Quantum Metrology Assisted by Abstention

Abstract

The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an inconclusive answer at readout, can drastically improve the measurement precision. We focus on phase estimation and quantify the required amount of abstention for a given precision. We also develop analytical tools to obtain the asymptotic behavior of the precision and required rate of abstention for arbitrary pure qubit states.In its simplest form, a quantum metrology problem has the following structure. A quantum system undergoes some physical interaction determined by some continuous parameters. The value of the parameters gets imprinted onto the evolved state and the task of the metrologist consists in uncovering it. For this purpose, she performs a suitable measurement on the system and, based on its outcome, she produces a guess for the unknown value of the parameters as accurate as possible [1]. The overall performance of the whole procedure can be quantified by the average of a figure of merit (typically the fidelity) over some a priori distribution of the parameters and over all possible outcomes. Phase estimation with pure states of N qubits is a paradigmatic example of this endeavor and will serve as the main exemplification of our findings.In standard parameter estimation protocols [2-5] the experimentalist is expected to produce a conclusive answer (maybe not right or accurate enough), at each run of the experiment. Here we show that there are situations where the ultimate precision of the standard approach can be improved substantially if one allows for a number of inconclusive responses, where the metrologist abstains from producing a guess. This is especially relevant in situations where she can afford to re-run the experiment (i.e. she can easily prepare a new instance of the problem) or where she prioritises having high-quality estimates. Abstention has already been used in the context of state discrimination [6][7][8][9], where some fixed rate Q of inconclusive outcomes can lower the probability of error significantly (even down to zero, as in unambiguous discrimination [6]) and can be seen as a particular example of post-selection [10].In the estimation framework the effects of abstention have hardly been considered. In the cases studied so far in the literature [11,12], abstention has limited impact for large samples unless the experimentalist refrains from producing an estimate most of the time. Here we show that there are relevant cases where abstention has a dramatic effect. Though we focus on phase estimation, other problems, such as direction estimation, can be tackled in the same way (results will be given elsewhere [13]).In this letter, we also present a very general technique to obtain the asymptotic form of pure-state parameter estimation problems, with or without abstention. The main idea is that the components of the probe state can be viewed as a discretization of some continuous function ϕ(t) on...

Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes

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Optimal quantum operations at zero energy cost

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