Abstract
The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumness , a measure that has been recently introduced in order to quantify how incompatible the parameters to be estimated are. In particular, is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cramér-Rao bound (i.e., the matrix generalization of the single-parameter quantum Cramér-Rao bound). For all the estimation problems considered, we evaluate the quantumness and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model, it is equal to the difference between the bounds and, in general, their behavior qualitatively coincide. On the other hand, we also find evidence that, for certain quantum statistical models, the bound is not in tight, and thus may overestimate the degree of quantum incompatibility between parameters.
Highlights
Quantum sensing is the art of exploiting quantum features as coherence to improve the sensitivity of measuring devices [1,2,3,4,5,6,7,8,9,10,11]
Our results show that the two quantities always share the same qualitative behavior, and give evidence that the bound is not always tight, i.e., R is a useful quantity to characterize multiparameter estimation problems, but it may overestimate the incompatibility between parameters
We have studied in detail the quantumness of multiparameter quantum statistical models for qubit systems, defined as the incompatibility of the parameters to be jointly estimated
Summary
Quantum sensing is the art of exploiting quantum features as coherence (or decoherence) to improve the sensitivity of measuring devices [1,2,3,4,5,6,7,8,9,10,11]. The Holevo bound is regarded as the most fundamental scalar lower bound, as it is attainable by allowing collective measurements on an asymptotically large number of copies of the quantum state defining the quantum statistical model [40,41]. We take the simplest quantum systems, a qubit, and consider several multiparameter estimation problems involving unitary and noisy channels. For those models, we evaluate its quantumness R that has been recently introduced in [42].
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