Global Quantum Thermometry

Quantum Scale Estimation

“…Not only does this recover the optimal estimator found in Ref. [33], but, notably, it also provides a new rule to calculate the associated probability-operator measurement (POM) 2 that is optimal for any amount of prior knowledge and a given quantum state. Built on Bayesian principles 3 [24,59], this framework is universally valid, with no restrictions on numbers of resources or initial information [46,73].…”

“…Estimation-theoretic quantities can, in addition, transcend their statistical role to acquire a new function within physics. For example, mean errors are known to enable the derivation of generalised uncertainty relations [1,22,[25][26][27][28][29] such as those for phase-number [1,25,30], time-energy [25,31] and temperature-energy [32,33] variables, and they appear to lead to a more detailed account of the notion of compatibility in quantum mechanics [4,[34][35][36][37][38][39]. Moreover, estimators provide the means to construct quantum observables such as those for phase [9,40] and time [22] parameters.…”

“…The sine error D[ θ(x), θ] = 4 sin 2 {[ θ(x) − θ]/2}, in particular, is often employed for this purpose [1,[42][43][44][45], and while there are other functions D that are appropriate for phase estimation [22,42], it is instructive to appreciate that the square error D[ θ(x), θ] = [ θ(x) − θ] 2 is generally not one of them. 1 In the context of thermometry [33,[47][48][49][50][51][52][53][54][55][56], the unknown parameter, temperature, establishes an energy scale [47,57]. In turn, this motivates the search for an estimation technique that preserves scale invariance [24,58,59], an idea that has recently be shown to lead to a logarithmic deviation function D[ θ(x), θ] = log 2 [ θ(x)/θ] [33].…”

“…1 In the context of thermometry [33,[47][48][49][50][51][52][53][54][55][56], the unknown parameter, temperature, establishes an energy scale [47,57]. In turn, this motivates the search for an estimation technique that preserves scale invariance [24,58,59], an idea that has recently be shown to lead to a logarithmic deviation function D[ θ(x), θ] = log 2 [ θ(x)/θ] [33]. Its theoretical and practical implications for quantum thermometry are currently under analysis [54][55][56], including the role of adaptive measurements [54] and its connection with the notion of thermodynamic length [55].…”

“…Its theoretical and practical implications for quantum thermometry are currently under analysis [54][55][56], including the role of adaptive measurements [54] and its connection with the notion of thermodynamic length [55]. However, it is to be noted that the logarithmic error transcends its thermometric origin, in the sense that it is valid for any other quantity that plays the role of a scale in physics [33]. For example, to estimate decay rates [60,Ch.…”