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...

We address the problem of estimating pure qubit states with nonideal (noisy) measurements in the multiple-copy scenario, where the data consist of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are illustrated for small values of N . For asymptotically large N , we derive analytical expressions of the fidelity and the probability of abstention and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback-Leibler (relative) entropy. As a byproduct, we give an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides the most significant improvement as compared to the standard approach.

The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.

We design an efficient and constructive algorithm to decompose any generalized quantum measurement into a convex combination of extremal measurements. We show that if one allows for a classical post-processing step only extremal rank-1 POVMs are needed. For a measurement with N elements on a d-dimensional space, our algorithm will decompose it into at most (N −1)d+1 extremals, whereas the best previously known upper bound scaled as d 2 . Since the decomposition is not unique, we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property.

It has recently been shown that probabilistic protocols based on postselection boost the performances of the replication of quantum clocks and phase estimation. Here we demonstrate that the improvements in these two tasks have to match exactly in the macroscopic limit where the number of clones grows to infinity, preserving the equivalence between asymptotic cloning and state estimation for arbitrary values of the success probability. Remarkably, the cloning fidelity depends critically on the number of rationally independent eigenvalues of the clock Hamiltonian. We also prove that probabilistic metrology can simulate cloning in the macroscopic limit for arbitrary sets of states when the performance of the simulation is measured by testing small groups of clones.