Fundamental limits to frequency estimation: a comprehensive microscopic perspective

Any frequency estimation task, and more generally any metrology scheme of figure 1, can be viewed at the abstract level as a quantum channel estimation protocol [49, 50], depicted in figure 2. In particular, the consecutive stages described in figure 1 can be formalised in the following way. Initially, the N probes are prepared in a potentially entangled quantum state ρ^(N)(0). Subsequently, the frequency ω₀ is encoded onto each of the probes via the action of a completely positive and trace preserving (CPTP) map [51, 52]—a quantum channel ${{\rm{\Lambda }}}_{{\omega }_{0}}(t)$—that specifies each probe dynamics. As a result, the global state of the probes after the frequency encoding stage of duration t reads

$$\begin{eqnarray}&&{\rho }_{{\omega }_{0}}^{(N)}(t)={{\rm{\Lambda }}}_{{\omega }_{0}}{(t)}^{\otimes N}\,[{\rho }^{(N)}(0)].\end{eqnarray} \tag{ 2 }$$

In the absence of noise the probe channel corresponds to the unitary ω₀-encoding introduced in equation (1), i.e., ${{\rm{\Lambda }}}_{{\omega }_{0}}(t)={{ \mathcal U }}_{{\omega }_{0}}(t)$. However, in general, it may incorporate the impact of any type of local noise that affects each probe in an uncorrelated fashion.

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Note that in this work we do not consider the effects of global decoherence mechanisms that disturb all the probes in a correlated manner. Such noise processes are known to impose fixed lower bounds on the achievable precision in metrology tasks that cannot be circumvented by any choice of probe states and measurements, even in the asymptotic limit of $N\to \infty$ [53–56]. As our motivation here is the investigation of the asymptotic precision scaling in frequency estimation—in particular, its potential quantum enhancement beyond SQL despite dissipative dynamics—we want to consider schemes in which the error asymptotically vanishes with N and it is this scaling that unambiguously quantifies the performance.

The measurement stage of any metrology scheme of figure 1 corresponds in figure 2 to a generalised quantum measurement, either local or global, that is performed on the final state (2) yielding an outcome x. It is formally defined by a positive operator valued measure (POVM), ${\{{M}_{x}\}}_{x}$, whose elements constitute positive-semidefinite operators, M_x ≥ 0, that sum to identity, ${\sum }_{x}{M}_{x}={\mathbb{1}}$ [51]. The measurement outcome x is then associated with its corresponding POVM element, so that the outcomes are distributed according to ${p}_{{\omega }_{0}}(x)=\mathrm{tr}\{{\rho }_{{\omega }_{0}}^{(N)}(t){M}_{x}\}$. Given that the protocol is repeated ν = T/t times over the total experiment duration T, a dataset of measurement outcomes ${{\boldsymbol{x}}}_{\nu }=\{{x}_{1},\,\ldots ,\,{x}_{\nu }\}$ is collected. Then, based on the data, an estimator $\tilde{\omega }({{\boldsymbol{x}}}_{\nu })$ is constructed whose value is aimed to most accurately reproduce the estimated frequency ω₀. Moreover, as the experiment is assumed to last much longer than a single protocol round, T ≫ t, the measurement data collected can always be taken to be sufficiently large for the asymptotic ($\nu \to \infty$) statistical analysis to apply.

Notice that the measurement outcomes are independently distributed with ${p}_{{\omega }_{0}}({{\boldsymbol{x}}}_{\nu })={\prod }_{i=1}^{\nu }{p}_{{\omega }_{0}}({x}_{i})$, as we have, for simplicity, disregarded the possibility of conducting adaptive strategies in which one adjusts the measurement, i.e., the POVM, in each protocol round based on the outcomes previously collected [57]. We are allowed to do so, as all the precision bounds discussed in the following sections are guaranteed to be saturated without the need of adaptive measurements in the so-called 'local estimation regime' [45], i.e., when sensing deviations of ω₀ from a known value⁷ . As such a scenario is the most optimistic one, the precision limits it provides can be considered fundamental—being applicable to all the more conservative approaches as $\nu \to \infty$ [58]. However, let us stress that the above requirement of 'estimation locality' may, indeed, be relaxed by allowing for the measurements to be adaptive, given the promise that the true value of ω₀ lies within a fixed, yet narrow enough window [59]. Nevertheless, if one was to consider the value of ω₀ to be largely unpredictable, one must explicitly follow Bayesian inference approaches to frequency estimation [60] in which the notions of SQL and HL must also be redefined [61].

Finally, the performance of the estimation protocol is quantified by the MSE of the estimator constructed, i.e.

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{2}\tilde{\omega }:= \displaystyle \sum _{{{\boldsymbol{x}}}_{\nu }}{p}_{{\omega }_{0}}({{\boldsymbol{x}}}_{\nu }){(\tilde{\omega }({{\boldsymbol{x}}}_{\nu })-{\omega }_{0})}^{2},\end{eqnarray} \tag{ 3 }$$

which must be minimised by optimising the initial state, ρ^(N)(0), and the measurement POVM, ${\{{M}_{x}\}}_{x}$, used in each round of the protocol of figure 2.

The minimal MSE (3) which can be attained by any consistent and unbiased estimator is determined by the Cramér-Rao bound (CRB) [45]:

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{2}\tilde{\omega }\geqslant \displaystyle \frac{1}{\nu \,{F}_{\mathrm{cl}}[{p}_{{\omega }_{0}}]},\quad {\rm{where}}\quad {F}_{\mathrm{cl}}[{p}_{{\omega }_{0}}]:= \displaystyle \sum _{x}\displaystyle \frac{{\dot{p}}_{{\omega }_{0}}{(x)}^{2}}{{p}_{{\omega }_{0}}(x)}\end{eqnarray} \tag{ 4 }$$

is the FI that is fully defined by the probability distribution of measurement outcomes, ${p}_{{\omega }_{0}}(x)$, and its dependence on the estimated ω₀. Here, and throughout the manuscript, we use the dot symbol to denote the derivative with respect to the estimated parameter, so that $\dot{\bullet }=\tfrac{{\rm{d}}\bullet }{{\rm{d}}{\omega }_{0}}$. As ${p}_{{\omega }_{0}}(x)=\mathrm{tr}\{{\rho }_{{\omega }_{0}}^{(N)}(t)\,{M}_{x}\}$ with the measured state given in equation (2), the CRB constitutes the ultimate limit on the precision attained by the protocol of figure 2 given a particular: initial state ρ^(N)(0), POVM ${\{{M}_{x}\}}_{x}$, and protocol duration t. Importantly, the optimization of equation (4) over measurements can be completely avoided in the quantum setting, as one may first explicitly maximise the FI over all POVMs by defining the Quantum-Fisher-Information (QFI) as [23]:

$$\begin{eqnarray}&&{F}_{Q}[{\rho }_{{\omega }_{0}}^{(N)}(t)]:= {\max }_{{\{{M}_{x}\}}_{x}}{F}_{\mathrm{cl}}[{p}_{{\omega }_{0}}]=\mathrm{tr}\{{\rho }_{{\omega }_{0}}^{(N)}(t){L}_{{\omega }_{0}}^{2}\},\end{eqnarray} \tag{ 5 }$$

which is now fully determined by the state ${\rho }_{{\omega }_{0}}^{(N)}(t)$ of equation (2) with ${L}_{{\omega }_{0}}$ being its symmetric logarithmic derivative (SLD) satisfying ${\dot{\rho }}_{{\omega }_{0}}^{(N)}(t)=\tfrac{1}{2}({L}_{{\omega }_{0}}{\rho }_{{\omega }_{0}}^{(N)}(t)+{\rho }_{{\omega }_{0}}^{(N)}(t){L}_{{\omega }_{0}})$.

In general, the evaluation of the SLD and, hence, the QFI (5) requires the explicit eigendecomposition of the state ${\rho }_{{\omega }_{0}}^{(N)}(t)$, which becomes rapidly intractable due to its dimension growing exponentially with the probe number N. However, in the absence of noise this is not the case, as the evolution of the probes is fully dictated by their Hamiltonians. Recalling equations (1) and (2) we may then write

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{{\omega }_{0}}^{\otimes N}(t)={U}_{{\omega }_{0}}^{(N)}(t)\,\bullet \,{U}_{{\omega }_{0}}^{(N)\dagger }(t)\quad {\rm{with}}\quad {U}_{{\omega }_{0}}^{(N)}(t)={{\rm{e}}}^{-{{\rm{i}}{H}}_{{\omega }_{0}}^{(N)}t},\end{eqnarray} \tag{ 6 }$$

where, ${H}_{{\omega }_{0}}^{(N)}={\sum }_{n=1}^{N}{H}_{{\omega }_{0}}^{[n]}={\omega }_{0}{\sum }_{n=1}^{N}{\hat{h}}^{[n]}$ is the effective global frequency-encoding Hamiltonian with n indexing the probes. Thus, when considering pure initial states ${\rho }^{(N)}(0)=| {\psi }^{(N)}\rangle \langle {\psi }^{(N)}|$ in the protocol⁸ , the QFI (5) simplifies to [4]

$$\begin{eqnarray}&&{F}_{Q}[{{ \mathcal U }}_{{\omega }_{0}}^{\otimes N}(t)[{\psi }^{(N)}]]=4{t}^{2}\,{{\rm{\Delta }}}^{2}{H}_{{\omega }_{0}}^{(N)}{| }_{{\psi }^{(N)}},\end{eqnarray} \tag{ 7 }$$

where ${{\rm{\Delta }}}^{2}{H}_{{\omega }_{0}}^{(N)}{| }_{{\psi }^{(N)}}$ is just variance of the frequency-encoding Hamiltonian for the state $| {\psi }^{(N)}\rangle$.

Combining equations (4) and (5), we arrive at the quantum Cramér-Rao bound (QCRB) [21] that we utilise throughout this work as the benchmark dictating the ultimate achievable precision:

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{2}\tilde{\omega }\geqslant \mathop{\min }\limits_{t\in [0,T]}\displaystyle \frac{1}{\nu {F}_{Q}[{\rho }_{{\omega }_{0}}^{(N)}(t)]}=\displaystyle \frac{1}{T}\,\mathop{\min }\limits_{t\in [0,T]}\displaystyle \frac{t}{{F}_{Q}[{\rho }_{{\omega }_{0}}^{(N)}(t)]}.\end{eqnarray} \tag{ 8 }$$

However, in the setting of frequency estimation, as indicated above, it must also be optimized over the duration time t of each protocol repetition. Then, as long as T ≫ t, the QCRB (8) sets the fundamental limit on precision for a given initial state ρ^(N)(0), which is utilized in each round of the protocol of figure 2, while the probes evolve according to particular dynamics specified by equation (2).

Now, stemming from equations (7) and (8), we can formally define the notions of SQL and HL in frequency estimation when the noise is absent as, respectively,

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\mathrm{SQL}}T=\displaystyle \frac{1}{t}\,\displaystyle \frac{1}{N}\quad {\rm{and}}\quad {{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\mathrm{HL}}T=\displaystyle \frac{1}{t}\,\displaystyle \frac{1}{{N}^{2}}.\end{eqnarray} \tag{ 9 }$$

The above MSEs correspond to the minimal values of the QCRB (8) attained when optimising the protocol over all separable and entangled initial states ρ^(N)(0), respectively. In particular, ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\mathrm{SQL}}$ is achieved by preparing the probes in a product ${| \phi \rangle }^{\otimes N}$ with $| \phi \rangle ={{\rm{argmax}}}_{\psi }{{{\rm{\Delta }}}^{2}\hat{h}| }_{\psi }$ , while ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\mathrm{HL}}$ is attained with $| {\psi }^{(N)}\rangle =\tfrac{1}{\sqrt{2}}(| {\lambda }_{\min }\rangle +| {\lambda }_{\max }\rangle )$, where $| {\lambda }_{\min /\max }\rangle$ are the eigenvectors corresponding to minimal/maximal eigenvalues of the Hamiltonian ${H}_{{\omega }_{0}}^{(N)}$ in equation (7) [63].

However, things change quite drastically if noise is taken into account. The first results in this direction were obtained in [26], which deals with a purely dephasing noise acting at rate γ independently and identically on each of the probes, such that the resulting evolution is given by a quantum dynamical semigroup, i.e., it is fixed by a Lindblad equation [64, 65]. The probes are described as qubits, as we will do from now on. Even with the preparation of entangled probes, one unavoidably recovers the SQL scaling, no matter how weak the dephasing is, with at most a constant factor of improvement. While this analysis was dedicated to a specific initial preparation and measurement (a generalized Ramsey scheme), it was afterwards extended to arbitrary preparations and measurements and other kinds of semigroup dynamics [27, 28, 34]: in the presence of pure dephasing, spontaneous emission, depolarization and loss, if one considers independent and identical noise and the dynamics is given by a semigroup, the asymptotic scaling is unavoidably bounded to the SQL.

Importantly, all the dynamics for which such limitation was proven are characterized by the fact that the action of the noise commutes with the unitary encoding of the parameter. In other terms, the dynamics of the probes, besides being independent and identical, is PC [38–40], which means that at any time t the quantum channel ${{\rm{\Lambda }}}_{{\omega }_{0}}(t)$ can be decomposed into the unitary encoding term and a noise term, and these two commute. More precisely, the dynamics of a two-level system is said to be PC, if for any rotation by an angle $\phi ,{{ \mathcal U }}_{\phi }={\rm{\exp }}\{{\rm{i}}\phi {\sigma }_{z}\}\bullet {\rm{\exp }}\{-{\rm{i}}\phi {\sigma }_{z}\}$, it holds

$$\begin{eqnarray}&&[{{ \mathcal U }}_{\phi },{{\rm{\Lambda }}}_{{\omega }_{0}}(t)]=0,\qquad \forall \phi ,t.\end{eqnarray} \tag{ 10 }$$

This is easily shown to be equivalent to⁹  ${{\rm{\Lambda }}}_{{\omega }_{0}}(t)={{ \mathcal U }}_{{\omega }_{0}}(t)\,\circ \,{{\rm{\Gamma }}}_{{\omega }_{0}}(t)={{\rm{\Gamma }}}_{{\omega }_{0}}(t)\,\circ \,{{ \mathcal U }}_{{\omega }_{0}}(t)$, where ${{\rm{\Gamma }}}_{{\omega }_{0}}(t)$ is a noise term, i.e the quantum channel acting on the probe that can associated purely with the noise.

By going beyond the assumptions of the above no-go theorems, i.e. the semigroup and PC properties, asymptotic precision scalings beyond SQL can be observed despite uncorrelated noise. On the one hand, by breaking the PC and considering noise that is perfectly transversal to the frequency encoding, the ultimate lower bound on precision has been derived [37]:

$$\begin{eqnarray}{{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\perp \&\mathrm{semi}}T\gtrsim \displaystyle \frac{1}{{N}^{5/3}}\end{eqnarray} \tag{ 11 }$$

and shown to be asymptotically attainable up to a constant factor (denoted by ≳). On the other, by circumventing the semigroup assumption and considering pure dephasing noise fixed by a time dependent dephasing rate γ(t), a scaling ∝1/N^3/2 has been found [42, 43, 66]. The super-classical 1/N^3/2 scaling was named Zeno limit due to it being dictated by the quadratic decay of the survival probabilities for short times, analogously to the Zeno effect [67, 68].

Recently [44], an achievable lower bound to the estimation error for the whole class of PC dynamics, including both semigroup and non-semigroup evolutions was derived. The maximal estimation precision is fixed by the power-law decay of the short-time expansion of the noise parameters and it goes beyond the SQL if and only if the semigroup composition law is violated at short times. Note that memory effects in the probes dynamics, i.e., non-Markovianity [69, 70], do not provide any improvement of the estimation precision (apart from the unrealistic case of a full revival of coherences). In particular, for any PC dynamics with linear decay of the noise parameters, which corresponds to the semigroup evolution, one gets

$$\begin{eqnarray}{{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\mathrm{PC}\&\mathrm{semi}}T\gtrsim \displaystyle \frac{1}{N},\end{eqnarray} \tag{ 12 }$$

while a quadratic decay yields the Zeno scaling

$$\begin{eqnarray}{{\rm{\Delta }}}^{2}{\tilde{\omega }}_{\mathrm{PC}\&\mathrm{Zeno}}T\gtrsim \displaystyle \frac{1}{{N}^{3/2}}.\end{eqnarray} \tag{ 13 }$$

These two scaling behaviors, along with that in (11) for purely transversal noise, provide the optimal asymptotic estimation precision achievable in the presence of different kinds of noise. Here, we will connect these results to the microscopic description of the probe-environment interaction, but we will also go beyond them by treating the cases of NC and non-semigroup noise.

To obtain a closed form of the master equation ruling the evolution of the probe subject to the noise fixed by equation (24), we exploit a perturbative approach, assuming that the system is weakly coupled to the environment. In particular, we use the time-convolutionless (TCL) master equation up to the second order [41, 78]. Its general form in the interaction picture is given by (denoting as $\tilde{\rho }(t)$ the system state in the interaction picture with respect to H₀ + H_B)

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\tilde{\rho }(t)}{{\rm{d}}t}=-{\int }_{0}^{t}{\rm{d}}\tau \,{\mathrm{tr}}_{E}\{[{H}_{I}(t),[{H}_{I}(\tau ),\tilde{\rho }(t)\otimes {\rho }_{E}]]\},\end{eqnarray} \tag{ 25 }$$

where H_I(t) is the interaction Hamiltonian H_I in the interaction picture. In appendix B, we describe how to get the desired master equation for the reduced system density matrix starting from the equation (25). At this point, let us just briefly introduce the main required quantities to define such a master equation, along with their physical meaning. First, the interaction Hamiltonian in the interaction picture is given by

$$\begin{eqnarray}&&{H}_{I}(t)={{\rm{e}}}^{{{\rm{i}}{H}}_{0}t}\left(\cos \vartheta \displaystyle \frac{{\sigma }_{x}}{2}+\sin \vartheta \displaystyle \frac{{\sigma }_{z}}{2}\right){{\rm{e}}}^{-{{\rm{i}}{H}}_{0}t}\otimes B(t),\end{eqnarray} \tag{ 26 }$$

where

$$\begin{eqnarray}&&B(t)=\displaystyle \sum _{n}({g}_{n}{{\rm{e}}}^{-{\rm{i}}{\omega }_{n}t}{a}_{n}+{g}_{n}^{* }{{\rm{e}}}^{{\rm{i}}{\omega }_{n}t}{a}_{n}^{\dagger })\end{eqnarray} \tag{ 27 }$$

is the interaction picture of the environmental operator appearing in the interaction Hamiltonian, see equation (24). The partial trace over the environment introduces the two-time correlation function ${\mathrm{tr}}_{E}[B(t)B(\tau ){\rho }_{E}]$ of the environment under its free dynamics, along with its complex conjugate ${\mathrm{tr}}_{E}[B(\tau )B(t){\rho }_{E}]$. This function encompasses the whole relevant information about the environment needed to characterize the open-system evolution in the weak coupling regime: as we will see, it fixes each coefficient of the master equation. In addition, if the initial state of the bath is thermal, i.e.

$$\begin{eqnarray}&&{\rho }_{E}(0)=\displaystyle \frac{\exp \{-\beta {H}_{B}\}}{Z},\end{eqnarray} \tag{ 28 }$$

with the inverse temperature β and $Z=\mathrm{tr}\{\exp \{-\beta {H}_{B}\}\}$, since [H_B, ρ_E(0)] = 0 the correlation function only depends on the difference of its time arguments t − τ. Therefore we can define the correlation function C(t) via

$$\begin{eqnarray}&&{\mathrm{tr}}_{E}[B(t)B(\tau ){\rho }_{E}]={\mathrm{tr}}_{E}[B(t-\tau )B(0){\rho }_{E}]\equiv C(t-\tau ).\end{eqnarray} \tag{ 29 }$$

Using the definition of B(t) in equation (27), this expression can be written as

$$\begin{eqnarray}&&C(t)=\displaystyle \sum _{n}{g}_{n}^{2}[N({\omega }_{n}){{\rm{e}}}^{{\rm{i}}{\omega }_{n}t}+(N({\omega }_{n})+1){{\rm{e}}}^{-{\rm{i}}{\omega }_{n}t}],\end{eqnarray} \tag{ 30 }$$

where $N({\omega }_{n})={\mathrm{tr}}_{E}\{{a}_{n}^{\dagger }{a}_{n}{\rho }_{E}\}$ represents the average number of excitations in the bath mode n. For the considered thermal state it is given by

$$\begin{eqnarray}&&N(\omega )=\displaystyle \frac{1}{{{\rm{e}}}^{\beta \omega }-1}=\displaystyle \frac{1}{2}\left[\coth \left(\displaystyle \frac{\beta \omega }{2}\right)-1\right].\end{eqnarray} \tag{ 31 }$$

The bath correlation function C(t) is conveniently expressed in terms of the spectral density of the environment, which is defined by

$$\begin{eqnarray}&&J(\omega )=\displaystyle \sum _{n}{g}_{n}^{2}\delta (\omega -{\omega }_{n}).\end{eqnarray} \tag{ 32 }$$

This quantity describes the density of the bath modes weighted with the square of their individual coupling strength to the system. In fact, the bath correlation function (30) can be written as

$$\begin{eqnarray}C(t) & = & {\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega J(\omega )[N(\omega ){{\rm{e}}}^{{\rm{i}}\omega t}+(N(\omega )+1){{\rm{e}}}^{-{\rm{i}}\omega t}]\\ & = & {\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\omega {{\rm{e}}}^{{\rm{i}}\omega t}N(\omega )[J(\omega )\,{\rm{\Theta }}\,(\omega )-J(-\omega )\,{\rm{\Theta }}\,(-\omega )]\\ & \equiv & {\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\omega {{\rm{e}}}^{{\rm{i}}\omega t}j(\omega ).\end{eqnarray} \tag{ 33 }$$

In the second line we used the formal identity −N(−ω) = N(ω) + 1 (see equation (31)) in order to introduce the function j(ω), i.e., the anti-Fourier transform of the bath correlation function. The Heaviside stepfunction ${\rm{\Theta }}(\omega )$ keeps track of the fact that J(ω) is defined only for positive frequencies. Finally, the relation in equation (33) allows us to perform the continuum limit straightforwardly by replacing the spectral density in equation (32) with a smooth function of the frequency bath modes [41].

As said, the bath correlation function C(t) or, equivalently, the bath spectral density J(ω) along with the initial state of the bath fix the reduced master equation in the weak coupling regime: since we are dealing with the second order perturbative (TCL) expansion, only the two-time correlation function C(t) is involved, while the bath multi-time correlation functions would only be involved in higher order terms (see also the recent [79]). As shown in appendix B, the master equation (back in the Schrödinger picture) is then given by

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\rho (t)}{{\rm{d}}t} & = & { \mathcal L }(t)[\rho (t)]=-{\rm{i}}[{H}_{0}+{H}^{\mathrm{LS}}(t),\rho (t)]\\ & & +\displaystyle \sum _{j,k=\pm ,z}{b}_{{kj}}(t)\left({\sigma }_{k}\rho (t){\sigma }_{j}^{\dagger }-\displaystyle \frac{1}{2}\{{\sigma }_{j}^{\dagger }{\sigma }_{k},\rho (t)\}\right),\end{eqnarray} \tag{ 34 }$$

where we introduced the function

$$\begin{eqnarray}&&{\rm{\Gamma }}(\varsigma ,t)={\int }_{0}^{t}{\rm{d}}\tau {{\rm{e}}}^{{\rm{i}}\varsigma \tau }C(\tau )\end{eqnarray} \tag{ 35 }$$

for ς = ±ω₀, 0, such that

$$\begin{eqnarray}{b}_{{zz}}(t) & = & \displaystyle \frac{{\sin }^{2}\vartheta }{2}{\rm{Re}}\{{\rm{\Gamma }}(0,t)\}\\ {b}_{++}(t) & = & \displaystyle \frac{{\cos }^{2}\vartheta }{2}{\rm{Re}}\{{\rm{\Gamma }}(-{\omega }_{0},t)\}\\ {b}_{--}(t) & = & \displaystyle \frac{{\cos }^{2}\vartheta }{2}{\rm{Re}}\{{\rm{\Gamma }}({\omega }_{0},t)\}\\ {b}_{+-}(t) & = & {b}_{-+}^{* }(t)=\displaystyle \frac{{\cos }^{2}\vartheta }{4}({\rm{\Gamma }}(-{\omega }_{0},t)+{{\rm{\Gamma }}}^{* }({\omega }_{0},t))\\ {b}_{z+}(t) & = & {b}_{+z}^{* }(t)=\displaystyle \frac{\sin \vartheta \cos \vartheta }{4}({\rm{\Gamma }}(0,t)+{{\rm{\Gamma }}}^{* }(-{\omega }_{0},t))\\ {b}_{z-}(t) & = & {b}_{-z}^{* }(t)=\displaystyle \frac{\sin \vartheta \cos \vartheta }{4}({\rm{\Gamma }}(0,t)+{{\rm{\Gamma }}}^{* }({\omega }_{0},t)),\end{eqnarray} \tag{ 36 }$$

while the Hamiltonian correction is fixed by the elements:

$$\begin{eqnarray}{H}_{11}^{\mathrm{LS}}(t) & = & \displaystyle \frac{{\cos }^{2}\vartheta }{4}{\rm{Im}}\{{\rm{\Gamma }}({\omega }_{0},t)\}\\ {H}_{10}^{\mathrm{LS}}(t) & = & {H}_{01}^{\mathrm{LS}* }(t)=-\displaystyle \frac{{\rm{i}}\cos \vartheta \sin \vartheta }{4}\\ & & \times \left({\rm{Re}}\{{\rm{\Gamma }}(0,t)\}-\displaystyle \frac{1}{2}({{\rm{\Gamma }}}^{* }(-{\omega }_{0},t)+{\rm{\Gamma }}({\omega }_{0},t))\right)\\ {H}_{00}^{\mathrm{LS}}(t) & = & \displaystyle \frac{{\cos }^{2}\vartheta }{4}{\rm{Im}}\{{\rm{\Gamma }}(-{\omega }_{0},t)\},\end{eqnarray} \tag{ 37 }$$

where ${H}_{{ij}}^{\mathrm{LS}}(t)=\langle {i}| {H}^{\mathrm{LS}}(t)| j\rangle$ for i, j = 1, 2.

Let us stress that we did not invoke the Born-Markov approximation [41] in our derivation—the above time-local master equation includes fully general non-Markovian effects and it will provide us with a satisfactory description of the noisy evolution of the probes as long as the interaction with the environment is weak enough (i.e., the higher orders of the TCL expansion can be neglected). In addition, we are taking into account the dependence of the coefficients of the dissipative part of the master equation on the free system frequency ω₀, see equation (36), i.e., on the parameter to be estimated. This is a natural consequence of the detailed microscopic derivation of the system dynamics [41], in contrast with the phenomenological approaches, where the master equation is postulated on the basis of the noise effects to be described. Let us emphasize that only in the case of pure dephasing, for which $\vartheta =\pi /2$ and all dissipative terms in (36) apart from b_zz(t) vanish, the dissipative part of the master equation can be assured not to depend on ω₀. Otherwise, this is not generally the case unless a special choice of J(w) is made (e.g., discussed later in section 5.2).

Finding an explicit solution to the master equation in equation (34) is in general a complicated task, even after fixing the explicit form of the spectral density of the bath modes. On the other hand, the structure of the dynamics can be simplified considerably by making the so-called secular approximation [41, 78, 80, 81], which relies on a time-scale separation between the system free-evolution time τ₀ and the relaxation time τ_R of the system subject to the interaction with the environment. Whenever the free dynamics is much faster than the dissipative one, i.e., ${\tau }_{0}\sim {\omega }_{0}^{-1}\ll {\tau }_{R}$, one can neglect terms oscillating with ${{\rm{e}}}^{\pm {\rm{i}}{\omega }_{0}t}$ because they will be averaged out to 0 over a time interval of the order of τ_R. If we apply this approximation to the weak coupling master equation, see in particular equation (B.7), all off-diagonal coefficients in equation (34) and the off-diagonal elements of the Hamiltonian in equation (37) vanish, so that one is left with the master equation

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\rho (t)}{{\rm{d}}t} & = & -{\rm{i}}\left[{H}_{0}+\displaystyle \frac{{H}_{11}^{\mathrm{LS}}(t)}{2}{\sigma }_{z},\rho (t)\right]\\ & & +{b}_{++}(t)\left({\sigma }_{+}\rho (t){\sigma }_{-}-\displaystyle \frac{1}{2}\{{\sigma }_{-}{\sigma }_{+},\rho (t)\}\right)\\ & & +{b}_{--}(t)\left({\sigma }_{-}\rho (t){\sigma }_{+}-\displaystyle \frac{1}{2}\{{\sigma }_{+}{\sigma }_{-},\rho (t)\}\right)\\ & & +{b}_{{zz}}(t)({\sigma }_{z}\rho (t){\sigma }_{z}-\rho (t)),\end{eqnarray} \tag{ 38 }$$

where all the non-zero coefficients are still those of equation (36). This master equation can be explicitly solved for generic coefficients ${b}_{++}(t),{b}_{--}(t),{b}_{{zz}}(t)$ and ${H}_{11}^{\mathrm{LS}}(t)$ (see, e.g., [44, 82]).

Crucially, we see how the secular master equation in equation (38) precisely corresponds to the most general form of a master equation associated with a PC qubit dynamics recalled in section 3, see equation (21). Hence the difference between secular and non-secular dynamics provides us with a direct physical explanation of the difference between PC and NPC dynamics. The complete (weak-coupling) dynamics described by the master equation in equation (34) will generally lead to NPC dynamical maps, represented by generic matrices ${{\mathsf{D}}}^{{\rm{\Lambda }}(t)}$ as in equation (17) and corresponding to a completely general affine transformation of the Bloch sphere. Instead, if one applies the secular approximation, thus getting the master equation in equation (38), the resulting dynamics is PC and will be then characterized by dynamical maps with a structure as in equation (20). In other words, within this framework, the distinction between PC and NPC dynamics precisely corresponds to the distinction between dynamics within or outside the secular regime, i.e., the regime τ₀≪ τ_R where the secular approximation is well-justified. Needless to say, and as we will see explicitly in the next sections, the two kinds of dynamics describe also qualitatively different open-system evolutions. As a paradigmatic example, one can easily see how for any secular master equation the populations and coherences are decoupled, while the inclusion of non-secular terms leads to a coupling between them. The latter can be relevant for different phenomena, such as exitonic transport [83, 84], or the speed of the evolution in non-Markovian dynamics [85, 86]. Finally, note that general constraints on the variation of the coherences for a given variation of the populations in the presence of a generic completely positive PC map have been recently derived in [87].

Using the Dyson expansion of equation (23) we obtain the short-time solution of master equation (34) as

$$\begin{eqnarray}{{\mathsf{D}}}_{(3)}^{{\rm{\Lambda }}(t)}=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1-\tfrac{{\omega }_{0}^{2}{t}^{2}}{2}-\tfrac{1}{2}\alpha {t}^{2}{\sin }^{2}\vartheta & -{\omega }_{0}t+q & \tfrac{1}{2}\alpha {t}^{2}\cos \vartheta \sin \vartheta \\ 0 & {\omega }_{0}t-q & 1-\tfrac{{\omega }_{0}^{2}{t}^{2}}{2}-\tfrac{\alpha {t}^{2}}{2} & \tfrac{1}{3}\alpha {\omega }_{0}{t}^{3}\cos \vartheta \sin \vartheta \\ 0 & \tfrac{1}{2}\alpha {t}^{2}\cos \vartheta \sin \vartheta & -\tfrac{1}{3}\alpha {\omega }_{0}{t}^{3}\cos \vartheta \sin \vartheta & 1-\tfrac{1}{2}\alpha {t}^{2}{\cos }^{2}\vartheta \end{array}\right)\end{eqnarray} \tag{ 42 }$$

where

$$\begin{eqnarray}&&\alpha ={\int }_{-\infty }^{\infty }{\rm{d}}\omega j(\omega )\approx {\int }_{0}^{\infty }\displaystyle \frac{2J(\omega )}{\beta \omega }\quad {\rm{and}}\quad q=\displaystyle \frac{{\omega }_{0}{t}^{3}}{6}[\alpha (1+2{\sin }^{2}\vartheta )+{\omega }_{0}^{2}].\end{eqnarray} \tag{ 43 }$$

Truncating the Dyson series is justified due to the the weak-coupling approximation, while we have kept the terms up to the third order (and not only to the second order) for a reason which will become clear when we evaluate the QFI of the corresponding evolved state in section 6.1. The short time dynamical maps do not depend on the specific form of spectral density, but only on the global parameter α. Furthermore, evaluating the eigenvalues of the Choi matrix reveals that the map is CP [89].

Repeating the same calculations for the PC master equation in equation (38) in the secular approximation, we arrive at

$$\begin{eqnarray}{{\mathsf{D}}}_{(3),\mathrm{PC}}^{{\rm{\Lambda }}(t)}=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1-\tfrac{{\omega }_{0}^{2}{t}^{2}}{2}-\tfrac{\alpha {t}^{2}}{4}(1+{\sin }^{2}\vartheta ) & -{\omega }_{0}t+q & 0\\ 0 & {\omega }_{0}t-q & 1-\tfrac{{\omega }_{0}^{2}{t}^{2}}{2}-\tfrac{\alpha {t}^{2}}{4}(1+{\sin }^{2}\vartheta ) & 0\\ 0 & 0 & 0 & 1-\tfrac{1}{2}\alpha {t}^{2}{\cos }^{2}\vartheta \end{array}\right).\end{eqnarray} \tag{ 44 }$$

Here, in order to characterize the reduced dynamics at any time t, we focus on a specific spectral density of the bath—the Ohmic spectral density:

$$\begin{eqnarray}&&J(\omega )=\lambda \omega {{\rm{e}}}^{-\omega /{\omega }_{c}},\end{eqnarray} \tag{ 45 }$$

where λ quantifies the global strength of the system-environment interaction, while ω_c sets the cut-off frequency which defines the relevant environmental frequencies in the open-system dynamics. We further assume that ω_c ≫ ω₀, so that the dependence of Γ(ς, t) on ς can be neglected and Γ(ω₀, t) ≈ Γ(−ω₀, t) ≈ Γ(0, t), as then, see equations (33) and (35):

$$\begin{eqnarray}{\rm{\Gamma }}(\pm {\omega }_{0},t) & \approx & \displaystyle \frac{\lambda }{\beta }{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}\omega {{\rm{e}}}^{{\rm{i}}(\pm {\omega }_{0}+\omega )\tau }\\ & & \times \ ({{\rm{e}}}^{-\omega /{\omega }_{c}}\,{\rm{\Theta }}\,(\omega )+{{\rm{e}}}^{\omega /{\omega }_{c}}\,{\rm{\Theta }}\,(-\omega ))\\ & = & \displaystyle \frac{2\lambda }{\beta }{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{e}}}^{\pm {\rm{i}}{\omega }_{0}\tau }{\displaystyle \int }_{0}^{\infty }{\rm{d}}\omega {{\rm{e}}}^{-\displaystyle \frac{\omega }{{\omega }_{c}}}\cos (\omega \tau )\\ & = & \displaystyle \frac{2\lambda }{\beta }{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{e}}}^{\pm {\rm{i}}{\omega }_{0}\tau }\displaystyle \frac{{\omega }_{c}}{1+{\omega }_{c}^{2}{\tau }^{2}}\approx \displaystyle \frac{2\lambda }{\beta }{\displaystyle \int }_{0}^{t}{\rm{d}}\tau \displaystyle \frac{{\omega }_{c}}{1+{\omega }_{c}^{2}{\tau }^{2}},\end{eqnarray} \tag{ 46 }$$

where in the first and last approximated equalities we used the high-temperature condition and ω₀/ω_c ≪ 1, respectively. The coefficients of the master equation in equation (36) then simplify to

$$\begin{eqnarray}{b}_{{zz}}(t) & = & \displaystyle \frac{{\sin }^{2}\vartheta }{2}{\rm{\Gamma }}(0,t)\\ {b}_{++}(t) & = & {b}_{+-}(t)={b}_{-+}(t)={b}_{--}(t)=\displaystyle \frac{{\cos }^{2}\vartheta }{2}{\rm{\Gamma }}(0,t)\\ {b}_{z+}(t) & = & {b}_{+z}(t)={b}_{z-}(t)={b}_{-z}(t)=\displaystyle \frac{\sin \vartheta \cos \vartheta }{2}{\rm{\Gamma }}(0,t),\end{eqnarray} \tag{ 47 }$$

while the Hamiltonian correction, H^LS(t), vanishes. We stress that it is the specific choice of Ohmic spectral density that assures the coefficients of the master equation to be independent of ω₀—a fact, typically taken for granted in quantum metrology scenarios [4, 34, 37, 44, 90–92].

Now, using equation (47) one can easily see (e.g., by diagonalizing the matrix with elements given by the coefficients b_jk(t)) that the time-local master equation can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\rho (t)}{{\rm{d}}t}=-{\rm{i}}[{H}_{0},\rho (t)]+\gamma (t)(\bar{\sigma }\rho (t){\bar{\sigma }}^{\dagger }-\rho (t)),\end{eqnarray} \tag{ 48 }$$

where the rate γ(t) and the dissipative operator $\bar{\sigma }$ are given by

$$\begin{eqnarray}\gamma (t) & = & \displaystyle \frac{1}{2}{\rm{\Gamma }}(0,t)=\displaystyle \frac{\lambda }{\beta }\arctan ({\omega }_{c}t)\\ \bar{\sigma } & = & \cos \vartheta {\sigma }_{x}+\sin \vartheta {\sigma }_{z}.\end{eqnarray} \tag{ 49 }$$

It is worth noting that the dissipative part of the master equation is fixed by one single operator $\bar{\sigma }$, i.e., we have the most general qubit rank-one Pauli noise, recently proved to be correctable in the semigroup case (γ(t) = const) in quantum metrology by ancilla-assisted error-correction [36], which has been demonstrated experimentally for transversal coupling, ϑ = 0, in [93]. In addition, the only noise rate γ(t) is a positive function of time, which guarantees not only the CP of the dynamics, but also that the dynamical maps can be always split into CP terms. In this case, one speaks of (CP)-divisible dynamics, which coincides with the definition of Markovian quantum dynamics put forward in [94]; see also [69]. As expected, in the limit of an infinite cut-off, ${\omega }_{c}\to \infty$, the rate goes to a positive constant value, $\gamma (t)\to \pi \lambda /(2\beta )$, so that we recover a Lindblad time-homogeneous (semigroup) dynamics [78]; see appendix C.1, where we also give the explicit form of the corresponding dynamical maps for $\vartheta =0,\pi /2$, i.e., transversal and pure dephasing noise-types, respectively.

Finally, note that a purely transversal interaction Hamiltonian ($\vartheta =0$) yields a purely transversal master equation, i.e., the only dissipative operator $\bar{\sigma }={\sigma }_{x}$ in equation (48) is orthogonal to H₀, which is generally not guaranteed for arbitrary spectral densities. $\bar{\sigma }={\sigma }_{x}$ characterizes what is usually known in the literature [37, 91] as (and what we will here denote as) transversal noise.

Let us now consider the corresponding dynamics under the secular approximation that provides us with a PC dynamics. The coefficients in the third line of equation (47) along with ${b}_{+-}(t)$ are set to 0 and we are thus left with the PC master equation

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\rho (t)}{{\rm{d}}t}=-{\rm{i}}[{H}_{0},\rho (t)]+\gamma (t)\displaystyle \sum _{j=\pm ,z}{{\rm{d}}}_{j}\left({\sigma }_{j}\rho (t){\sigma }_{j}^{\dagger }-\displaystyle \frac{1}{2}\{{\sigma }_{j}^{\dagger }{\sigma }_{j},\rho (t)\}\right)\end{eqnarray} \tag{ 50 }$$

with ${d}_{+}={d}_{-}={\cos }^{2}\vartheta$, while ${d}_{z}={\sin }^{2}\vartheta$. Once again, the dynamics is CP and due to the positivity of γ(t) and the d_js it is even CP-divisible. Despite having now three different dissipative operators, these claims hold because there is only one single time-dependent function which defines all the rates.

In figure 4 we illustrate the different dynamics geometrically by comparing the different evolutions of the open system for the NPC dynamics described by equation (48) and the PC dynamics fixed by equation (50), respectively, see appendix C.1. In figures 4(a)–(c), we report the evolution for the same dynamics (i.e., the same ω₀, ω_c, λ, β and $\vartheta$), for the three different initial conditions which correspond to the three canonical orthogonal axes in the Bloch sphere. Of course, this is enough to detect all the possible linear transformations that the set of states undergoes during the evolution. In the PC dynamics we have contractions and rotations about the z-axis, as well as equal contractions along the x- and y-axes. These are all transformations commuting with the unitary rotation about the z-axis, as recalled in section 3. On the other hand, in the non-secular dynamics we can observe a rotation about an axis with components in the plane perpendicular to the z-axis, which clearly breaks the PC of the dynamics. Figure 4(d) is devoted to illustrate another NPC effect, which is already present in the dynamics in figures 4(a)–(c), but is not clearly observable due to the other transformations of the Bloch sphere. We consider a dynamics where $\vartheta =0$, thus excluding any rotation apart from that about the z-axis¹¹ . As we see, the NPC dynamics introduces different contractions along the x and y directions, contrary to the PC case. The effects on parameter estimation of the rotations about the x- and y-axes, as well as the different contractions along them will be investigated in section 6.2.

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Finally, note that although the non-secular terms introduce a transient behavior, which departs from the secular (i.e., PC) evolution, the system relaxes, in any case, to the fully mixed state. When probe systems can be interrogated within the transient dynamics, metrological advances may arise, as discussed in the following sections.

Thus, let us start by looking at the short-time expansion of the QFI in equation (51). The spherical parametrisation provides us with a clear relation among the short-time QFI for the NPC and PC dynamics, see equations (42) and (44), respectively. As a matter of fact, the first non-trivial term (i.e., the first contribution to F_Q which is induced by the noise and therefore the first contribution where F_Q differs between NPC and PC dynamics) in the QFI is of the order t⁴ and it is fixed by those terms up to t³ in ${V}_{{\omega }_{0}}(t)$ and ${\dot{V}}_{{\omega }_{0}}(t)$. After a straightforward calculation, we arrive in fact at

$$\begin{eqnarray}{F}_{Q,\mathrm{PC}}^{(4)} & = & {\sin }^{2}\theta \,{t}^{2}-\displaystyle \frac{1}{3}\alpha {\sin }^{2}\theta (1+{\sin }^{2}\vartheta ){t}^{4}\\ {F}_{Q}^{(4)} & = & {F}_{Q,\mathrm{PC}}^{(4)}+\alpha {t}^{4}\sin \theta \left(\displaystyle \frac{1}{3}\cos \theta \sin 2\vartheta \cos \phi \right.\\ & & \left.+\displaystyle \frac{{\sin }^{2}\phi {(\sin \vartheta \cos \theta +\cos \vartheta \cos \phi \sin \theta )}^{2}/4}{\cos \theta \cot \theta -2\tan \vartheta \cos \phi \cos \theta +({\cos }^{-2}\vartheta -{\cos }^{2}\phi )\sin \theta }\right).\end{eqnarray} \tag{ 52 }$$

The maximum value of the QFI for a PC dynamics is obtained for $\vartheta =0$, i.e., for a pure transversal Hamiltonian [37] and for θ = π/2, i.e., for a state lying on the equator of the Bloch sphere; moreover, the dephasing noise, i.e., $\vartheta =\pi /2$, is the most detrimental in this regime. Although the expression for ${F}_{Q}^{(4)}$ in the NPC case is too cumbersome to yield a comprehensible analytical solution for a state which maximizes the QFI in the short time limit, even for a fixed value of the parameter $\vartheta$, we report an approximated evaluation in appendix D.

As can be directly inferred comparing the two formulas in equation (52), a crucial difference between PC and NPC dynamics is that in the former case the QFI only depends on the initial distance of the Bloch vector from the z-axis and hence on the angle θ, while the NPC terms introduce a dependence of the QFI on the direction of the Bloch vector itself and therefore on the angle ϕ. Such a dependence is a consequence of the non-commutativity of the encoding Hamiltonian with the action of the noise. For any PC dynamical map ${{\rm{\Lambda }}}_{{\omega }_{0},\mathrm{PC}}$, if we rotate the state ρ about the z-axis by a certain angle ϕ, we have that

$$\begin{eqnarray}&&{F}_{Q}[{{\rm{\Lambda }}}_{{\omega }_{0},\mathrm{PC}}{{ \mathcal U }}_{\phi }[\rho ]]={F}_{Q}[{{ \mathcal U }}_{\phi }{{\rm{\Lambda }}}_{{\omega }_{0},\mathrm{PC}}[\rho ]]={F}_{Q}[{{\rm{\Lambda }}}_{{\omega }_{0},\mathrm{PC}}[\rho ]],\end{eqnarray} \tag{ 53 }$$

by virtue of equation (10) the invariance of the QFI under rotations independent from the parameter to be estimated.

Now, the contributions due to the NPC terms are able to enhance the QFI, as can be seen in figure 5(a), where we illustrate the behavior of the difference ${\rm{\Delta }}{F}_{Q}\equiv {F}_{Q}^{(4)}-{F}_{Q,\mathrm{PC}}^{(4)}$ as a function of the initial conditions. Besides the dependence on the initial phase ϕ, one can clearly observe the presence of several areas where the NPC terms do increase the QFI. Moreover, there are two maxima of the increment, one in the neighborhood of ϕ = 0 and one in the neighborhood of ϕ = π; we plot ΔF_Q for values ϕ ∈ [0, π], since it is a symmetric function under the reflection $\phi \to 2\pi -\phi$, see equation (52). In the plot, we fixed $\vartheta =\pi /4$ but the behavior is qualitatively the same for different values of $\vartheta$. Indeed, ΔF_Q goes to 0 for $\vartheta$ going to π/2 since for a pure dephasing Hamiltonian the secular approximation has no effect, so that the dynamical maps in equations (42) and (44) coincide.

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Moreover, the presence of NPC terms can enhance the value of the QFI maximized over all the initial conditions and hence enlarge the maximal achievable precision. This is explicitly shown by taking into account the states lying on the equatorial plane of the Bloch sphere, which as said maximize ${F}_{Q,\mathrm{PC}}^{(4)}$. For θ = π/2, the second relation in equation (52) reduces to

$$\begin{eqnarray}&&{F}_{Q}^{(4)}={F}_{Q,\mathrm{PC}}^{(4)}+\displaystyle \frac{\alpha {t}^{4}}{4}\left(\displaystyle \frac{{\cos }^{4}\vartheta {\cos }^{2}\phi {\sin }^{2}\phi }{{\sin }^{2}\vartheta {\cos }^{2}\phi +{\sin }^{2}\phi }\right),\end{eqnarray} \tag{ 54 }$$

which clearly shows that the maximum value of ${F}_{Q,\mathrm{PC}}^{(4)}$ can be actually overcome for any value of $\vartheta \ne \pi /2$. In figure 5(b), we plot the increase of the QFI due to the NPC terms for θ = π/2, while varying the initial phase ϕ and mixing angle $\vartheta$. The QFI with the NPC terms is always bigger or equal than ${F}_{Q,\mathrm{PC}}^{(4)}$ and the maximum enhancement occurs for ϕ close to kπ with k = 0, 1, 2 and the pure transversal noise corresponding to $\vartheta =0$. However, the latter condition depends on the specific choice of the initial state: for $\theta \ne \pi /2$ one can have the maximal amplification due to the NPC terms for non-zero values of $\vartheta$.

6.1.1. Different contributions to the QFI

In this paragraph we examine the behavior of the QFI for finite times, when the dynamics are dictated by the master equation expressed in equations (48) and (50). This will allow us to analyze more in detail the difference between the NPC and the PC contributions to the QFI. The results presented in this section are numeric, calculated using equation (51) and the same parametrization of the Bloch vector as before. Figure 6 contains the foundation of the following discussion.

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Let us first note that the dependence on the initial phase ϕ already mentioned above affects the whole time evolution of the NPC-QFI. Figures 6(a) and (b) show the evolution with $\vartheta =0$ for an initial state in an equally weighted superposition, i.e., a state in the $\hat{x}-\hat{y}$ plane of the Bloch sphere (θ = π/2), but with initial phases ϕ = π/2 and ϕ = 0, respectively. Comparing the two figures, one observes that the initial phase is of no relevance for PC dynamics on the whole timescale, while the NPC dynamics introduces a dependence on ϕ. The NPC contributions enhance the maximum value of the QFI and shift its position, depending on the value of the initial phase.

For the Ohmic spectral density considered here, the noise terms do not depend on ω₀, see section 5.2, so that ${\tilde{F}}_{Q}(t)={F}_{Q}(t)$ and the same result holds for the PC case. Hence, F_Q,PC(t) is directly fixed by the distance of the Bloch vector from the z-axis, along with the elapsed time t, see equation (D.3) in appendix D, while the further contributions within the NPC-QFI F_Q(t) can be fully ascribed to the non-commutativity of the Hamiltonian and dissipative part, see the discussion in the previous paragraph.

While the independence of the QFI from the parameter to be estimated in the PC case can be readily shown [34, 44], we can see from figure 6 that the NPC-QFI depends on ω₀. In particular, with growing values ω₀, the NPC-QFI converges to its PC counterpart: higher values of ω₀ imply a faster free dynamics of the system, which thus reduces the relevance of NPC terms and increases the validity of the secular approximation, see section 3.

We further observe that the overall effect of the NPC terms can yield an increase or a decrease of the QFI, depending on the time interval considered. On the one hand, the NPC terms induce a contraction in the x–y plane, which is no longer isotropic. Comparing the evolution of the QFIs in figures 6(a) and (b) with the evolution of the Bloch vector in the insets, it is clear how the non-isotropic contractions can bring the Bloch vector further or closer to the z-axis, thus increasing or decreasing the QFI. On the other hand, as mentioned in the previous paragraph, due to the non-commutativity of the dynamics additional information about ω₀ is enclosed in other features of the Bloch vector; the action of decoherence itself adds some information about ω₀ to the information imprinted by the rotation about the z-axis given by the Hamiltonian encoding.

The delicate interplay of the different mechanisms of production and annihilation of the QFI is also illustrated in figures 6(c) and (d). Here we consider values of $\vartheta$ different from 0, so that the states initially on the equator of the Bloch sphere are no longer confined to the xy-plane. Comparing figures 6(b) and (c), we see how the introduced NPC rotation partially counterbalances the oscillations due to the non-isotropic contraction. Furthermore, the role of the different NPC terms strongly depends on the initial state. As an example, figure 6(d) shows the strongest (relative) enhancement of the maximum value of the QFI due to the action of both the NPC rotations and contractions.

The starting point is the master equation given by equation (48). In particular, we considered three different NPC noise scenarios: the first two cases of a purely transversal noise, i.e., $\vartheta =0$, for a non-semigroup (see figure 7(a)) and for a semigroup (figure 7(b)) dynamics. As a third case, we chose noise with a (small) longitudinal component fixed by $\vartheta =\pi /100$ for a non-semigroup dynamics (see figure 7(c)). In figures 7(a)–(c) we report the numerical study of $t/{F}_{Q}^{\uparrow }[{\rho }^{(N)}(t)]$, which fixes a lower bound to the estimation error, see equations (8) and (56), along with the estimation error for the parity measurement, ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$, see equation (57). As clearly observed in figure 7, the two quantities have the same asymptotic scaling, therefore the bound is achievable, at most up to a constant factor. Hence we can infer the scaling with respect to N of the error for the optimal estimation strategy. Denoting the latter as ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0}T$, we can in fact write the lower and upper bounds as

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T\geqslant {{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0}T\geqslant \displaystyle \frac{t}{{F}_{Q}^{\uparrow }[{\rho }^{(N)}(t)]}\ \Rightarrow \ {{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0}T\propto \displaystyle \frac{1}{{N}^{\eta }},\end{eqnarray} \tag{ 59 }$$

where the implication follows from the fact that since both the lower and the upper bound approach 0 for $N\to \infty$ as N^−η with the same value of η, this will be the case also for ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0}T$.

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Table 1 contains the values of the optimal scaling η for the different NPC noise scenarios as inferred from our numerical analysis, along with the corresponding PC scaling behavior (i.e., those for the dynamics after the secular approximation, see equation (50)) taken from [34, 44]. The optimal scaling of the estimation error for the full NPC dynamics is fixed by two key features: whether we have a semigroup or a non-semigroup evolution and the direction of the noise fixed by the angle $\vartheta$. The presence of a time-dependent rate γ(t) as in equation (49) always leads to an improved scaling, with respect to the constant rate γ of the semigroup evolution; in particular, for any $\vartheta \ne 0$ we have the Zeno η = 3/2 scaling, associated with the linear increase of the rate γ(t) for short times [43, 44]. Moreover, we numerically find the novel η = 7/4 scaling for a non-semigroup, purely transversal noise.

Table 1.  Ultimate scaling exponent, η in equation (59), of the optimal estimation error ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0}T$ for different types of noise in the asymptotic limit of $N\to \infty$.

η	NPC	PC	NPC, semigroup	PC, semigroup
$\vartheta =0$	7/4	3/2	5/3	1
$\vartheta \ne 0$	3/2	3/2	1	1

We stress that for any value of $\vartheta$ different from 0 the full NPC dynamics leads to the same scaling behavior as in the corresponding PC case. We can say that the transversal noise represents a special case of NPC noise, which might be seen as a 'purely NPC noise'. For any $\vartheta \ne 0$, the dissipative part of the master equation given in equation (48) together with the resulting dynamical maps, will have a component longitudinal to the parameter imprinting, fixing the asymptotic scaling to the less favorable one proper to PC dynamics and hence extending the Zeno regime recalled in section 2.3 to the scenario governed by NPC noise. This result, already known for the semigroup regime [37] (see also section 2.3), is here extended to the non-semigroup case. Summarizing, we can conclude that the ultimate achievable estimation precision can overcome the SQL whenever we have a non-semigroup (short-time) evolution, irrespective of the direction of the noise, or in the cases where we have a purely transversal noise, irrespective of whether we have a semigroup or not.

Interestingly, similar results have been derived recently [96] for rather different, infinite dimensional probing systems. The probes are prepared in Gaussian states and undergo a Gaussian dynamics, possibly non-semigroup and NPC. The NPC contributions are induced by the presence of squeezing in the initial bath state. Also there the optimal asymptotic scaling of the error is found to be the same for PC and NPC dynamics, going from the SQL for a semigroup to the Zeno limit for a linear increase of the dissipative rates. Such a transition for a PC evolution of a Gaussian system has been shown also in [97].

The plots in figures 7(a)–(c) allow us to get some interesting information also about the behavior of the estimation error for a finite number of probes, showing that the asymptotic scaling is approached in a possibly non-trivial way.

First of all, we note that for smaller values of N, the lower bound to the estimation precision $t/{F}_{Q}^{\uparrow }[{\rho }^{(N)}(t)]$ and the error under parity ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$ seem to follow the SQL and then, only for intermediate and high values of N, the two quantities converge to the asymptotic behavior, approaching it always from above. This was already shown for a semigroup NPC noise, also with a longitudinal component (see [37], in particular figure 3) and here we see how the same happens for a non-semigroup NPC noise. Actually, the effect is even more pronounced for a non-semigroup non-transversal noise, where the asymptotic behavior emerges only if almost 10⁴ probes are used, see figure 7(c). Additional numerical studies (not reported here) show that the asymptotic scaling is approached earlier when the coupling to the bath is increased. Even if it is clear that the finite-N behavior do not spoil the validity of the different scalings pointed out in the previous paragraph, it should also be clear the relevance of such behavior in many experimental frameworks, when, indeed, the high-N regime might be not achievable. In such situations, the experimental data would follow a scaling which is different from the asymptotic one for all practical purposes.

In addition, the behavior of the estimation error for finite values of N provides us with a more complete understanding of the specific role played by the geometry of the noise, i.e. the coupling angle $\vartheta$. In figure 7(e) we study $t/{F}_{Q}^{\uparrow }[{\rho }^{(N)}(t)]$ and ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$, but now for different values of $\vartheta \in [0,\pi /2]$ and a fixed number of probes N = 160. For this value of N and $\vartheta =0$ the two quantities have essentially already reached their asymptotic values, see figure 7(a), while this is not the case for $\vartheta \ne 0$, see figure 7(c). Now, figure 7(d) shows how both $t/{F}_{Q}^{\uparrow }[{\rho }^{(N)}(t)]$ and ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$ change continuously with the variation of $\vartheta$. They increase from $\vartheta =0$ up to $\vartheta =\pi /2$, with the increment being more pronounced for values of $\vartheta$ close to 0. The sudden transition between different scalings for, respectively, $\vartheta =0$ and $\vartheta \ne 0$ is a peculiarity of the asymptotic limit, $N\to \infty$. Furthermore, this also confirms that noise in the direction of the parameter imprinting is more detrimental than any other direction, if the absolute noise strength is kept identical.

As a final remark, note that the optimal time of the estimation error for a parity measurement as a function of N has discrete jumps between smooth periods, see the lower insets in figures 7(a)–(c). These jumps originate from the fact that ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$ does possess multiple local maxima instead of one global maxima as $t/{F}_{Q}^{\uparrow }[{\rho }^{(N)}(t)]$ does, see figure 7(d). The jump occurs when the global maximum of ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$ changes to a different peak, which was only a local maximum before. On the other hand, for large values of N ${{\rm{\Delta }}}^{2}{\tilde{\omega }}_{0,P}T$ will converge to a function with only one local maximum, as the following ones have been damped off, so that the optimal time will stay a smooth function of N. The jumps in the optimal evaluation time for a parity measurement can be observed also in the polar plot in figure 7(e), in terms of non-smooth variation as a function of $\vartheta$.

C.1.1. Differential equations for the density matrix elements

Due to the simple master equations in the Ohmic regime described in section 5.2, equations (49) and (50), it is more convenient to solve the dynamics taking into account the evolution of the elements of the system's density matrix, ${\rho }_{{ij}}(t)=\langle {\rm{i}}| \rho (t)| j\rangle$ for i, j = 1, 2.

For the NPC dynamics, the master equation in equation (49) is equivalent to the following system of equations (of course, ${\rho }_{01}(t)={\rho }_{10}^{* }(t)$ and ρ₀₀(t) = 1 − ρ₁₁(t)):

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{11}(t) & = & {{\rm{\cos }}}^{2}\vartheta \gamma (t)(1-2{\rho }_{11}(t))+2{\rm{\cos }}\,\vartheta {\rm{\sin }}\,\vartheta \gamma (t){\rm{Re}}\{{\rho }_{10}(t)\}\\ \displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{10}(t) & = & -{\rm{i}}{\omega }_{0}{\rho }_{10}(t)-{\rm{\sin }}\,\vartheta {\rm{\cos }}\,\vartheta \gamma (t)(1-2{\rho }_{11}(t))\\ & & -(1+{{\rm{\sin }}}^{2}\vartheta )\gamma (t){\rho }_{10}(t)+{{\rm{\cos }}}^{2}\vartheta \gamma (t){\rho }_{10}^{* }(t),\end{eqnarray} \tag{ C.6 }$$

which can be easily solved numerically. For the PC dynamics, equation (50) leads us to

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{11}(t) & = & {{\rm{\cos }}}^{2}\vartheta \gamma (t)(1-2{\rho }_{11}(t))\\ \displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{10}(t) & = & (-{\rm{i}}{\omega }_{0}-(2-{{\rm{\cos }}}^{2}\vartheta )\gamma (t)){\rho }_{10}(t).\end{eqnarray} \tag{ C.7 }$$

Contrary to the NPC case, populations and coherences are decoupled. Indeed, the solution of this system of equations reads

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{11}(t) & = & {{\rm{e}}}^{-2{{\rm{\cos }}}^{2}\vartheta {\displaystyle \int }_{0}^{t}{\rm{d}}\tau \gamma (\tau )}{\rho }_{11}(0)\\ & & +{{\rm{\cos }}}^{2}\vartheta {\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{e}}}^{-2{{\rm{\cos }}}^{2}\vartheta {\displaystyle \int }_{\tau }^{t}{\rm{d}}\tau ^{\prime} \gamma (\tau ^{\prime} )}\gamma (\tau )\\ \displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{10}(t) & = & {{\rm{e}}}^{-{\rm{i}}{\omega }_{0}t-(2-{{\rm{\cos }}}^{2}\vartheta ){\displaystyle \int }_{0}^{t}{\rm{d}}\tau \gamma (\tau )}{\rho }_{10}(0).\end{eqnarray} \tag{ C.8 }$$

In figure 4 we reported the evolution of the Bloch vector ${\boldsymbol{r}}(t)$ for different initial conditions for ρ(0). The components of the vector ${\boldsymbol{r}}(t)$ are directly related to the matrix elements of the corresponding state, see section 3. Finally, the CP of the dynamics is guaranteed by the master equations themselves, as mentioned in the main text.

C.1.2. Semigroup limit

Taking the limit ${\omega }_{C}\to \infty$, the decay rate given by equation (49) becomes time independent,

$$\begin{eqnarray}&&{\gamma }_{s}={\mathrm{lim}}_{{\omega }_{C}\to \infty }\displaystyle \frac{\lambda }{\beta }\arctan ({\omega }_{C}t)=\displaystyle \frac{\pi }{2}\displaystyle \frac{\lambda }{\beta }.\end{eqnarray} \tag{ C.9 }$$

In the NPC case, this yields the generator

$$\begin{eqnarray}{{\mathsf{D}}}_{\mathrm{NPC}}^{{ \mathcal L }}(t)=\left(\begin{array}{cccc}0 & 0 & 0 & 0\\ 0 & -2{\gamma }_{s}{\sin }^{2}(\vartheta ) & -{\omega }_{0} & {\gamma }_{s}\sin (2\vartheta )\\ 0 & {\omega }_{0} & -2\gamma & 0\\ 0 & {\gamma }_{s}\sin (2\vartheta ) & 0 & -2{\gamma }_{s}{\cos }^{2}(\vartheta )\end{array}\right).\end{eqnarray} \tag{ C.10 }$$