Quantum metrology beyond the asymptotic regime

In quantum metrology, one of the major applications of quantum technologies, the ultimate precision limits with which an unknown parameter encoded in a quantum state can be estimated is often stated in terms of the Cramer-Rao bound. Yet, the Cramer-Rao bound implicitly assumes that sufficiently many independent and identically distributed copies of the state are available to estimate the expectation value of a suitable observable that serves as an estimator. The operational relevance of the Cramer-Rao bound can be cast in doubt in regimes in which this expectation value depends significantly on events that happen with vanishing probability, and when only a limited number of samples are available. In this work, we consider the fundamental limits on estimation when only a finite number of copies of the state are available. Our approach is to formulate the task of parameter estimation analogously to the task of multiple hypothesis testing, thereby opening up the extensively-developed toolbox of hypothesis testing for use in the realm of quantum metrology. We start by defining a single-shot, operational definition of a measure of sensitivity, which is formulated as a semi-definite program, and derive bounds relating it to well-known one-shot entropy measures. We then give an upper bound on the asymptotic decay rate of the error in terms of Chernoff divergences. Finally, when restricted to the case of a pure state evolving under a fixed Hamiltonian, we find explicit formulas for the success probability and characterize the optimal probe states. This perspective on quantum metrology opens up a plethora of new directions of research into the non-asymptotic setting of quantum metrology.