Observations on noisy coins are unreliable -- much like soliciting sensitive information such as ones tax preparation habits. So, like a quantum system, it cannot be sampled directly. Unlike standard coins or dice, whose worst-case estimation \emph{risk} scales as $1/N$ for all states, noisy coins (and quantum states) have a worst-case risk that scales as $1/\sqrt{N}$ and is overwhelmingly dominated by nearly-pure states. The resulting optimal estimation strategies for noisy coins are surprising and counterintuitive. We demonstrate some important consequences for quantum state estimation -- in particular, that adaptive tomography can recover the $1/N$ risk scaling of classical probability estimation.