# Restoring Heisenberg scaling in noisy quantum metrology by monitoring the
environment

Francesco Albarelli  Quantum Technology Lab, Dipartimento di Fisica “Aldo
Pontremoli”, Università degli Studi di Milano, IT-20133, Milan, Italy
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
Matteo A. C. Rossi  Quantum Technology Lab, Dipartimento di Fisica “Aldo
Pontremoli”, Università degli Studi di Milano, IT-20133, Milan, Italy  QTF
Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics
and Astronomy, University of Turku, FI-20014 Turun Yliopisto, Finland  Dario
Tamascelli  Quantum Technology Lab, Dipartimento di Fisica “Aldo Pontremoli”,
Università degli Studi di Milano, IT-20133, Milan, Italy  Marco G. Genoni
Quantum Technology Lab, Dipartimento di Fisica “Aldo Pontremoli”, Università
degli Studi di Milano, IT-20133, Milan, Italy<EMAIL_ADDRESS>
###### Abstract

We study quantum frequency estimation for $N$ qubits subject to independent
Markovian noise via strategies based on time-continuous monitoring of the
environment. Both physical intuition and the extended convexity of the quantum
Fisher information (QFI) suggest that these strategies are more effective than
the standard ones based on the measurement of the unconditional state after
the noisy evolution. Here we focus on initial GHZ states subject to parallel
or transverse noise. For parallel, i.e., dephasing noise, we show that
perfectly efficient time-continuous photodetection allows us to recover the
unitary (noiseless) QFI, and hence obtain Heisenberg scaling for every value
of the monitoring time. For finite detection efficiency, one falls back to
noisy standard quantum limit scaling, but with a constant enhancement due to
an effective reduced dephasing. In the transverse noise case, Heisenberg
scaling is recovered for perfectly efficient detectors, and we find that both
homodyne and photodetection-based strategies are optimal. For finite detector
efficiency, our numerical simulations show that, as expected, an enhancement
can be observed, but we cannot give any conclusive statement regarding the
scaling. We finally describe in detail the stable and compact numerical
algorithm that we have developed in order to evaluate the precision of such
time-continuous estimation strategies, and that may find application in other
quantum metrology schemes.

##  1 Introduction

Quantum metrology is one of the most promising fields within the realm of
quantum technologies, with applications ranging from spectroscopy and
magnetometry to interferometry and gravitational waves detection [1, 2, 3, 4].
While $N$ uncorrelated (_classical_) probe states lead to an estimation
precision scaling as $1/\sqrt{N}$, typically referred to as _standard quantum
limit_ (SQL), quantum probes made of $N$ entangled particles allow to design
schemes with precision scaling as $1/N$, reaching the so-called _Heisenberg
limit_ (HL) [5].

The above result relies on the assumption of noiseless unitary dynamics, but
the unavoidable interaction with the surrounding environment can have dramatic
consequences on the performances of these protocols. When the dynamics is
described by a dephasing dynamical semigroup, the estimation precision is
bounded to follow a SQL scaling, and thus only a constant enhancement can be
obtained by exploiting quantum resources [6, 7, 8, 9]. Several attempts to
circumvent these no-go theorems and obtain a superclassical scaling in the
presence of noise have been pursued, exploiting time-inhomogeneous dynamics
[10, 11, 12, 13, 14], noise with a particular geometry [15, 16], dynamical
decoupling [17], and quantum error-correction protocols (or, more generally,
the possibility to implement control Hamiltonians) [18, 19, 20, 21, 22, 23,
24, 25, 26].

Our goal is to attack the problem of noisy quantum metrology by exploiting
time-continuous monitoring [27, 28]. Time-continuous measurements have been
often studied as a tool for quantum parameter/state estimation [29, 30, 31,
32, 33, 34, 35, 36], with a particular focus on classical time-dependent
signals [37, 38, 39]. More recently, methods to calculate classical and
quantum Cramér-Rao bounds for parameter estimation via time-continuous
monitoring have been proposed [40, 41, 42, 43, 44], and put into action [45,
46, 47, 48].

In this paper, we apply such techniques to the problem of frequency estimation
for an ensemble of two-level atoms (qubits), each subjected to independent
Markovian noise. In particular, we consider estimation strategies based on
time-continuous (_weak_) measurements of each qubit via the monitoring of the
corresponding environment, and a final (_strong_) measurement on the
conditional state of the $N$ qubits. Few similar attempts in this sense have
already been discussed in the literature, but they all present substantial
differences compared to our approach. For instance, in [49, 50, 48] the same
problem of frequency estimation (magnetometry) is addressed, but in the
presence of collective transverse noise. There, a single _environment_
collectively interacts with all the qubits and the corresponding noise is not
detrimental for the Heisenberg scaling; time-continuous monitoring is however
extremely promising as, thanks to the corresponding back-action on the system,
a Heisenberg-limited precision can be achieved even by starting the dynamics
with an uncorrelated spin-coherent state. An estimation problem more similar
to ours is presented in [51], where independent environments interact with
each probe, but the analysis is restricted to a single (discrete) step of the
dynamics and to interaction Hamiltonians commuting with the generator of the
phase rotation (i.e., in the presence of pure dephasing only).

Here we consider a proper continuous evolution, described by a Markovian
master equation in the Lindblad form, leading to a dynamics satisfying the
semigroup property. We focus in particular on independent noise, either
parallel or transverse to the generator of the phase rotation to be estimated.
In the former case, which physically corresponds to pure dephasing, the
unconditional dynamics leads to a standard quantum limited precision, even for
an infinitesimal amount of noise [8]. In the latter, it was shown that, by
optimizing over the evolution time, it is possible to restore a super-
classical scaling between SQL and HL [15, 16].

Our goal is to study whether in both cases time-continuous monitoring will
allow for the restoration of the HL, and to analyze in detail the effect of
the monitoring efficiency on the performance of the estimation schemes. First,
we will derive the ultimate limit on estimation precision, optimizing over the
most general measurements on the joint degrees of freedom of system and
environment. We will then restrict ourselves to the strategies briefly
described above, focusing on time-continuous photodetection and homodyne-
detection.

To achieve those aims, we develop a stable and compact numerical algorithm
that allows us to calculate the effective quantum Fisher information
characterizing this kind of measurement strategy, and that will find
application in quantum metrology problems beyond the ones considered in this
paper.

The manuscript is structured as follows: In Sec. 2 we introduce the basic
concepts of quantum estimation theory, with a particular focus on measurement
strategies based on time-continuous measurements. In Sec. 3 we first introduce
the problem of noisy quantum frequency estimation and then we present our
original results for parallel and transverse Markovian noise. The following
sections are devoted to describing the methods exploited to obtain such
results. In Sec. 4 we show how to evaluate analytically the ultimate limits
posed by quantum mechanics for strategies optimizing over all the the possible
global measurements on system and environment. In Sec. 5 we present in detail
the numerical algorithm we have developed for the calculation of the effective
quantum Fisher information, pertaining to strategies based on time-continuous
monitoring of the environment. Sec. 6 concludes the paper with some final
remarks and discussion of possible future directions.

##  2 Quantum estimation via time-continuous measurements

A classical estimation problem is typically described in terms of a
conditional probability $p(x|\omega)$ of obtaining the measurement outcome
$x$, given the value of the parameter $\omega$ that one wants to estimate. The
classical Cramér-Rao bound states that the precision of any unbiased estimator
is lower bounded as

$\delta\omega\geq\frac{1}{\sqrt{M\mathcal{F}[p(x|\omega)]}}\,,$ (1)

where $M$ is the number of measurements performed,
$\mathcal{F}[p(x|\omega)]=\mathbbm{E}_{p}[(\partial_{\omega}\ln
p(x|\omega))^{2}]$ is the classical Fisher information, and
$\mathbbm{E}_{p}[\cdot]$ denotes the average over the probability distribution
$p(x|\omega)$.

In the quantum setting, the probability is obtained via the Born rule,
$p(x|\omega)=\hbox{Tr}[\varrho_{\omega}\Pi_{x}]$, where $\varrho_{\omega}$ is
a family of quantum states parametrized by $\omega$, and $\Pi_{x}$ is an
element of the positive-operator valued measure (POVM) describing the
measuring process. It is then possible to optimize over all the possible
POVMs, obtaining the quantum Cramér-Rao inequality [52]

$\displaystyle\delta\omega\geq\frac{1}{\sqrt{M\mathcal{F}[p(x|\omega)]}}\geq\frac{1}{\sqrt{M\mathcal{Q}[\varrho_{\omega}]}}\,,$
(2)

where

$\displaystyle\mathcal{Q}[\varrho_{\omega}]=\lim_{\epsilon\rightarrow
0}\frac{8\left(1-F\left[\varrho_{\omega},\varrho_{\omega+\epsilon}\right]\right)}{\epsilon^{2}}$
(3)

is the quantum Fisher information (QFI), expressed in terms of the fidelity
between quantum states
$F\left[\varrho_{1},\varrho_{2}\right]=\hbox{Tr}\left[\sqrt{\sqrt{\varrho_{1}}\varrho_{2}\sqrt{\varrho_{1}}}\right]$.
The QFI $\mathcal{Q}[\varrho_{\omega}]$ clearly depends only on the quantum
statistical model, namely the $\omega$-parametrized family of quantum states
$\varrho_{\omega}$, and one can always find the optimal POVM saturating the
bound., i.e., such that
$\mathcal{F}[p(x|\omega)]=\mathcal{Q}[\varrho_{\omega}]$.

In several quantum parameter estimation problems, including the case of
standard frequency estimation, $\varrho_{\omega}$ corresponds to the
_unconditional_ state evolved according to a Markovian master equation of the
form

$\frac{d\varrho}{dt}=\mathcal{L}_{\omega}\varrho=-i[\hat{H}_{\omega},\varrho]+\sum_{j}\mathcal{D}[\hat{c}_{j}]\varrho\,,$
(4)

where the parameter is encoded in the Hamiltonian $\hat{H}_{\omega}$, the
operators $\hat{c}_{j}$ denote different independent noisy channels, and we
have defined the superoperator
$\mathcal{D}[\hat{A}]\varrho=\hat{A}\varrho\hat{A}^{\dagger}-(\hat{A}^{\dagger}\hat{A}\varrho+\varrho\hat{A}^{\dagger}\hat{A})/2$.

The master equation above can be obtained assuming that the system is
interacting with a sequence of input operators
$\hat{a}^{(j)}_{\mathsf{in}}(t)$, one for each noise operator $\hat{c}_{j}$,
that satisfy the bosonic commutation relation
$[\hat{a}^{(j)}_{\mathsf{in}}(t),\hat{a}^{(k){\dagger}}_{\mathsf{in}}(t^{\prime})]=\delta_{jk}\delta(t-t^{\prime})$.
Some information on the parameter $\omega$ is then contained in the
corresponding output operators $\hat{a}^{(j)}_{\mathsf{out}}(t)$, i.e., the
environmental modes, just after the interaction with the system. One can
imagine to perform a joint measurement on all output modes and the quantum
system itself. The ultimate limit on the estimation of $\omega$ that takes
into account all these strategies based on measurements of system and output,
is quantified by the QFI introduced in [42]. It depends only on the initial
state of the system and on the superoperator $\mathcal{L}_{\omega}$ describing
the evolution. In general, it can be evaluated as

$\displaystyle\mathcal{\overline{Q}}_{\mathcal{L}_{\omega}}=4\partial_{\omega_{1}}\partial_{\omega_{2}}\log\left|\hbox{Tr}[\bar{\varrho}]\right|\big{|}_{\omega_{1}=\omega_{2}=\omega}\,\,,$
(5)

where, in the case we are considering (i.e., a Hamiltonian parameter), the
operator $\overline{\varrho}$ is the one solving the generalized master
equation

$\displaystyle\frac{d\bar{\varrho}}{dt}$
$\displaystyle=\mathcal{L}_{\omega_{1},\omega_{2}}\bar{\varrho}$
$\displaystyle=-i\left(\hat{H}_{\omega_{1}}\bar{\varrho}-\bar{\varrho}\hat{H}_{\omega_{2}}\right)+\sum_{j}\mathcal{D}[\hat{c}_{j}]\bar{\varrho}\,.$
(6)

This QFI corresponds to an optimization of the classical FI over all the
possible measurement strategies described above, including the possibility of
performing non-separable (entangled) measurements over the system and all the
output modes at different times.

However, one can restrict to more feasible strategies, where the outputs are
sequentially measured continuously in time, and a final _strong_ measurement
is performed on the conditional state of the system. Depending on the type of
measurement performed on the outputs, one obtains different stochastic master
equations for the conditional state of the system. In the following we will
focus on the two paradigmatic cases of photodetection (PD) and homodyne
detection (HD).

In the first case, assuming time-continuous PD on each output with efficiency
$\eta_{j}$, the evolution is described by the stochastic master equation [27]

$\displaystyle d\varrho^{(c)}$
$\displaystyle=-i[\hat{H}_{\omega},\varrho^{(c)}]\,dt+\sum_{j}(1-\eta_{j})\mathcal{D}[\hat{c}_{j}]\varrho^{(c)}\,dt$
$\displaystyle-\frac{\eta_{j}}{2}(\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}\varrho^{(c)}+\varrho^{(c)}\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j})\,dt+\eta_{j}\hbox{Tr}[\varrho^{(c)}\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}]\varrho^{(c)}\,dt$
$\displaystyle+\left(\frac{\hat{c}^{\vphantom{{\dagger}}}_{j}\varrho^{(c)}\hat{c}_{j}^{\dagger}}{\hbox{Tr}[\varrho^{(c)}\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}]}-\varrho^{(c)}\right)dN_{j}\,,$
(7)

where $dN_{j}$ denote Poisson increments taking value $0$ (no-click event) or
$1$ (detector click event), and having average value
$\mathbbm{E}[dN_{j}]=\eta_{j}\hbox{Tr}[\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}\varrho^{(c)}]\,dt$.

Likewise, for time-continuous HD on each output, the stochastic master
equation reads [27, 53]

$\displaystyle d\varrho^{(c)}={}$
$\displaystyle-i[\hat{H}_{\omega},\varrho^{(c)}]\,dt+\sum_{j}\mathcal{D}[\hat{c}_{j}]\varrho^{(c)}\,dt$
$\displaystyle+\sum_{j}\sqrt{\eta}_{j}\mathcal{H}[\hat{c}_{j}]\varrho^{(c)}\,dw_{j}\>,$
(8)

where
$\mathcal{H}[\hat{c}]\varrho=\hat{c}\varrho+\varrho\hat{c}^{\dagger}-\hbox{Tr}[\varrho(\hat{c}+\hat{c}^{\dagger})]\varrho$,
and
$dw_{j}=dy_{j}-\sqrt{\eta}_{j}\hbox{Tr}[\varrho^{(c)}(\hat{c}^{\vphantom{{\dagger}}}_{j}+\hat{c}_{j}^{\dagger})]$
represents a standard Wiener increment (s.t. $dw_{j}dw_{k}=\delta_{jk}dt$),
operationally corresponding to the difference between the measurement result
$dy_{j}$ and the rescaled average value of the operator
$(\hat{c}^{\vphantom{{\dagger}}}_{j}+\hat{c}_{j}^{\dagger})$.

In general, different measurement strategies mathematically correspond to
different possible unravellings of the Markovian master equation: any quantum
state, solution of Eq. (4) at a certain time $t$, can be written as
$\varrho_{\mathsf{unc}}=\sum_{\mathsf{traj}}p_{\mathsf{traj}}\varrho^{(c)}$,
where $p_{\mathsf{traj}}$ is the probability of a certain trajectory leading
to the conditional state $\varrho^{(c)}$, and where the sum is replaced by an
integral in the case of time-continuous measurements with a continuous
spectrum (e.g. homodyne).

One can then define an effective QFI, which poses the ultimate bounds for
these kinds of estimation strategies [51, 39, 48], and that depends both on
the specific unravelling and on the monitoring efficiencies $\eta_{j}$:

$\displaystyle\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta_{j}}=\mathcal{F}[p_{\mathsf{traj}}]+\sum_{\mathsf{traj}}p_{\mathsf{traj}}\mathcal{Q}[\varrho^{(c)}]\,.$
(9)

As it is apparent from the formula,
$\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta}$ is equal to the sum of the
classical Fisher information of the trajectories probability distribution
$\mathcal{F}[p_{\mathsf{traj}}]=\sum_{\mathsf{traj}}\frac{(\partial_{\omega}p_{\mathsf{traj}})^{2}}{p_{\mathsf{traj}}}$,
plus the average of the QFIs of the corresponding conditional states
$\mathcal{Q}[\varrho^{(c)}]$. The first term quantifies the amount of
information obtained from the measurement of the output modes after the
interaction with the system, while the second term quantifies the amount of
information obtained from the final strong measurement on the conditional
states of the system itself. We point out that an analogous figure of merit
has been recognized as the appropriate one for metrology with post-selection
[54, 55, 56].

A method for the calculation of $\mathcal{F}[p_{\mathsf{traj}}]$ has been
firstly proposed in [41] for generic quantum states, and then in [44] for
continuous-variable Gaussian states. In Sec. 5, we describe in detail a more
compact and stable numerical algorithm for the calculation of the terms
present in Eq. (9), for both homodyne and photodetection measurements.

It is reasonable to expect that, thanks to the information obtained from the
time-continuous monitoring, and thanks to the, typically beneficial, effects
of quantum measurements on the conditional states, one will obtain more
information on the parameter via these strategies, than by solely measuring
the unconditional state solution of the Markovian master equation (4). This
intuition is confirmed by the extended convexity property of the QFI, recently
proved in [57, 39]. In fact, the following chain of inequalities holds:

$\displaystyle\mathcal{Q}[\varrho_{\mathsf{unc}}]$
$\displaystyle\leq\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta_{j}}\leq\overline{\mathcal{Q}}_{\mathcal{L_{\omega}}}\,.$
(10)

We also conjecture that, for a fixed unravelling, and for fixed efficiency on
all the output channels (i.e., for $\eta_{j}=\eta\,\forall j$), the effective
QFI is monotonic with respect to the efficiency parameter, that is:

$\displaystyle\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta}\leq\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta^{\prime}}\quad\Longleftrightarrow\quad\eta\leq\eta^{\prime}\qquad\textrm{(conjecture)}\,.$
(11)

##  3 Quantum frequency estimation in the presence of noise

We consider a system of $N$ qubits, described by a Hamiltonian
$\hat{H}_{\omega}=(\omega/2)\sum_{j=1}^{N}\sigma_{z}^{(j)}$, where $\omega$ is
the unknown frequency to be estimated, and $\sigma_{z}^{(j)}$ is the Pauli-z
operator acting on the $j$-th qubit. The system is interacting also with a
Markovian environment, so that the evolution is described by the master
equation

$\displaystyle\frac{d\varrho}{dt}$
$\displaystyle=\mathcal{L}_{\omega}\varrho=-i[\hat{H}_{\omega},\varrho]+\frac{\kappa}{2}\sum_{j=1}^{N}\mathcal{D}[\sigma_{\alpha}^{(j)}]\varrho\,,$
(12)

where $\alpha\in\\{x,z\\}$, i.e., we only consider noise parallel or
transverse to the Hamiltonian. The master equation (12) can be easily mapped
to Eq. (4) by considering the noise operators
$\hat{c}_{j}=\sqrt{\kappa/2}\sigma_{\alpha}^{(j)}$. In what follows we will
study all the quantities described in the previous section in order to assess
the estimation of the parameter $\omega$.

In quantum frequency estimation strategies, one considers the number of qubits
$N$ and the total time of the experiment $T$ as the resources of the protocol.
The quantum Cramér-Rao bound (2) is then more efficiently rewritten as

$\displaystyle\delta\omega\sqrt{T}\geq\frac{1}{\sqrt{\mathcal{Q}/t}}\geq\frac{1}{\sqrt{\max_{t}[\mathcal{Q}/t]}}$
(13)

where $t=T/M$ corresponds to the duration of each round, over which one can
perform a further optimization, and where $\mathcal{Q}$ corresponds here to
the proper QFI charaterizing the particular estimation strategy considered.

In the rest of the manuscript we are restricting to initial entangled GHZ
states $|\psi_{\mathsf{GHZ}}\rangle=(|0\rangle^{\otimes N}+|1\rangle^{\otimes
N})/\sqrt{2}$. It is well known that in the noiseless case, i.e., for
$\kappa=0$, the corresponding QFI is Heisenberg limited,
$\mathcal{Q}_{\mathsf{HL}}=N^{2}t^{2}$. This leads to a quadratic enhancement
w.r.t. the “standard quantum limited” QFI, $\mathcal{Q}_{\mathsf{SQL}}=Nt^{2}$
(obtained in the case of a factorized _coherent-spin_ initial state
$|\psi_{\mathsf{coh}}\rangle=[(|0\rangle+|1\rangle)/\sqrt{2}]^{\otimes N}$).

In what follows, we will consider the noisy case ($\kappa>0$); the generic QFI
$\mathcal{Q}$ in Eq. (13) will correspond to either: i) the QFI of the
unconditional state $\mathcal{Q}[\varrho_{\mathsf{unc}}]$ corresponding to the
master equation (12); ii) the ultimate QFI
$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}$ obtained optimizing over all
the possible measurements on system and environmental outputs; iii) the
effective QFI $\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta}$ corresponding to a
specific time-continuous (sequential) measurement of the output modes and a
final strong measurement on the conditional state of the system. In particular
we are will focus on time-continuous photodetection and homodyne detection,
with measurement efficiency $\eta$ and we are labelling the respective
effective QFIs as $\widetilde{\mathcal{Q}}_{\mathsf{pd,\eta}}$ and
$\widetilde{\mathcal{Q}}_{\mathsf{hom,\eta}}$. It is important to remark that,
given the master equation (12), one is assuming that $N$ different (homodyne
or photo-) detectors are monitoring the environment of each qubit. We will
however find instances where this assumptions may be relaxed.

In the next subsections we address separately the two different cases of
parallel and transverse noise. For each case we start by reviewing known
results for the QFI of the unconditional states; then we will present our
original results, regarding the ultimate QFI
$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}$ and effective QFIs
$\widetilde{\mathcal{Q}}_{\mathsf{pd,\eta}}$ and
$\widetilde{\mathcal{Q}}_{\mathsf{hom,\eta}}$ (corresponding to the schemes
pictorially represented in Fig. 1), without dwelling into the details of their
calculation, which will be left to Secs. 4 and 5.

Figure 1: Schematic representation of the metrological approach we consider in
this paper. A $N$-qubit (possibly entangled) input state $|\psi_{0}\rangle$
interacts with $N$ independent environments which are monitored by $N$
detectors, either by photodetection (PD) or homodyne detection (HD). In the
former case each output is a binary valued detection record $N_{j}(t)$, in the
latter a real valued photo-current $y_{j}(t)$. The map
$\mathcal{E}^{(c)}_{\omega,j}(t)$ represents the corresponding conditional
dynamics, i.e., the solution of the stochastic master equation for the single
$j$-th qubit only, either for PD (7) or HD (8), and thus depending on the
corresponding measurement output $N_{j}(t)$ or $y_{j}(t)$. The $N$-qubit
quantum state $\varrho^{(c)}(t)$, conditioned on all the measurement outcomes
acquired during the evolution, is further collectively measured at the end of
the conditional dynamics.

###  3.1 Parallel noise

Parallel noise, corresponding to the master equation (12) with
$\sigma_{\alpha}^{(j)}=\sigma_{z}^{(j)}$, is typically considered the most
detrimental noise for frequency estimation since the evolution induces
dephasing in the eigenbasis of the Hamiltonian $\hat{H}_{\omega}$. For an
initial GHZ state, the QFI of the unconditional state can be evaluated
analytically [6, 58], obtaining
$\mathcal{Q}[\varrho_{\mathsf{unc}}]=N^{2}t^{2}e^{-2\kappa Nt}$.
By optimizing over the single-shot duration $t$, one obtains that the optimal
QFI is standard quantum limited,

$\displaystyle\max_{t}\left[\frac{\mathcal{Q}[\varrho_{\mathsf{unc}}]}{t}\right]=\frac{N}{2e\kappa}\,.$
(14)

This result is equivalent to the one obtainable via a factorized initial
state. Only a constant enhancement can in fact be gained, by optimizing over
the initial entangled state [6]; by doing this one saturates the ultimate
noisy bound derived in [7, 8], which dictates that, as soon as some parallel
noise is present in the dynamics, the ultimate precision is standard quantum
limited.

On the other hand, the ultimate QFI
$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}$ can be easily evaluated (see
Sec. 4 for details), and it turns out to be equal to the noiseless QFI, i.e.

$\displaystyle\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\parallel}=N^{2}t^{2}\,.$
(15)

This shows that, by measuring also the output modes, it is in principle
possible to recover not only a Heisenberg scaling for the error
$\delta\omega$, but also the whole information on the parameter.

For both time-continuous homodyne and photodetection, the evolution of an
initial GHZ state, under parallel noise and conditioned on the measurement
results, is restricted to a two-dimensional Hilbert space. As a consequence,
the results obtained for $N=1$ qubit can be readily used to infer the results
for generic $N$ qubits, by simply rescaling the evolution time $t\rightarrow
Nt$ (see Sec. 5 for details). We can thus obtain numerically exact values of
the effective QFIs for any value of $N$; in particular we have been able to
numerically verify that the strategy based on time-continuous photodetection
with perfect efficiency $\eta=1$ is indeed optimal, i.e.

$\displaystyle\widetilde{\mathcal{Q}}^{\parallel}_{\mathsf{pd},\eta=1}=\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\parallel}=N^{2}t^{2}\,,$
(16)

showing that the noiseless Heisenberg-limited result can be recovered, without
the need to perform complicated (non-local in time) measurement strategies on
system and environment.

The physical explanation of this result can be easily understood by studying
the action of the two Kraus operators describing the conditional dynamics for
initial GHZ states (see Sec. 5 for more details on the Kraus operators). The
_no-jump_ evolution is ruled by the infinitesimal Kraus operator
$M_{0}=\mathbbm{1}-i\hat{H}_{\omega}\,dt+(\kappa N/4)\mathbbm{1}dt$; as a
consequence it easy to check that the GHZ state evolves as in the unitary
case, apart from an irrelevant normalization factor that can be ignored.

On the other hand, when a jump occurs due to a photon detected in the output
corresponding to one of the $N$ qubits, the only effect of the corresponding
Kraus operator, $M_{1}^{(j)}=\sigma_{z}^{(j)}\sqrt{(\kappa/2)dt}$, is to
induce a relative minus sign between the $|0\rangle^{\otimes N}$ and the
$|1\rangle^{\otimes N}$ components of the quantum conditional state,
independently on the index $j$ of the corresponding qubit. In general, after a
time $t$ and $m$ detected photons from all the the output channels, the
conditional state reads:

$\displaystyle|\psi_{m\lvert\mathsf{GHZ}}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle^{\otimes
N}+e^{i(N\omega t+m\pi)}|1\rangle^{\otimes N}\right)\,.$ (17)

It is important to remark that, thanks to the symmetry of the GHZ state, and
given that jumps may occur only one by one [27], it is not necessary to know
exactly from which qubit channel the photon has been detected; as a
consequence one may obtain the same result by using a single perfect photo-
detector monitoring jointly all the $N$ output modes. Once the number of
detected photons $m$ is known, and the corresponding “GHZ-equivalent”
conditional state is prepared, one can estimate the frequency $\omega$ at the
Heisenberg limit.

As soon as the photodetection monitoring is not perfectly efficient (we always
consider equal efficiency for all the qubits, $\eta_{j}=\eta<1,\,\,\forall
j$), the Heisenberg scaling is immediately lost. Our numerical results clearly
show that the effective QFI is equal to
$\widetilde{\mathcal{Q}}^{\parallel}_{\mathsf{pd},\eta}=N^{2}t^{2}e^{-2\kappa(1-\eta)Nt}$,
and the optimized QFI reads

$\displaystyle\max_{t}\left[\frac{\widetilde{\mathcal{Q}}^{\parallel}_{\mathsf{pd},\eta}}{t}\right]=\frac{N}{2e\kappa(1-\eta)}\,.$
(18)

Time-continuous inefficient photodetection thus leads to a constant
enhancement, compared to the unconditional/classical case (14), which
physically corresponds to have a reduced effective dephasing parameter
$\kappa_{\mathsf{eff}}=\kappa(1-\eta)$. We remark that also in this case it is
not necessary to monitor every output channel separately, and that a single-
photo detector can be employed. Moreover, in this picture, the overall
efficiency parameter $\eta$ corresponds to the product between the factual
efficiency of the detectors and the fraction of qubits that are effectively
monitored.

We also notice that, for every value of $\omega$, all the information is
contained in the conditional quantum states: the classical Fisher information
$\mathcal{F}[p_{\mathsf{traj}}]$ is in fact identically equal to zero. This
follows from the unitarity of the Pauli matrices, leading to
$\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}=\kappa\mathbbm{1}_{2}/2$,
and thus to a parameter-independent Poisson increment with average value
$\mathbbm{E}[dN_{j}]=(\eta\kappa/2)\,dt$ in Eq. (7). Nevertheless, we remark
that the output from the photodetection measurement is in fact essential to
know the corresponding conditional state, and thus to extract the whole
information on $\omega$ via the final strong measurement.

These results can be readily extended to the scenario of non-linear quantum
metrology, where $k$-body Hamiltonian and $p$-body dissipators are considered
[59]. If we focus on the case where $k$ and $p$ are odd numbers, by preparing
the initial state in a GHZ state, and by monitoring via photodetection each
dissipation channel, one can show by a simple rescaling of the variables
$\omega$ and $\kappa$ that for unit monitoring efficiency, the ultimate
(noiseless) scaling $\sim N^{k}$ can be restored; on the other hand, as soon
as the efficiency is smaller than one, one goes back to the noisy scaling
$\sim N^{k-p/2}$, with only a constant enhancement, due to the finite
monitoring efficiency.

The results for the case of continuous homodyne monitoring are not reported
here since the corresponding effective QFI, at fixed efficiency $\eta$, is
always lower than the one obtained for continuous photodetection, and
Heisenberg scaling is not recovered even in the case of perfect monitoring.

###  3.2 Transverse noise

Thanks to the symmetry of the GHZ state, the QFI for the unconditional
dynamics with arbitrary collapse operators can be obtained without the need to
diagonalize the full density matrix [15]. The corresponding optimized QFI can
be then numerically obtained and the scaling is found to be intermediate
between SQL and Heisenberg:
$\max_{t}\left[\mathcal{Q}\left[\varrho_{\mathsf{unc}}^{\bot}\right]/t\right]\approx
N^{5/3}$.

On the other hand, the ultimate QFI can be computed analytically (see Sec. 4
for details on the calculation), yielding

$\displaystyle\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}=\frac{N^{2}\left(1-e^{-\kappa
t}\right)^{2}+N\left[2\kappa t+1-\left(2-e^{-\kappa
t}\right)^{2}\right]}{\kappa^{2}}.$ (19)

Two main observations are in order here: on the one hand we observe that
$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}$ depends on the noise
parameter $\kappa$ and is always smaller than the noiseless QFI
$\mathcal{Q}_{\sf HL}=N^{2}t^{2}$. This shows how, unlike in the parallel
noise case, for transverse (non-commuting) noise, part of the information
leaking into the environment is irretrievably lost, and cannot be recovered
even if one has at disposal all the environmental degrees of freedom. On the
other hand, this expression does explicitly show Heisenberg scaling, and can
be further optimized over the evolution time $t$. In contrast to the
unconditional case, the optimal time $t_{\mathsf{opt}}(N)$ does not go to zero
for $N\to\infty$, instead it tends to a constant:
$\lim_{N\to\infty}t_{\mathsf{opt}}(N)=c/\kappa$, where $c\approx 1.26$ . The
very same results can be obtained by computing the unconditional QFI in the
limit $\omega\to 0$ [60], which is already known to give rise to Heisenberg
scaling (see Appendix D of [16]), i.e.

$\displaystyle\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}$
$\displaystyle=\lim_{\omega\to
0}\mathcal{Q}\left[\varrho^{\bot}_{\mathsf{unc}}\right].$ (20)

Notice that it is necessary to take the limit, instead of using directly
$\omega=0$, because for this value of the parameter the density matrix changes
its rank and this gives rise to a discontinuity in the QFI [61, 62]. The
effective QFI for photodetection and homodyne detection is obtained
numerically with the methods presented in Sec. 5. From our results we observe
that for unit efficiency $\eta=1$ the effective QFI saturates the ultimate
bound in both cases

$\displaystyle\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{hd},\eta=1}=\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{pd},\eta=1}=\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}.$
(21)

This has been checked up to $N=14$, but we conjecture that this equality holds
in general. The two terms that contribute to
$\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{hd},\eta=1}$ in Eq. (9) always sum up
to $\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}$ but with
$\omega$-dependent behaviors. This is shown for a particular set of parameters
in Fig. 2. On the other hand, as explained before for the parallel case,
$\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{pd},\eta=1}$ is only equal to the
average QFI of the conditional states, since the classical FI
$\mathcal{F}[p_{\mathsf{traj}}]$ is always identical to zero.

Figure 2: Contributions of the classical FI $\mathcal{F}[p_{\mathsf{traj}}]$
and the QFI of the final strong measurement $\mathcal{Q}[\varrho^{(c)}]$ to
the effective QFI $\tilde{\mathcal{Q}}_{\mathsf{hd|\eta=1}}$ for homodyne with
$N=7$ and $\omega=1$, in the transverse noise case.

As expected, for $\eta<1$, we observe that the effective QFI lies between the
unconditional QFI and the ultimate QFI, confirming the chained inequalities
(10) for both detection strategies and also the conjecture regarding the
monotonicity with the efficiency $\eta$.

Figure 3: The figure shows the effective QFI for photodetection
$\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{pd},\eta}/t$ (top row) and for
homodyne $\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{ph},\eta}/t$ (bottom row)
with transverse noise over time, for different efficiencies $\eta$ and for
different values of $N$. The effective QFI is compared with the ultimate bound
$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}$ (black dashed line),
and the QFI for the unconditional evolution
$\mathcal{Q}[\varrho^{\bot}_{\mathsf{unc}}]$ (black dot-dashed line). Here we
take $\omega=\kappa$. The colored curves are obtained numerically as explained
in Sec. 5, by simulating a large number $>10\,k$ trajectories.

As an example, in Fig. 3 we show the effective QFI over time for
photodetection and homodyne detection at different efficiencies, for three
different values of $N$. We can see that at lower efficiencies the curves tend
to the unconditional QFI $\mathcal{Q}[\varrho_{\mathsf{unc}}^{\bot}]$, while
for perfect efficiency they coincide with
$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}$. Notice that in general one
cannot define a hierarchy between the two strategies, and that in particular,
in the case of homodyne detection, the curves become constant at large $t$,
due to the non-vanishing contribution of the classical Fisher information
$\mathcal{F}[p_{\mathsf{traj}}]$ that is linear in $t$.

Since the numerical method for non-unit efficiency requires using the full
Hilbert space, the complexity of the algorithm is exponential in $N$ and we
have been able to obtain results only up to $N=7$. Consequently, we cannot
explicitly witness a different scaling from the unconditional case. As a
matter of fact the difference between
$\max_{t}\left[\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\bot}/t\right]$
and
$\max_{t}\left[\mathcal{Q}\left[\varrho^{\bot}_{\mathsf{unc}}\right]/t\right]$
is not very significant for $N\leq 7$. This is shown in Fig. 4 for
photodetection, in the case $\omega=\kappa$. As we can see, the two quantities
have a similar scaling in this range of $N$, with the effective QFI lying
between them with optimal values that are monotonous with $\eta$. The optimal
measurement time decreases with $N$, and it increases with increasing $\eta$.

Figure 4: The maximum over time of the quantity $\mathcal{Q}/t$ (left panel)
and the corresponding optimal time (right panel) for $\omega=\kappa$, where
$\mathcal{Q}=\widetilde{\mathcal{Q}}^{\bot}_{\mathsf{pd|\eta}}$ for various
values of $\eta$ for the colored dots,
$\mathcal{Q}=\overline{\mathcal{Q}}^{\bot}_{\mathcal{L}_{\omega}}$ for the
black circles, and $\mathcal{Q}=\mathcal{Q}[\varrho_{\mathsf{unc}}]$ for the
black stars. The gray shade shows the region individuated by the bounds in Eq.
(10). Notice that $\overline{\mathcal{Q}}^{\bot}_{\mathcal{L}_{\omega}}/t$ is
monotonically increasing for $N\leq 3$, and it reaches its maximum value for
$t\to\infty$, and that the optimal measurement time $N\leq 2$ for $\eta=0.99$
is out of scale.

##  4 Evaluation of the ultimate QFI

In this section we show how to derive the analytical formulas for the ultimate
QFI $\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}$ for both parallel and
transverse noise.

###  4.1 Parallel noise

We start by showing that, for parallel noise, the ultimate QFI is equal to the
QFI of the noiseless case: the proof is however more general and is valid
whenever the collapse operators $\hat{c}_{j}$ commute with the free
Hamiltonian $\hat{H}_{\omega}$.

The master equation (4) is obtained by considering the following interaction
Hamiltonian between the system and the input modes:
$\hat{H}_{\text{int}}(t)=\sum_{j=1}^{N}\left(\hat{c_{j}}\hat{a}^{(j){\dagger}}_{\mathsf{in}}(t)+\hat{c}^{\dagger}_{j}\hat{a}^{(j)}_{\mathsf{in}}(t)\right)$.
We remark that $t$ is merely a parameter that labels which mode interacts with
the system at time $t$ and for each $t$ we have a different operator acting on
a different Hilbert space. The total time-dependent Hamiltonian for the system
and environment is thus
$\hat{H}_{\mathsf{SE}}(t)=\hat{H}_{\omega}+\hat{H}_{\mathsf{int}}(t)$ and each
input mode interacts with the main system for an infinitesimal time $dt$.
However, for the sake of clarity, we will consider a discretization with a
finite interaction time $\delta t$, so that the evolution over a total time
$T=M\delta t$ involves a finite number $M$ of input modes. We also assume that
the state of the input modes is the vacuum $|0\rangle$ and that the initial
state of the system $|\psi_{0}\rangle$ is pure.

Under these assumptions the joint state of system and environment evolves as

$|\psi_{\mathsf{SE}}(\omega)\rangle=\hat{U}_{t_{M}}\dots\hat{U}_{t_{2}}\hat{U}_{t_{1}}\left(|\psi_{0}\rangle\otimes|0\rangle^{\otimes
M}\right),$ (22)

where $t_{j}=j\cdot\delta t$ and $U_{t_{j}}=\exp\left[-i\delta
t\left(\hat{H}_{\omega}+\hat{H}_{\text{int}}(t_{j})\right)\right]$.

This joint state is connected to the operator $\bar{\varrho}$ appearing in Eq.
(2), since its trace represents the fidelity for two different values of the
parameter [42, 43], i.e.

$\langle\psi_{\mathsf{SE}}(\omega_{1})|\psi_{\mathsf{SE}}(\omega_{2})\rangle=\hbox{Tr}\left[\bar{\varrho}\right].$
(23)

When all the collapse operators $\hat{c}_{j}$ commute with the free
Hamiltonian $\hat{H}_{\omega}$, $\hat{H}_{\mathsf{int}}$ commutes as well and
we have

$\hat{U}_{t_{i}}=\exp\left[-i\delta
t\hat{H}_{\mathsf{int}}(t_{i})\right]\cdot\exp\left[-i\delta
t\hat{H}_{\omega}\right].$ (24)

Therefore, in the computation of the overlap (23) the terms due to the
interaction cancel out and we have

$\langle\psi_{\mathsf{SE}}(\omega_{1})|\psi_{\mathsf{SE}}(\omega_{2})\rangle=\langle\psi_{0}|\exp\left[-iT\left(H_{\omega_{2}}-H_{\omega_{1}}\right)\right]|\psi_{0}\rangle,$

so that Eq. (5) gives the QFI of the unitary case. In particular, this is true
for parallel noise, i.e., for $\hat{c}_{j}=\sqrt{k/2}\sigma_{z}^{(j)}$, and,
for any pure initial state of the system $|\psi_{0}\rangle$, we have:

$\displaystyle\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}^{\mathsf{\parallel}}$
$\displaystyle=\mathcal{Q}\left[e^{-i\hat{H}_{\omega}t}|\psi_{0}\rangle\\!\langle\psi_{0}|e^{i\hat{H}_{\omega}t}\right]$
(25)
$\displaystyle=4\left[\langle\psi_{0}|\left(\partial_{\omega}\hat{H}_{\omega}\right)^{2}|\psi_{0}\rangle-\langle\psi_{0}|\left(\partial_{\omega}\hat{H}_{\omega}\right)|\psi_{0}\rangle^{2}\right]\,.$

###  4.2 Transverse noise

When the collapse operators do not commute with the generator, as in the case
of frequency estimation with transverse noise, we can simplify the computation
by using the assumption of an identical and independent noise acting on each
qubit. Since Eq. (2) is linear in $\bar{\varrho}$ and the coefficients are
time-independent, we can still write the solution as a linear map, formally
$\mathcal{\tilde{E}}_{\omega_{1},\omega_{2}}(t)=\exp\left(t\mathcal{\tilde{L}}_{\omega_{1},\omega_{2}}\right)$.
This map is only guaranteed to be linear and in general it is not even
positive.

Since the map acts independently on every qubit, we can still write the global
action on the $N$-qubit state as the tensor product
$\mathcal{\tilde{E}}^{N}_{\omega_{1},\omega_{2}}(t)=\mathcal{\tilde{E}}_{\omega_{2},\omega_{2}}(t)^{\otimes
N}$, where $\mathcal{\tilde{E}}_{\omega_{2},\omega_{2}}(t)$ is the single-
qubit solution. The ultimate bound is thus obtained as

$\overline{\mathcal{Q}}_{\mathcal{L}_{\omega}}\left[|\psi_{0}\rangle\right]=4\left.\partial_{\omega_{1}}\partial_{\omega_{2}}\log\hbox{Tr}[\mathcal{\tilde{E}}^{N}_{\omega_{1},\omega_{2}}(t)\,\varrho_{0}]\right|_{\omega_{1}=\omega_{2}=\omega},$
(26)

where $\varrho_{0}$ is the initial pure state. Given our choice
$\varrho_{0}=|\psi_{\mathsf{GHZ}}\rangle\\!\langle\psi_{\mathsf{GHZ}}|$, the
computation can be greatly simplified. We find that

$\displaystyle\mathcal{\tilde{E}}^{N}_{\omega_{1},\omega_{2}}(t)\,|\psi_{\mathsf{GHZ}}\rangle\\!\langle\psi_{\mathsf{GHZ}}|=$
(27)
$\displaystyle=\frac{1}{2}\Biggl{[}\left(\mathcal{\tilde{E}}_{\omega_{1},\omega_{2}}(t)\,|0\rangle\\!\langle
0|\right)^{\otimes
N}+\left(\mathcal{\tilde{E}}_{\omega_{1},\omega_{2}}(t)\,|1\rangle\\!\langle
1|\right)^{\otimes N}$
$\displaystyle\quad\;\,+\left(\mathcal{\tilde{E}}_{\omega_{1},\omega_{2}}(t)\,|0\rangle\\!\langle
1|\right)^{\otimes
N}+\left(\mathcal{\tilde{E}}_{\omega_{1},\omega_{2}}(t)\,|1\rangle\\!\langle
0|\right)^{\otimes N}\Biggr{]}.$

For transverse noise and for a single qubit, the equation to solve to compute
$\overline{\mathcal{Q}}$ is the following

$\begin{split}\frac{d\bar{\varrho}}{dt}&=\mathcal{\tilde{L}}^{\bot}_{\omega_{1},\omega_{2}}\left[\bar{\varrho}\right]=\\\
&=-\frac{i}{2}\left(\omega_{1}\sigma_{z}\bar{\varrho}-\omega_{2}\tilde{\varrho}\sigma_{z}\right)+\frac{\kappa}{2}\left(\sigma_{x}\bar{\varrho}\sigma_{x}-\bar{\varrho}\right).\end{split}$
(28)

The solution is obtained in the same way as for canonical master equations: by
choosing a basis of operators and using a matrix representation of
superoperators [63] (see also the Supplementary Material of [42]). By making
use of the normalized Pauli operators $\tilde{\sigma}_{i}=\sigma_{i}/\sqrt{2}$
(where $\sigma_{0}=\mathbbm{1}$), so that
$\hbox{Tr}\left[\tilde{\sigma}_{i}\tilde{\sigma}_{j}\right]=\delta_{ij}$, we
can find the matrix associated to the single qubit map
$\mathcal{\tilde{E}}_{\omega_{1},\omega_{2}}(t)$, obtained by matrix
exponentiation. We can write the generalized density operator $\bar{\varrho}$
in Bloch form as
$\bar{\varrho}=\frac{1}{\sqrt{2}}\left(a_{0}\tilde{\sigma}_{0}+\vec{a}\cdot\vec{\tilde{\sigma}}\right)$
such that its trace is simply $\hbox{Tr}\left[\bar{\varrho}\right]=a_{0}$. In
this notation, the initial states $|0\rangle\\!\langle 0|$ and
$|1\rangle\\!\langle 1|$ correspond to the vectors
$e_{00}=\frac{1}{\sqrt{2}}(1,0,0,1)^{\mathsf{T}}$ and
$e_{11}=\frac{1}{\sqrt{2}}(1,0,0,-1)^{\mathsf{T}}$, while the off-diagonal
elements are $e_{01/10}=\frac{1}{\sqrt{2}}(0,1,\pm i,0)^{\mathsf{T}}$.

Since we are only interested in the coefficient $a_{0}$, we only need the
first row of the matrix representation of
$\mathcal{\tilde{E}}^{\bot}_{\omega_{1},\omega_{2}}(t)$, which turns out to be

$\leavevmode\resizebox{390.25534pt}{}{$\left[\mathcal{\tilde{E}}^{\bot}_{\omega_{2},\omega_{2}}(t)\right]_{0,*}=e^{-\frac{\kappa
t}{2}}\left(\cosh\left(\frac{t}{2}\sqrt{\kappa^{2}-\left(\omega_{1}-\omega_{2}\right)^{2}}\right)+\frac{\kappa\sinh\left(\frac{t}{2}\sqrt{\kappa^{2}-\left(\omega_{1}-\omega_{2}\right)^{2}}\right)}{2\sqrt{\kappa^{2}-\left(\omega_{1}-\omega_{2}\right)^{2}}},0,0,\frac{i\left(\omega_{2}-\omega_{1}\right)\sinh\left(\frac{t}{2}\sqrt{\kappa^{2}-\left(\omega_{1}-\omega_{2}\right)^{2}}\right)}{\sqrt{\kappa^{2}-\left(\omega_{1}-\omega_{2}\right)^{2}}}\right)$}.$
(29)

We can see that the off-diagonal terms are kept off-diagonal by the map,
therefore they do not contribute to the trace and we can further simplify the
calculation as

$\displaystyle\hbox{Tr}\left[\mathcal{\tilde{E}}^{\bot
N}_{\omega_{1},\omega_{2}}(t)|\psi_{\mathsf{GHZ}}\rangle\\!\langle\psi_{\mathsf{GHZ}}|\right]=$
(30)
$\displaystyle=\frac{1}{2}\left(\hbox{Tr}\left[\mathcal{\tilde{E}}^{\bot}_{\omega_{1},\omega_{2}}(t)|0\rangle\\!\langle
0|\right]^{N}+\hbox{Tr}\left[\mathcal{\tilde{E}}^{\bot}_{\omega_{1},\omega_{2}}(t)|1\rangle\\!\langle
1|\right]^{N}\right)$
$\displaystyle=\frac{1}{\sqrt{2}}\Biggl{\\{}\left([\mathcal{\tilde{E}}^{\bot}_{\omega_{1},\omega_{2}}(t)]_{0,*}\cdot
e_{00}\right)^{N}+\left([\mathcal{\tilde{E}}^{\bot}_{\omega_{1},\omega_{2}}(t)]_{0,*}\cdot
e_{11}\right)^{N}\Biggr{\\}}.$

We can thus plug this result into (26) and finally obtain Eq. (19).

##  5 Numerical algorithm for the calculation of the effective QFI for time-
continuous strategies

We can now describe the numerical algorithm we have implemented to calculate
the effective QFI $\widetilde{\mathcal{Q}}_{\mathsf{unr,\eta}}$. The numerical
results presented in this manuscript are obtained with the code available
online at [64], written in the Julia language [65].

We will focus on the case of homodyne detection and then we will discuss the
small changes needed for photodetection. We first review the existing method
to calculate one of the two key quantities in Eq. (9), namely the classical
Fisher information $\mathcal{F}[p_{\mathsf{traj}}]$. In [41] it is shown that
it can be calculated as

$\displaystyle\mathcal{F}[p_{\mathsf{traj}}]=\mathbbm{E}\left[\hbox{Tr}[\tau]^{2}\right],$
(31)

where we have defined the operator

$\displaystyle\tau=\frac{\partial_{\omega}\tilde{\varrho}^{(c)}}{\hbox{Tr}[\tilde{\varrho}^{(c)}]}\>,$
(32)

in terms of the unormalized conditional state $\tilde{\varrho}^{(c)}$ evolving
according to the SME

$\displaystyle d\tilde{\varrho}^{(c)}={}$
$\displaystyle-i[\hat{H}_{\omega},\tilde{\varrho}^{(c)}]\,dt+\sum_{j}\mathcal{D}[\hat{c}_{j}]\tilde{\varrho}^{(c)}\,dt$
$\displaystyle+\sum_{j}\sqrt{\eta}_{j}\left(\hat{c}_{j}\tilde{\varrho}^{(c)}+\tilde{\varrho}^{(c)}\hat{c}_{j}^{\dagger}\right)dy_{j}\,,$
(33)

and such that
$\varrho^{(c)}=\tilde{\varrho}^{(c)}/\hbox{Tr}[\tilde{\varrho}^{(c)}]$.

One can then numerically integrate simultaneously the two SMEs, the one for
the conditional state $\varrho^{(c)}$ in Eq. (8), and the one above for
$\tau$, and then evaluate the corresponding FI, by averaging over a certain
number of trajectories. However, it is computationally very expensive to
obtain a guaranteed Hermitian $\varrho^{(c)}$ by using standard methods, such
as Euler-Maruyama or Euler-Milstein, and still these methods do not guarantee
the positivity of corresponding density operator.

An alternative method to integrate the SME, which is able to circumvent these
problems, has been introduced in [66, 53], by exploiting the Kraus operators
corresponding to the (weak) measurement performed at each instant of time.
After an infinitesimal time $dt$, the evolved state corresponding to (8) can
in fact be written as

$\displaystyle\varrho^{(c)}_{t+dt}=\frac{M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\varrho^{(c)}_{t}M_{\mathbf{dy}}^{\dagger}+\sum_{j}(1-\eta_{j})\hat{c}^{\vphantom{{\dagger}}}_{j}\varrho_{t}^{(c)}\hat{c}_{j}^{\dagger}\,dt}{\hbox{Tr}[M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\varrho^{(c)}_{t}M_{\mathbf{dy}}^{\dagger}+\sum_{j}(1-\eta_{j})\hat{c}^{\vphantom{{\dagger}}}_{j}\varrho_{t}^{(c)}\hat{c}_{j}^{\dagger}\,dt]}\,,$
(34)

where we have explicitly put the time dependence of the density operators and
where we have defined the Kraus operator

$\displaystyle
M_{\mathbf{dy}}^{\vphantom{{\dagger}}}=\mathbbm{1}-i\hat{H}_{\omega}\,dt-\frac{1}{2}\sum_{j}\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}\,dt+\sum_{j}\sqrt{\eta}\hat{c}_{j}\,dy_{j}\,,$
(35)

with $\mathbf{dy}=\\{dy_{j}\\}$ being a vector of measurement results,
corresponding to each output channel,

$dy_{j}=\sqrt{\eta_{j}}\,\hbox{Tr}[\varrho_{t}^{(c)}(\hat{c}^{\vphantom{{\dagger}}}_{j}+\hat{c}_{j}^{\dagger})]\,dt+dw_{j}.$
(36)

Equation (34) can be used for numerical purposes, where the infinitesimal time
$dt$ is replaced by a finite time step $\Delta t$, the Wiener increments
$dw_{j}$ are replaced by Gaussian random variables $\Delta w_{j}$ centered in
zero and with variance equal to $\Delta t$, and where one can also implement
the second order Euler-Milstein corrections. The (numerical) Kraus operators
in this case read

$\displaystyle M_{\mathbf{\Delta y}}={}$
$\displaystyle\mathbbm{1}-i\hat{H}_{\omega}\,\Delta
t-\frac{1}{2}\sum_{j}\hat{c}_{j}^{\dagger}\hat{c}^{\vphantom{{\dagger}}}_{j}\,\Delta
t+\sum_{j}\sqrt{\eta_{j}}\,\hat{c}_{j}\,\Delta y_{j}$
$\displaystyle+\sum_{j,k}\frac{\eta}{2}\hat{c}_{j}\hat{c}_{k}(\Delta
y_{j}\Delta y_{k}-\delta_{j,k}\Delta t)\,,$ (37)

with

$\displaystyle\Delta
y_{j}=\sqrt{\eta_{j}}\,\hbox{Tr}[\varrho_{t}^{(c)}(\hat{c}^{\dagger}_{j}+\hat{c}_{j}^{\vphantom{{\dagger}}})]\,\Delta
t+\Delta w_{j}\,,$ (38)

denoting the (finite) increments of the measurement records.

In the following we will show how to extend this method to obtain an efficient
and numerically stable calculation of both $\mathcal{F}[p_{\mathsf{traj}}]$
and $\mathcal{Q}[\varrho_{c}]$. We will prove everything in terms of the
“infinitesimal” Kraus operators, which will have to be replaced by Eq. (37)
when implementing the numerical algorithm. We will start by showing the
results for the most general case of SME and inefficient detection; we will
then describe the more efficient algorithm that can be implemented in the case
of perfect detection ($\eta_{j}=1$), i.e., when the dynamics can be described
by a stochastic Schrödinger equation.

###  5.1 Non-unit efficiency detection (stochastic master equation)

We start by observing that the evolution of the unnormalized conditional state
and of its derivative can be written in terms of the Kraus operators (in what
follows we will omit the superscript $(c)$ used to denote conditional states):

$\displaystyle\tilde{\varrho}_{t+dt}={}$ $\displaystyle
M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\tilde{\varrho}_{t}M_{\mathbf{dy}}^{\dagger}+\sum_{j}(1-\eta_{j})\hat{c}_{j}^{\vphantom{{\dagger}}}\tilde{\varrho}_{t}\hat{c}_{j}^{\dagger}\,dt\>,$
$\displaystyle\partial_{\omega}\tilde{\varrho}_{t+dt}={}$ $\displaystyle
M_{\mathbf{dy}}^{\vphantom{{\dagger}}}(\partial_{\omega}\tilde{\varrho}_{t})M_{\mathbf{dy}}^{\dagger}+(\partial_{\omega}M_{\mathbf{dy}}^{\vphantom{{\dagger}}})\tilde{\varrho}_{t}M_{\mathbf{dy}}^{\dagger}+$
$\displaystyle+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\tilde{\varrho}_{t}(\partial_{\omega}M_{\mathbf{dy}}^{\dagger})+$
$\displaystyle+\sum_{j}(1-\eta_{j})\hat{c}_{j}^{\vphantom{{\dagger}}}(\partial_{\omega}\tilde{\varrho}_{t})\hat{c}_{j}^{\dagger}\,dt\>,$
(39)

where
$\partial_{\omega}M_{\mathbf{dy}}^{\vphantom{{\dagger}}}=-i(\partial_{\omega}\hat{H}_{\omega})\,dt$.
The trace of the unnormalized state reads

$\displaystyle\hbox{Tr}[\tilde{\varrho}_{t+dt}]$
$\displaystyle=\hbox{Tr}[M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\tilde{\varrho}_{t}M_{\mathbf{dy}}^{\dagger}+\sum_{j}(1-\eta_{j})\hat{c}_{j}^{\vphantom{{\dagger}}}\tilde{\varrho}_{t}\hat{c}_{j}^{\dagger}\,dt]$
$\displaystyle=\hbox{Tr}[\tilde{\varrho}_{t}]\hbox{Tr}[M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\varrho_{t}M_{\mathbf{dy}}^{\dagger}+\sum_{j}(1-\eta_{j})\hat{c}_{j}^{\vphantom{{\dagger}}}{\varrho}_{t}\hat{c}_{j}^{\dagger}\,dt]\,,$
(40)

where we have used the relation
$\varrho_{t}=\tilde{\varrho}_{t}/\hbox{Tr}[\tilde{\varrho}_{t}]$. We can now
use these formulas to obtain the evolution for the operator $\tau_{t+dt}$ just
in terms of the operators $\varrho_{t}$ and $\tau_{t}$ at the previous time
step:

$\displaystyle\tau_{t+dt}={}$
$\displaystyle\frac{1}{\hbox{Tr}[\tilde{\varrho}_{t+dt}]}\left[(\partial_{\omega}M_{\mathbf{dy}}^{\vphantom{{\dagger}}})\tilde{\varrho}_{t}M_{\mathbf{dy}}^{\dagger}+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}(\partial_{\omega}\tilde{\varrho}_{t})M_{\mathbf{dy}}^{\dagger}+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\tilde{\varrho}_{t}(\partial_{\omega}M_{\mathbf{dy}}^{\dagger})+\sum_{j}(1-\eta_{j})\hat{c}^{\vphantom{{\dagger}}}_{j}(\partial_{\omega}\tilde{\varrho}_{t})\hat{c}_{j}^{\dagger}\,dt\right]$
$\displaystyle={}$
$\displaystyle\frac{(\partial_{\omega}M_{\mathbf{dy}}^{\vphantom{{\dagger}}})\varrho_{t}M_{\mathbf{dy}}^{\dagger}+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\tau_{t}M_{\mathbf{dy}}^{\dagger}+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\varrho_{t}(\partial_{\omega}M_{\mathbf{dy}}^{\dagger})+\sum_{j}(1-\eta_{j})\hat{c}^{\vphantom{{\dagger}}}_{j}\tau_{t}\hat{c}_{j}^{\dagger}\,dt}{\hbox{Tr}\left[M_{\mathbf{dy}}^{\vphantom{{\dagger}}}\varrho_{t}M_{\mathbf{dy}}^{\dagger}+\sum_{j}(1-\eta_{j})\hat{c}^{\vphantom{{\dagger}}}_{j}{\varrho}_{t}\hat{c}_{j}^{\dagger}\,dt\right]}\,.$
(41)

One can thus evaluate the trace of this operator at each time $t$, and
evaluate accordingly the classical Fisher information
$\mathcal{F}[p_{\mathsf{traj}}]$ as in Eq. (31).

Notice that the evolution for the derivative operator
$\partial_{\omega}\varrho_{t}$ can be now written in terms of the renormalized
operators $\varrho_{t}$ and $\tau_{t}$:

$\displaystyle\partial_{\omega}\varrho_{t}$
$\displaystyle=\frac{\partial_{\omega}\tilde{\varrho}_{t}}{\hbox{Tr}[\tilde{\varrho}_{t}]}-\frac{\hbox{Tr}[\partial_{\omega}\tilde{\varrho}_{t}]}{\hbox{Tr}[\tilde{\varrho}_{t}]^{2}}\tilde{\varrho}_{t}=\tau_{t}-\hbox{Tr}[\tau_{t}]\varrho_{t}\,.$
(42)

The QFI $\mathcal{Q}[\varrho_{t}]$ can then be evaluated at each time $t$ by
using the formula [67]:

$\displaystyle\mathcal{Q}[\varrho_{t}]=2\sum_{\lambda_{s}+\lambda_{t}\neq
0}\frac{|\langle\psi_{s}|\partial_{\omega}\varrho_{t}|\psi_{t}\rangle|^{2}}{\lambda_{s}+\lambda_{t}}\,,$
(43)

upon writing $\varrho_{t}$ in its eigenbasis, i.e.,
$\varrho_{t}=\sum_{s}\lambda_{s}|\psi_{s}\rangle\\!\langle\psi_{s}|\,.$

We would like to underline the key features of our algorithm. The relevant
figures of merit could naively be derived from the evolution of the
unnormalized conditional state $\tilde{\varrho}_{t}$, described in Eq. (39).
However $\hbox{Tr}[\tilde{\varrho}_{t}]$ becomes very small during the
evolution, leading to numerical instabilities in the evaluation of both
$\mathcal{F}[p_{\mathsf{traj}}]$ and $\mathcal{Q}[\varrho_{t}]$. Thanks to
Eqs. (41) and (42), we are able to express the above quantities only in terms
of the numerically stable operators $\varrho_{t}$ and $\tau_{t}$. Besides, the
formulation in terms of Kraus operators, following [66, 53], ensures that the
density operator remains positive, as opposed to standard numerical
integration of the SME.

###  5.2 Unit efficiency detection (stochastic Schrödinger equation)

The above calculations are greatly simplified when the dynamics starts with a
pure initial state and the efficiency parameters are equal to one,
$\eta_{j}=1$. In fact the quantum conditional state $|\psi_{t}\rangle$ remains
pure during the whole evolution and the dynamics is described by a stochastic
Schrödinger equation. We can thus work with state vectors, instead of density
matrices, with a consequent reduction of complexity of the numerical
simulation, which allows us to reach higher values of $N$ with a given amount
of memory.

In terms of Kraus operators, the unnormalized and normalized conditional
states are obtained respectively as:

$\displaystyle|\widetilde{\psi}_{t+dt}\rangle$
$\displaystyle=M_{\mathbf{dy}}^{\vphantom{{\dagger}}}|\widetilde{\psi}_{t}\rangle\,,$
(44) $\displaystyle|\psi_{t+dt}\rangle$
$\displaystyle=\frac{M_{\mathbf{dy}}^{\vphantom{{\dagger}}}|\psi_{t}\rangle}{\sqrt{\langle\psi_{t}|M_{\mathbf{dy}}^{\dagger}M^{{\vphantom{{\dagger}}}}_{\mathbf{dy}}|\psi_{t}\rangle}}=\frac{M_{\mathbf{dy}}^{\vphantom{{\dagger}}}|\widetilde{\psi}_{t}\rangle}{\sqrt{\langle\widetilde{\psi}_{t}|M_{\mathbf{dy}}^{\dagger}M^{{\vphantom{{\dagger}}}}_{\mathbf{dy}}|\widetilde{\psi}_{t}\rangle}}\,.$

The operator $\tau_{t}$ in this case can be written as

$\displaystyle\tau_{t}$
$\displaystyle=\frac{\partial_{\omega}(|\widetilde{\psi}_{t}\rangle\\!\langle\widetilde{\psi}_{t}|)}{\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle}=\frac{|\partial_{\omega}\widetilde{\psi}_{t}\rangle\\!\langle\widetilde{\psi}_{t}|+|\widetilde{\psi}_{t}\rangle\\!\langle\partial_{\omega}\widetilde{\psi}_{t}|}{\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle}\,,$
(45)

and its trace is equal to

$\displaystyle\hbox{Tr}[\tau_{t}]$
$\displaystyle=\frac{\langle\widetilde{\psi}_{t}|\partial_{\omega}\widetilde{\psi}_{t}\rangle+\hbox{h.c.}}{\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle}=\langle\psi_{t}|\phi_{t}\rangle+\langle\phi_{t}|\psi_{t}\rangle\,.$
(46)

In the last equation we have introduced the vector

$\displaystyle|\phi_{t}\rangle=\frac{|\partial_{\omega}\widetilde{\psi}_{t}\rangle}{\sqrt{\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle}}\,.$
(47)

At time $t+dt$, the vector $|\phi_{t+dt}\rangle$ can be obtained as

$\displaystyle|\phi_{t+dt}\rangle$
$\displaystyle=\frac{(\partial_{\omega}M_{\mathbf{dy}}^{\vphantom{{\dagger}}})|\widetilde{\psi}_{t}\rangle+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}|\partial_{\omega}\widetilde{\psi}_{t}\rangle}{\sqrt{\langle\widetilde{\psi}_{t}|M_{\mathbf{dy}}^{\dagger}M^{{\vphantom{{\dagger}}}}_{\mathbf{dy}}|\widetilde{\psi}_{t}\rangle}}$
$\displaystyle=\frac{(\partial_{\omega}M_{\mathbf{dy}}^{\vphantom{{\dagger}}})|\psi_{t}\rangle+M_{\mathbf{dy}}^{\vphantom{{\dagger}}}|\phi_{t}\rangle}{\sqrt{\langle\psi_{t}|M_{\mathbf{dy}}^{\dagger}M^{{\vphantom{{\dagger}}}}_{\mathbf{dy}}|\psi_{t}\rangle}}\,,$
(48)

where we have exploited the identity

$\langle\widetilde{\psi}_{t}|M_{\mathbf{dy}}^{\dagger}M^{{\vphantom{{\dagger}}}}_{\mathbf{dy}}|\widetilde{\psi}_{t}\rangle=\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle\\!\langle\psi_{t}|M_{\mathbf{dy}}^{\dagger}M^{{\vphantom{{\dagger}}}}_{\mathbf{dy}}|\psi_{t}\rangle\,.$
(49)

We notice that as in Eq. (41), the evolution equation for the vector
$|\phi_{t+dt}\rangle$ depends only on the vectors $|\psi_{t}\rangle$ and
$|\phi_{t}\rangle$ and not on the unnormalized state
$|\widetilde{\psi}_{t}\rangle$, and one can readily evaluate the classical
Fisher information in terms of these two vectors via Eqs. (46) and (31).

Regarding the QFI of the conditional state, we first observe that

$\displaystyle|\partial_{\omega}\psi_{t}\rangle$
$\displaystyle=\frac{|\partial_{\omega}\widetilde{\psi}_{t}\rangle}{\sqrt{\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle}}-\frac{\langle\widetilde{\psi}_{t}|\partial_{\omega}\widetilde{\psi}_{t}\rangle+\langle\partial_{\omega}\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle|\widetilde{\psi}_{t}\rangle}{2\langle\widetilde{\psi}_{t}|\widetilde{\psi}_{t}\rangle^{3/2}}$
$\displaystyle=|\phi_{t}\rangle-\frac{\langle\psi_{t}|\phi_{t}\rangle+\langle\phi_{t}|\psi_{t}\rangle}{2}|\psi_{t}\rangle\,.$
(50)

As we are dealing with pure states, the QFI of the conditional state can be
simply evaluated as [67]

$\displaystyle\mathcal{Q}[|\psi_{t}\rangle]=4\left[\langle\partial_{\omega}\psi_{t}|\partial_{\omega}\psi_{t}\rangle+(\langle\partial_{\omega}\psi_{t}|\psi_{t}\rangle)^{2}\right]\,.$
(51)

The algorithms above have been derived for the case of time-continuous
homodyne detection. Nonetheless one can easily extend them to time-continuous
photodetection as follows.

The Kraus operators $M_{\mathbf{dy}}^{\vphantom{{\dagger}}}$ have to be
replaced at each time step with one of the following Kraus operators
corresponding to “no detector click” and “detector click” (in one of the
output channels) events, respectively

$\displaystyle M_{0}$
$\displaystyle=\mathbbm{1}-i\hat{H}_{\omega}\,dt-\frac{1}{2}\sum_{j}\hat{c}_{j}^{{\dagger}}\hat{c}^{{\vphantom{{\dagger}}}}_{j}\,dt\,,$
(52) $\displaystyle M_{1}^{(j)}$
$\displaystyle=\sqrt{\eta_{j}\,dt}\,\hat{c}_{j}\,.$ (53)

At each time step, each Kraus operator $M_{1}^{(j)}$ has to be applied with
probabilities
$p_{1}^{(j)}=\eta_{j}\hbox{Tr}[\varrho_{t}\hat{c}_{j}^{\dagger}\hat{c}^{{\vphantom{{\dagger}}}}_{j}]\,dt$
and, correspondingly, the Kraus operator $M_{0}$ has to be applied with
probability $p_{0}=1-\sum_{j}p_{1}^{(j)}$ [27].

We finally remark that in the case of frequency estimation with parallel noise
and initial GHZ state, the numerical calculations are incredibly simplified
thanks to the symmetry of the dynamics. In fact the whole (both conditional
and unconditional) evolution is equivalently described in two two-dimensional
Hilbert space spanned by the vectors $|\bar{0}\rangle=|0\rangle^{\otimes N}$
and $|\bar{1}\rangle=|1\rangle^{\otimes N}$. As regards continuous
photodetection, the Kraus operators (52) and (53) are replaced by the
effective Kraus operators

$\displaystyle\overline{M}_{0}$
$\displaystyle=\mathbbm{1}_{2}-iN\omega\overline{\sigma}_{z}\,dt-(N\kappa/4)\mathbbm{1}_{2}\,dt\,,$
(54) $\displaystyle\overline{M}_{1}$
$\displaystyle=\sqrt{(\eta\kappa/2)\,dt}\;N\overline{\sigma}_{z}\,.$ (55)

where $\mathbbm{1}_{2}$ and $\overline{\sigma}_{z}$ are respectively the
identity operator and the Pauli-z matrix in the Hilbert space spanned by
$|\bar{0}\rangle$ and $|\bar{1}\rangle$, and where the operators
$\overline{M}_{1}$ and $\overline{M}_{0}$ have to be applied respectively with
probabilities $\bar{p}_{1}=N(\eta\kappa/2)\,dt$ (i.e., the total probability
of observing a photon in one of the $N$ output channels) and
$\bar{p}_{0}=1-\bar{p}_{1}$. More practically the results for generic $N$
qubits can be readily obtained by exploiting the results obtained for one
qubit, and rescaling the time as $t\rightarrow Nt$. In this case it is also
easy to show that the efficiency parameter $\eta$ corresponds to the product
between the factual efficiency of the photo-detectors (that we assume to be
equal) and the fraction of qubits that are effectively monitored.

##  6 Conclusions and remarks

We have discussed for the first time the usefulness of time-continuous
monitoring in the context of noisy quantum metrology. We have proven that the
desired Heisenberg scaling can in fact be restored by exploiting these schemes
and we have obtained several conceptually and practically relevant results.
We have shown the fundamental difference between parallel and transverse noise
with regards to the ultimate limit achievable when the environmental degrees
of freedom can be measured. In the first case, i.e., when the Hamiltonian and
the noise generator commute, having access to every degree of freedom of the
environment allows us to restore the unitary (noiseless) Heisenberg limit. In
the latter case, i.e., in the presence of transverse noise, the ultimate QFI
still presents a Heisenberg (quadratic) scaling in the number of qubits $N$;
however, it is always lower than the unitary QFI, thus showing that some
information is irremediably lost due to the interaction with the environment.
These findings complement the results obtained for frequency estimation with
open quantum systems [68], where the geometry of the noise is indeed crucial
in determining the different achievable scalings.

The second non-trivial and conceptually relevant result is that estimation
strategies based on (sequential) time-continuous monitoring and final strong
measurements on the system are optimal, i.e., the corresponding effective QFIs
$\widetilde{\mathcal{Q}}_{\mathsf{unr},\eta_{j}}$ are equal to the ultimate
QFIs $\overline{\mathcal{Q}}_{\mathcal{L_{\omega}}}$. This is true for both
parallel and transverse noise, showing that no complicated estimation
strategies based on “entangled measurements” on the environment and system are
needed to achieve the ultimate limit on the precision. This result, which was
not suggested by any mathematical or physical intuition, is particularly
important from a practical point of view, as it greatly relaxes the
assumptions and demands that are requested to achieve the ultimate limits.

We have also discussed in detail the case of finite efficiency monitoring. In
the presence of parallel noise we have shown that the estimation precision is
subject to the standard quantum limit, with a behaviour ruled by a reduced
effective dephasing: one can still obtain a constant enhancement, which can be
made arbitrarily high by increasing the monitoring efficiency. In the presence
of transverse noise, we have observed the expected enhancement with respect to
the optimized (super-classical) unconditional results [15]. However, due to
the computational complexity of the algorithm needed to obtain the effective
QFI, we could not infer any conclusion regarding the scaling with the number
of probes $N$.

It is important to remark that the use of continuous measurements and feedback
techniques has long been recognized as a useful tool for fighting decoherence
and to prepare non-classical quantum states [27, 69, 70, 71, 72, 73, 74, 75].
Our metrological scheme follows this line of thought, but with the great
advantage that it is based on continuous monitoring only, without error
correction steps (or feedback), differing it from other recent approaches in
noisy quantum metrology [21, 22].

Moreover, while most of the literature on parameter estimation with continuous
measurements focuses on the information gained from the continuous signal, the
crucial part that makes our protocol able to recover Heisenberg scaling is the
final strong measurement on the conditional state. A great deal of effort has
been also devoted to studying the asymptotic properties of estimation via
repeated/continuous measurements [76, 77, 78]. In this approach one is usually
interested in performing a single run of the experiment and thus observes the
system for a long time. Our point of view is radically different and rooted in
previous works on quantum frequency estimation. As a matter of fact, we are
interested in the initial part of the dynamics, long before reaching the
steady state. For this reason the asymptotic regime of the statistical model
is reached by running the experiment many times, instead of observing the
system for a long time.

In order to obtain these results we have developed a stable algorithm for the
evaluation of the effective quantum Fisher information that quantifies the
performance of the proposed strategies. The flexibility of this algorithm and
the originality of our protocols pave the way for a series of future
investigations that will exploit techniques and ideas that so far have only
been applied to standard noisy metrology schemes. For example, one can
investigate the performances of these protocols with optimized [79] and noisy
quantum probes [14], or by considering entangled ancillary systems [80, 81,
82]. Furthermore, one can also study in detail the role played by quantum
coherence [83] and the possibility of considering unravellings of non-
Markovian master equations [84].

The authors acknowledge useful discussions with A. Del Campo, M. Paris, P.
Rouchon, A. Smirne and T. Tufarelli. MACR was supported by the Horizon 2020 EU
collaborative project QuProCS (Grant Agreement No. 641277) and by the Academy
of Finland Centre of Excellence program (project 312058). MGG acknowledges
support from Marie Skodowska-Curie Action H2020-MSCA-IF-2015 (project
ConAQuMe, grant no. 701154) and from a Rita Levi-Montalcini fellowship of
MIUR.
_FA and MACR have equally contributed to this work._

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