Quantum estimation via sequential measurements

Abstract

The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number N of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies (Guţă M 2011 Phys. Rev. A 83 062324; van Horssen M and Guţă M 2015 J. Math. Phys. 56 022109) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.See figure 1(a). In this context, the ultimate limits on the attainable precision in the estimation of g, optimized with respect to the general detection strategy, can be computed, resulting in the so-called quantum Cramér-Rao bound, which exhibits the functional dependence upon g r via the quantum Fisher information. See e.g. [2,3,[7][8][9][10][11][12].In many situations of physical interest, however, the possibility of reinitializing the setup to the same state is not necessarily guaranteed. In the present study we are going to consider a different scheme, in which a single probing system undergoes multiple applications of g L while being monitored during the process without being

Estimating temperature via sequential measurements

“…In the next section we shall study the behavior of (15) and (18) for fixed choices of the POVM operators M s and for fixed values of the number of the iterations n. In particular, we shall focus on the functional dependence of these quantities with respect to the input state ρ 0 of the probe.…”

“…which reads as in (24) ]. Replacing (22) into (15), and (25) into (18) we can now study the FI of the two procedures for different choices of the POVM parameter ϕ and for different values of the iterations n. In particular in the following subsections we shall focus on the dependence upon the input state ρ 0 of the probe Q. For the sake of simplicity and without loss of generality, the times t (or τ ) will be parametrized in units of the coupling constant γ, and the bath temperature T in units of the qubit energy gap Ω/k B .…”

“…In particular, the arbitrary initialization of independent probing systems at the beginning of the estimation procedure, or of a single probe at disposal after each measurement process, might encounter some obstructions, due to fundamental or practical reasons. One way to circumvent such difficulty is to rely on sequential measurement schemes (SMSs) [15][16][17][18], where repeated consecutive measurements are performed on a single probe while it is still in interaction with the bath without reinitializing it. The performance of SMS will be therefore compared with the IID protocol in different specific situations, taking the Fisher informations (FIs) [19][20][21][22][23] as the figure of merit for the corresponding temperature estimation accuracies.…”

Collisional Quantum Thermometry

“…by preparing an initially entangled N-ancilla state [1,2]. Alternatively, one may perform N sequential measurements on the same ancilla [19][20][21][22], in such a way that the outcomes are correlated. Specifically in the context of thermometry, this method would generally not improve the accuracy of temperature estimation [21], but this could in principle change using adaptive schemes [22].…”

Pulsed Quantum-State Reconstruction of Dark Systems

“…However, even with the rapid progress in the coherent manipulation and quantum-state tomography of several quantum systems, such as photons [6,7], electron spins [8][9][10], atomic qubits [11], superconducting circuits [12,13], and mechanical resonators [14,15], many quantum systems still remain difficult to access for a direct observation of their state, systems we will refer to as dark. In order to circumvent the requirement of such a direct access, a promising technique is to employ an auxiliary quantum system as a measurement probe, on which measurements as well as coherent manipulations can be performed [16][17][18][19][20][21][22][23]. Interferometry [24] based on such a measurement probe allows us to extract information on a target system [25][26][27][28][29][30].…”