# When does dissipative evolution preserve and amplify metrological
information?

Jing Yang  Nordita, KTH Royal Institute of Technology and Stockholm
University, Hannes Alfvéns vag 12, 106 91 Stockholm, Sweden

###### Abstract

We study quantum metrology with post-selection measurement encoding, which
includes post-selected non-Hermitian sensing as a special case. As in the weak
value amplification, the average metrological precision over the post-
selection probability does not increase compared to the post-selection free
unitary encoding. Nevertheless, we identify the conditions under which
significant amplification effects can be achieved without the loss of the
average precision. These conditions can help design quantum measurements as a
lossless information transducer. When these conditions are violated in post-
selected non-Hermitian sensing, we obtain a universal formula for the loss of
the average precision. Hence our findings offer a unified metrological
framework to clarify the pros and cons of non-Hermitian sensing.

Introduction.— Physics is all about observations. Quantum metrology and
sensing is the subject of reducing noise in experimental observations [1, 2,
3, 4, 5, 6] by utilizing quantum resources, such as coherences [7],
entanglement [8, 9] or many-body interactions [10, 11, 12, 13, 14, 15]. In
many scenarios, the information is encoded in the quantum sensor by a unitary
operation that depends on the estimation parameter. The precision for such a
unitary encoding under the environmental decoherence has been widely studied
[16, 17, 18, 19, 20, 9, 21, 12, 22, 23, 24, 25, 26, 27]. According to
Uhlmann’s theorem [28, 29], it is expected that the precision cannot beat the
decoherence-free case. On the other hand, another metrological scheme, non-
unitary encoding through parameter-dependent quantum measurements with post-
selections is less explored. Note that non-unitary encoding through a post-
selection measurement channel, while related to unitary encoding under
decoherence, is fundamentally different. In the former case, the experimenter
has access to the measurement outcomes, whereas in the latter case, they do
not.

Recently, numerous progress has been made on non-Hermitian physics in both
classical and quantum systems [30, 31, 32, 33, 34]. In particular, sensing
with non-Hermitian physics has attracted a lot of attention [35, 36, 37, 38,
39, 40, 41, 42, 43] as well as debates [44, 45]. On one hand, complex level
splitting due to small perturbations near the non-Hermitian exceptional points
are amplified compared to the Hermitian diabolic point. On the other hand, it
was shown that the scaling of measurable response with the small perturbation
is the same for both exceptional and diabolic points [44]. Furthermore, with
the error propagation formula, Ref. [46] demonstrated that for a single-qubit
pseudo-Hermitian and exceptional-point based sensors, their respective
precision cannot outperform their Hermitian counterparts.

In this work, we develop a framework for quantum sensing with post-selection
encoding. Post-selection was originally introduced in weak value amplification
[47, 48] and recently further generalized in quantum metrology [49, 50, 51,
52]. It is worth noting that the non-Hermitian dynamics of quantum systems
also involves the post-selection of the non-jump evolution in the Markovian
GKSL (Gorini–Kossakowski–Sudarshan–Lindblad) master equation [53, 30]. Such a
connection enables us to address the confusion in the current literature on
non-Hermitian sensing and identify the advantages and disadvantages: Indeed,
the large sensitivity due to post-selection may be counter-balanced by the
small success probability and the overall average precision cannot increase,
as in the weak-value amplification [54, 55, 56, 57]. Nevertheless, post-
selection can bring the benefits of reducing the number of measurements hence
may lower the measurement cost and technical noise. On the fundamental aspect,
we give the necessary and sufficient conditions that preserve the precision,
quantified by the Quantum Fisher Information (QFI) averaged over the post-
selection possibility, while amplifying the sensitivity of the post-selected
states and compress all the precision to a small number of samples thanks to
the very low success probability. When these conditions are violated in post-
selected non-Hermitian sensing, we give a universal formula of estimating the
loss of the average precision.

Refined post-selection and convexity inequalities.— Throughout this work,
operators with superscripts “SE” or “E” indicate they act on the system and
environment jointly or only on the environment, respectively. Operators with
no superscripts indicate that they act only on the primary system. We consider
an initial parameter-independent pure state of the system
$\varrho_{i}\\!=\\!\ket{\psi_{i}}\bra{\psi_{i}}$ that undergoes a generic
parameter-dependent quantum measurement channel described by
$\\{M_{\omega,\,x}\\}_{\omega\in\Omega}$ [58, 29], where $\Omega$ denotes the
set of all measurement outcomes. Upon purification [29], one can dilate the
evolution into a unitary channel, $U_{x}^{\text{SE}}$ that acts on both the
system and the environment where
$M_{\omega,\,x}\\!\equiv\\!\braket{\pi_{\omega}^{\text{E}}\big{missing}}{U_{x}^{\text{SE}}\big{missing}}{\phi_{i}^{\text{E}}}.$
Two scenarios were compared in the previous literature [16, 17, 18, 19, 20, 9,
21, 59, 12, 22, 23, 24, 25, 26, 27]: (i) The decoherence-free scenario—The
experimenter has access to both the system and the environment so that they
can be regarded as a whole closed system. The joint state is described by
$\ket{\Psi_{x}^{\text{SE}}}\\!\equiv\\!U_{x}^{\text{SE}}\ket{\psi_{i}}\otimes\ket{\phi_{i}^{\text{E}}}$.
(ii) Decoherent scenario—Only the system’s degree of freedom is accessible.
Then the decoherence effects induced by the environment are characterized by
the set of Kraus operators $\\{M_{\omega,\,x}\\}_{\omega\in\Omega}$. The state
after decoherence becomes
$\rho_{x}\\!=\\!\sum_{\omega\in\Omega}M_{\omega,\,x}\varrho_{i}M_{\omega,\,x}^{\dagger}\\!=\\!\sum_{\omega\in\Omega}p(\omega|x)\sigma_{x|\omega}$,
where
$p(\omega|x)\\!=\\!\text{Tr}(M_{\omega,\,x}^{\dagger}M_{\omega,\,x}\varrho_{i})$,
$\sigma_{x|\omega}\\!=\\!\tilde{\sigma}_{x|\omega}/p(\omega|x)$,
$\tilde{\sigma}_{x|\omega}\\!\equiv\\!M_{\omega,\,x}\varrho_{i}M_{\omega,\,x}^{\dagger}$.

As is clear, scenario (ii) is equivalent to the case where quantum
measurements are performed but the outcomes are not accessible. Environmental
decoherence leads to a mixture of the post-measurement states. Yet, a third
scenario was largely ignored previously: (iii) The measurement encoding
scenario—A given quantum measurement specified by $\\{M_{\omega,\,x}\\}$ are
performed with the outcomes accessible, which is the main focus of this work.
In addition, one may further post-select the measurement outcomes to a
specific set denoted as $\checkmark$.

The QFIs associated with each scenarios are denoted as
$I(\ket{\Psi_{x}^{\text{SE}}})$, $I(\rho_{x})$ and $I(\sigma_{x|\omega})$. We
shall study the fundamental inequalities between these QFIs and their
saturation conditions, which play an important role in tracking the QFI in the
dissipative evolution. To begin with, let us note after the quantum
measurement specified by $\\{M_{\omega,\,x}\\}$, the state becomes
$\sigma_{x}^{\text{SE}}\\!=\\!\sum_{\omega\in\Omega}p(\omega|x)\sigma_{x|\omega}\otimes\ket{\pi_{\omega}^{\text{E}}}\bra{\pi_{\omega}^{\text{E}}}$.
The QFI associated with $\sigma_{x}^{\text{SE}}$ is
$I(\sigma_{x}^{\text{SE}})\\!=\\!\sum_{\omega\in\Omega}I_{\omega}(\sigma_{x}^{\text{SE}})$,
where
$I_{\omega}(\sigma_{x}^{\text{SE}})\\!\equiv\\!p(\omega|x)I(\sigma_{x|\omega})\\!+\\!I_{\omega}^{\text{cl}}\left(p(\omega|x)\right)$
and
$I_{\text{$\omega$}}^{\text{cl}}\left(p(\omega\big{|}x)\right)\\!\equiv\\!\left[\partial_{x}p(\omega\big{|}x)\right]^{2}/p(\omega\big{|}x)$
[56, 49]. Since $\sigma_{x}^{\text{SE}}$ is connected to
$\ket{\Psi_{x}^{\text{SE}}}$ through the parameter-independent projective
measurement $\\{\ket{\pi_{\omega}^{\text{E}}}\\}$ acting on the environment,
intuitively the QFI cannot increase. As a result, we obtain the following
inequality regarding the post-selection,

$I^{Q}\equiv I(\ket{\Psi_{x}^{\text{SE}}})\geq I(\sigma_{x}^{\text{SE}}).$ (1)

A rigorous justification of Eq. (1) can be found in Ref. [49]. On the other
hand, the convexity of the QFI [60, 61, 62] dictates that

$I(\sigma_{x}^{\text{SE}})\geq I(\rho_{x}).$ (2)

A few comments in order: Combing Eq. (1) and Eq. (2), we recover the Uhlmann
inequality [28, 29], $I(\ket{\Psi_{x}^{\text{SE}}})\\!\geq\\!I(\rho_{x}).$
Ref. [16, 17] employs this inequality to derive the bounds for dissipative
sensing by optimizing over all possible Kraus operators. Nevertheless,
saturating the Uhlmann inequality is directly very challenging. As manifested
from the above expression for $I(\sigma_{x}^{\text{SE}})$, further post-
selecting the states $\sigma_{x|\omega}$ cannot increase the QFI, i.e.,
$I(\sigma_{x}^{\text{SE}})\\!\geq\\!\sum_{\omega\in\checkmark}p(\omega|x)I(\sigma_{x|\omega})$.
Combining with inequality (1), we obtain

$I^{Q}\geq\sum_{\omega\in\checkmark}p(\omega|x)I(\sigma_{x|\omega}),$ (3)

which clearly indicates that the average QFI over the post-selection
probability cannot outperform the one of post-selection free unitary encoding.
Ref. [46] analyzed the average precision for non-Hermitian sensing through
error propagation formula. Here, we would like to emphasize that their claim
naturally follows from the inequality (3), which holds for generic post-
selection measurement encoding even beyond post-selected non-Hermitian
sensing.

However, there are several fundamental issues behind inequalities (2) and (3),
which can lead to interesting yet uncharted physical consequences. The
saturation of (2) and (3) indicates the existence of dissipative and post-
selected non-unitary evolutions that are “dissipationless” in the metrological
information. In particular, the saturation of Eq. (3) can illuminate the
(dis)advantages of non-Hermitian sensing. Fundamentally, the saturation of Eq.
(3) further generalizes the theory of optimal compression channels proposed in
Ref. [49]. In Ref. [49], the post-selection channel is assumed to be
parameter-independent, while in the current case, we consider a parameter-
dependent post-selection channel. To this end, we start with a refined version
of the post-selection inequality (1) [63],

$I_{\omega}(\ket{\Psi_{x}^{\text{SE}}})\geq
I_{\omega}(\sigma_{x}^{\text{SE}}),\,\forall\omega,$ (4)

where $I_{\omega}(\ket{\Psi_{x}^{\text{SE}}})$ and
$I_{\omega}(\sigma_{x}^{\text{SE}})$ physically denote the QFIs associated
with the particular measurement operator $M_{\omega,\,x}$ before and after
post-selection, respectively. More specifically,
$I_{\omega}(\ket{\Psi_{x}^{\text{SE}}})\\!\equiv\\!\braket{\partial_{x}^{\perp}\Psi_{x}^{\text{SE}}}{\left(\mathbb{I}\otimes\ket{\pi_{\omega}^{\text{E}}}\bra{\pi_{\omega}^{\text{E}}}\right)\partial_{x}^{\perp}\Psi_{x}^{\text{SE}}}$
and $\ket{\partial_{x}^{\perp}\Psi_{x}^{\text{}}}$ is defined as [64, 65, 66,
67]$\ket{\partial_{x}^{\perp}\Psi_{x}^{\text{SE}}}\\!\equiv\\!\ket{\partial_{x}\Psi_{x}^{\text{SE}}}\\!-\\!\ket{\Psi_{x}^{\text{SE}}}\braket{\Psi_{x}\big{missing}}{\partial_{x}\Psi_{x}^{\text{SE}}}.$
Summing over all $\omega$ in Eq. (4) leads to Eq. (1). Similarly, a refined
convexity inequality can be also established
[63]$J_{\mu}(\sigma_{x}^{\text{SE}})\geq J_{\mu}(\rho_{x}),$where
$J_{\mu}(\sigma_{x}^{\text{SE}})\\!\equiv\\!\text{Tr}\left(\sum_{\omega}\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right)$,
$J_{\mu}(\rho_{x})\\!\equiv\\!\text{Tr}(\rho_{x}L_{x}E_{\mu}L_{x})$, and
$\\{E_{\mu}\\}$ is the set of positive operator-value measure (POVM) operators
acting on the system. $L_{x}$ and $\tilde{L}_{x|\omega}$ are the symmetric
logarithmic derivative operator [68] associated with the $\rho_{x}$ and
$\tilde{\sigma}_{x|\omega}$, respectively, defined as
$\partial_{x}\rho_{x}\\!\equiv\\!(L_{x}\rho_{x}+\rho_{x}L_{x})/2$ and
$\partial_{x}\tilde{\sigma}_{x|\omega}\\!\equiv\\!\left(\tilde{L}_{x|\omega}\tilde{\sigma}_{x|\omega}+\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}\right)/2$.
In particular, $\tilde{L}_{x|\omega}\\!=\\!\partial_{x}\ln
p(\omega|x)+2\partial_{x}\sigma_{x|\omega}$. Again, summing over all $\mu$
leads to Eq. (2).

Lossless measurement encoding.— While the saturation of the refined convexity
is left for the future, we shall focus on the saturation of post-selected
inequality (3), which leads to no loss in the measurement encoding. It is
straightforward to check that under the $U(1)$ parameter-dependent gauge
transformation $U_{x}^{\text{SE}}\to e^{\text{i}\theta_{x}}U_{x}^{\text{SE}}$,
$I(\ket{\Psi_{x}^{\text{SE}}})$, $I(\sigma_{x}^{\text{SE}})$ and $I(\rho_{x})$
do not change. Such a gauge transformation leads to $M_{\omega,\,x}\to
e^{\text{i}\theta_{x}}M_{\omega,\,x}$. For fixed initial state
$\ket{\psi_{i}}$ and $\ket{\phi_{i}^{\text{E}}}$, we can alway choose to work
in the perpendicular gauge such that
$\langle\\!\langle\partial_{x}U_{x}^{\dagger\text{SE}}U_{x}^{\text{SE}}\rangle\\!\rangle=\langle\\!\langle
U_{x}^{\dagger\text{SE}}\partial_{x}U_{x}^{\text{SE}}\rangle\\!\rangle=0$,
where $\langle\\!\langle\bullet\rangle\\!\rangle$ denotes the average over the
joint system-environment state
$\ket{\psi_{i}}\otimes\ket{\phi_{i}^{\text{E}}}$. In this gauge,
$\ket{\partial_{x}\Psi_{x}^{\text{SE}}}=\ket{\partial_{x}^{\perp}\Psi_{x}^{\text{SE}}}$.
Using the generalized optimal measurement condition proposed in Ref. [49], the
saturation condition of Eq. (3) can be found, which is our first main result
[63]:

######  Theorem 1.

In the perpendicular $U(1)$ gauge, measurement encoding is lossless if

$\displaystyle\langle\tilde{\psi}_{x|\omega}\big{|}\partial_{x}\tilde{\psi}_{x|\omega}\rangle$
$\displaystyle=0,\omega\in\checkmark,$ (5)
$\displaystyle\ket{\partial_{x}\tilde{\psi}_{x|\omega}}$
$\displaystyle=0,\,\omega\in\times,$ (6)

where $\ket{\tilde{\psi}_{x|\omega}}\equiv M_{\omega,\,x}\ket{\psi_{i}}$. In
addition, for all $\omega\in\checkmark$, if
$\ket{\partial_{x}\tilde{\psi}_{x|\omega}}\neq 0$ then $p(\omega|x)=$ must be
strictly positive.

A few comments in order: First, when Eq. (5) does not exactly but
approximately vanishes, it includes two possibilities: The norm of
$\ket{\tilde{\psi}_{x|\omega}}$ or $\ket{\partial_{x}\tilde{\psi}_{x|\omega}}$
is small or $\ket{\tilde{\psi}_{x|\omega}}$ and
$\ket{\partial_{x}\tilde{\psi}_{x|\omega}}$ are almost orthogonal to each
other up to a small error. We emphasize here that only the latter case leads
to a small loss of the average QFI. Secondly, regardless of lossless or not,
in the general gauge

$I^{Q}\\!=\\!4(\langle G_{\Omega,\,x}\rangle-\langle F_{\Omega,\,x}\rangle),$
(7)

where
$G_{\omega,\,x}\\!\equiv\\!\partial_{x}M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}$,
$F_{\omega,\,x}\\!\equiv\\!\text{i}\partial_{x}M_{\omega,\,x}^{\dagger}M_{\omega,\,x}$
and $\langle\bullet\rangle$ denotes the expectation averaged over the system’s
initial state $\ket{\psi_{i}}$ [16, 63]. The corresponding saturation
condition for the general gauge can be derived from Theorem 1, see the
Supplemental Material [63] for details. In the perpendicular gauge, $I^{Q}$
takes the simple form $I^{Q}\\!=4\langle G_{\Omega,\,x}\rangle$ since
$F_{\Omega,\,x}\rangle=0$. When only Eq. (5) is satisfied, the average QFI
over the post-selection probability and its loss is

$p(\omega|x)I(\sigma_{x|\omega})=4\langle
G_{\omega,\,x}\rangle,\omega\in\checkmark,\quad\kappa\equiv\langle
G_{\times,\,x}\rangle/\langle G_{\Omega,\,x}\rangle.$ (8)

Quantum measurements as an information transducer.—When lossless post-
selection is applied, amplification of the QFI of the post-selected states can
be achieved [49, 54, 50, 51, 41, 55]. In fact, when Eq. (S38) is satisfied for
all $\omega\in\Omega$, we have

$\sum_{\omega\in\Omega}I(\sigma_{x|\omega})\geq\sum_{\omega\in\Omega}p(\omega|x)I(\sigma_{x|\omega})=I^{Q}.$
(9)

In the case of $p(\omega|x)\in(0,\,1)$ $\forall\omega\in\Omega$, the
inequality is strict, meaning that the total QFI of the split states
$\sigma_{x|\omega}$ is higher than the original QFI. Close to but not exact at
the boundary of $p(\omega|x)=0$, the amplification is significant. It is worth
noting that such an amplification can be achieved as long as measurement
encoding is lossless and the measurement outcomes are accessible.

Previous works on post-selected metrology concern with either a single quantum
system [49, 50, 51] or the estimation of the interaction strength between two
subsystems as in the case of weak-value amplification [55, 49]. One use of the
lossless measurement encoding discovered here is that it can be employed as an
information transducer, which transfers the metrological information from one
subsystem (the environment) to another(the primary system), as schematically
shown in Fig. 1(a). When the estimation parameter $x$ initially lies in the
environmental degrees of freedom, a measurement consisting of rank$-1$
projective operators acting on the environment results in post-measurement
environmental states that does not depend on the estimation parameter $x$.
Hence, after the measurement, no information is left in the environment.
Furthermore, if Theorem 1 is satisfied, system now must contain all the
metrological information, as the measurement does not dissipate information.

Figure 1: (a) Schematic picture of metrological information transduction from
the environment to the system through a proper quantum measurement. (b)
Numerical calculation of various QFIs in the two-qubit example in the main
text. Value of parameters: $x=10^{-5}$ and the sensing time $T=1$. The black
solid line corresponds to the total average QFI, i.e.
$\sum_{\omega\in\Omega}p(\omega|x)I(\sigma_{x|\omega})$ which is almost equal
to $I(\ket{\Psi_{x}^{\text{SE}}})=4$ in this case. The blue dashed and the
purple dotted-dashed lines correspond to $I(\sigma_{x|\omega=1})$ and
$I(\sigma_{x|\omega=1})$ respectively while the red solid line is the total
sum of both.

We consider the following generic protocol: The environment first evolves
under the Hamiltonian $H^{\text{E}}\\!=\\!xH_{0}^{\text{E}}$ for some $T$,
followed by an entangling gate $U_{I}$. Then a rank-$1$ projective measurement
formed by the basis $\\{\ket{\pi_{\omega}^{\text{E}}}\\}$ is performed such
that all the metrological information is transferred to the system. Throughout
the process, we freeze the system’s free evolution, i.e.,
$H_{\text{S}}\\!=\\!0$ for simplicity. It can readily calculated that
$U_{x}^{\text{SE}}\\!=\\!U_{\text{I}}e^{-\text{i}xH_{0}^{\text{E}}T}$,
$U_{x}^{\dagger\text{SE}}\partial_{x}U_{x}^{\text{SE}}\\!=\\!-\text{i}H_{0}^{\text{E}}T$
and
$I^{Q}\\!=\\!4T^{2}\text{Var}(H_{0}^{\text{E}})_{\ket{\phi_{i}^{\text{E}}}}$.
One can always shift $H_{0}^{\text{E}}$ by a constant such that
$\braket{\phi_{i}^{\text{E}}\big{missing}}{H_{0}^{\text{E}}\big{missing}}{\phi_{i}^{\text{E}}}\\!=\\!0$
to move in the perpendicular gauge. Furthermore, with prior knowledge of the
estimation parameter and adaptive control [9, 69, 70, 71, 22, 14], we can
assume $x$ is a very weak signal without loss of generality. Then Eq. (5)
leads to

$\langle\psi_{i}|\otimes\langle\phi_{i}^{\text{E}}|U^{\text{I}\dagger}(\mathbb{I}\otimes\ket{\pi_{\omega}^{\text{E}}}\bra{\pi_{\omega}^{\text{E}}})U^{\text{I}}\ket{\psi_{i}}\otimes\ket{\phi_{i}^{\text{E}\perp}}=0,$
(10)

where we have used the weak signal approximation to set
$U_{x}^{\text{SE}}\\!=\\!U_{\text{I}}$ and
$\ket{\phi_{i}^{\text{E}\perp}}\\!\equiv\\!H_{0}^{\text{E}}\ket{\phi_{i}^{\text{E}}}/\sqrt{\text{Var}(H_{0}^{\text{E}})_{\ket{\phi_{i}^{\text{E}}}}}$
is orthogonal to $\ket{\phi_{i}^{\text{E}}}$ in the perpendicular gauge.
Intuitively, one can construct the following control-$U$ gate,

$U^{\text{I}}=\mathbb{I}\otimes\left(\mathbb{I}^{\text{E}}-\ket{\phi_{i}^{\text{E}\perp}}\bra{\phi_{i}^{\text{E}\perp}}\right)+\mathbb{U}\otimes\ket{\phi_{i}^{\text{E}\perp}}\bra{\phi_{i}^{\text{E}\perp}},$
(11)

where $U\ket{\psi_{i}}$ is orthogonal to $\ket{\psi_{i}}$. Then Eq. (10)
becomes
$\langle\psi_{i}|\otimes\langle\phi_{i}^{\text{E}}|\mathbb{I}\otimes\ket{\pi_{\omega}^{\text{E}}}\langle\pi_{\omega}^{\text{E}}\ket{\psi_{i}^{\perp}}\otimes\ket{\phi_{i}^{\text{E}\perp}}\\!=\\!0$,
which can satisfies by any rank-$1$ projective measurements since
$\braket{\psi_{i}}{\psi_{i}^{\perp}}\\!=\\!0$. The simplest choice one can
construct is the following:

$\ket{\pi_{1}^{\text{E}}(\varepsilon)}=\frac{\ket{\phi_{i}^{\text{E}}}+\varepsilon\ket{\phi_{i}^{\text{E}\perp}}}{\sqrt{1+\varepsilon^{2}}},\,\ket{\pi_{2}^{\text{E}}(\varepsilon)}=\frac{\ket{\phi_{i}^{\text{E}\perp}}-\varepsilon\ket{\phi_{i}^{\text{E}}}}{\sqrt{1+\varepsilon^{2}}},$
(12)

where $\varepsilon$ is an arbitrary real number and other projectors are
formed by orthonormal basis outside the subspace spanned by
$\ket{\phi_{i}^{\text{E}}}$ and $\ket{\phi_{i}^{\text{E}\perp}}$. Since
$U_{x}^{\text{SE}}\\!\approx\\!U^{\text{I}}$ and
$\partial_{x}U_{x}^{\text{SE}}\\!=\\!-\text{i}TU^{\text{I}}H_{0}^{\text{E}}$ ,
we obtain
$M_{\omega}(\varepsilon)\\!=\\!\braket{\pi_{\omega}^{\text{E}}(\varepsilon)\big{missing}}{U^{\text{I}}\big{missing}}{\phi_{i}^{\text{E}}}$,
$\partial_{x}M_{\omega}(\varepsilon)\\!=\\!-\text{i}\sqrt{I^{Q}}\braket{\pi_{\omega}^{\text{E}}(\varepsilon)\big{missing}}{U^{\text{I}}\big{missing}}{\phi_{i}^{\text{E}\perp}}/2$.
Thus it is straightforward to show
$M_{1}(\varepsilon)\\!\approx\\!-M_{2}(\varepsilon)/\varepsilon\\!\approx\\!\mathbb{I}/\sqrt{1+\varepsilon^{2}}$
and
$\partial_{x}M_{1}(\varepsilon)/\varepsilon\\!\approx\\!\partial_{x}M_{2}(\varepsilon)\\!\approx\\!-\text{i}\sqrt{I^{Q}}\mathbb{U}/\left(2\sqrt{1+\varepsilon^{2}}\right)$.
Note that although the magnitude of these measurement operators can be small,
their sensitivity to change of the estimation parameter can be large.

For example, consider a minimum two-qubit model with $H^{\text{S}}\\!=\\!0$,
$H^{\text{E}}\\!=\\!x\sigma_{z}$, $\ket{\phi_{i}^{\text{E}}}\\!=\\!\ket{+x}$,
where $\ket{0}$ and $\ket{1}$ denotes the excited and ground states of
$\sigma_{z}$ and $\ket{\pm x}\\!=\\!(\ket{0}\\!\pm\\!\ket{1})/\sqrt{2}$. The
initial state of the system is $\ket{\psi_{i}}\\!=\\!\ket{0}$. The
controlled-$U$ gate then becomes
$U^{\text{I}}\\!=\\!\mathbb{I}\otimes\ket{+x}\bra{+x}\\!+\\!\sigma_{x}\otimes\ket{-x}\bra{-x}$.
The plots of the various QFIs for post-measurement states for small $x$ are
shown in Fig. 1(b). In the perpendicular gauge, the QFI for post-measurement
states is the norm of $\ket{\partial_{x}\tilde{\psi}_{x|\omega}}$. In the
limit $\varepsilon\to 0$ or $\varepsilon\to\infty$, one of the post-
measurement state will carry no information and thus can be safely discarded,
as shown in Fig. 1(b).

Post-selected non-Hermitian sensing. —Now we are in a position to how non-
Hermitian sensing is connected to the measurement encoding problem above.
Consider the non-Hermitian dynamics,

$\dot{\rho}_{x}(t)=-\text{i}\left(H_{x}^{\text{nh}}(t)\rho_{x}(t)-\rho_{x}(t)H_{x}^{\text{nh}\dagger}(t)\right),$
(13)

where $H_{1}(t)$ is some control Hamiltonian,

$H_{x}^{\text{nh}}(t)\equiv
H_{0x}(t)+H_{1}(t)-\frac{\text{i}}{2}\sum_{j}\gamma_{j}(t)L_{j}^{\dagger}L_{j},$
(14)

and $\gamma_{j}(t)\geq 0$. It is well known that the GKSL Markovian master
equation [72, 73, 74, 75] describing open-system dynamics can be unravelled by
a sequel of measurements in continuous time, which leads to stochastic
evolution of the state [76, 77]. Non-Hermitian dynamics can be further
effectively realized by eliminating state trajectories with quantum jumps
[53].

To facilitate our subsequent discussion, we first consider the discrete time
evolution and take the continuum limit at the end. We denote the probe time as
$T$. Then discrete non-Hermitian evolution is given by
$M_{\checkmark,\,x}\\!\equiv\\!M_{\checkmark,\,x}^{(N)}\cdots
M{}_{\checkmark,\,x}^{(2)}M_{\checkmark,\,x}^{(1)}$, where
$M_{\checkmark,\,x}^{(n)}\\!=\\!\mathbb{I}-\text{i}H_{x}^{\text{nh}}(t_{n})\Delta
t$, $M_{j}^{(n)}\\!=\\!\sqrt{\gamma_{j}(t_{n})\Delta t}L_{j}$, and
$t_{n}\\!=\\!n\Delta t$.The discrete-time analog of quantum trajectories
correspond to the quantum collision model [78, 79, 80, 81] with a
concatenation of measurements, as depicted in Fig. 2(a). The discrete non-
Hermitian evolution is implemented by post-selecting the outcomes of these
measurements. We define
$\braket{\pi_{\checkmark}}{U_{x}^{\text{SE}(n)}}{0}\\!\equiv\\!M_{\checkmark,\,x}^{(n)}$,
$\,\braket{\pi_{j}}{U_{x}^{\text{SE}(n)}}{0}\\!\equiv\\!M_{j,\,x}^{(n)},\,$
and $U_{x}^{\text{SE}}\\!\equiv\\!U_{x}^{\text{SE}(N)}\cdots
U_{x}^{\text{SE}(1)}$. As the measurement operators satisfy the approximate
completeness relations
$M_{\checkmark,\,x}^{(n)^{\dagger}}M_{\checkmark,\,x}^{(n)}\\!+\\!\sum_{j\in\times}M_{j,\,x}^{(n)\dagger}M_{j,\,x}^{(n)}\\!=\\!\mathbb{I}\\!+\\!O(\Delta
t^{2})$ and
$\sum_{n=1}^{N}\sum_{j}M_{jn,\,x}^{\dagger}M_{jn,\,x}\\!+\\!M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}\\!=\\!\mathbb{I}\\!+\\!O(N\Delta
t^{2})$ [63], so are the unitary operators, i.e.,
$U_{x}^{\text{SE}\dagger(n)}U_{x}^{\text{SE}(n)}\\!=\\!\mathbb{I}+O(\Delta
t^{2})$ and
$U_{x}^{\text{SE}\dagger}U_{x}^{\text{SE}}\\!=\\!\mathbb{I}+O(N\Delta t^{2})$.

Figure 2: (a) Unravelling the discrete GKSL master equation with the collision
model(b) Deferred measurement principle for post-selected non-Hermitian
evolution.

According to the deferred measurement principle, the post-selection
measurement can be always deferred to the very end, as shown in Fig. 2(b).
With this observation, we recognize that the discrete non-Hermitian sensing
belongs to the measurement encoding problem with a single retained state. The
retained measurement outcome is $(\checkmark,\,\checkmark,\,\cdots\checkmark)$
corresponding to the measurement operator $M_{\checkmark,\,x}$. All the
remaining outcomes lies in the discarded set $\times$, with the measurement
operators $M_{jn,\,x}\\!\equiv\\!M_{j}^{(n)}M_{\checkmark,\,x}^{(n-1)}\cdots
M_{\checkmark,\,x}^{(1)}$, and the environment initial state
$\ket{\phi_{i}^{\text{E}}}=\ket{0}^{\otimes N}$. We can then apply Theorem 1
and take the continuous-time limit $\Delta t\to 0$, $N\to\infty$ $N\Delta t=T$
to identify QFI-preserving conditions for post-selected non-Hermitian sensing.
Note that in the continuous-time limit,
$M_{\checkmark,\,x}(T)\\!\equiv\\!\mathcal{T}e^{-\text{i}\int_{0}^{T}H_{x}^{\text{nh}}(\tau)d\tau}$
and the approximate completeness relation for the measurement operators in the
discrete case now becomes the exact integral completeness relation:

$\int_{0}^{T}M_{\checkmark,\,x}^{\dagger}(t)L^{2}(t)M_{\checkmark}(t)dt+M_{\checkmark,\,x}^{\dagger}(T)M_{\checkmark,\,x}(T)=\mathbb{I},$
(15)

where $L^{2}(t)\equiv\sum_{j}\gamma_{j}(t)L_{j}^{\dagger}L_{j}$. Eq. (15) can
be straightforwardly verified by taking derivatives with respect to $T$ and
then applying the non-Hermitian dynamics
$\text{i}M_{\checkmark,\,x}(t)=H_{\text{nh}}(t)M_{\checkmark,\,x}(t)$. Next,
we present our second main result:

######  Theorem 2.

Non-Hermitian sensing with is lossless compared with the quantum collision
model shown in Fig. 2(b) in the continuous-time limit, if and only if (a) The
post-selection probability is insensitive to the estimation parameter, i.e.,
$\partial_{x}\langle E_{\checkmark,\,x}\rangle\\!=\\!0$, where
$E_{\checkmark,\,x}\\!\equiv\\!M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}$.
(b) For all $j$ and $t\in[0,\,T]$,
$L_{j}\ket{\tilde{\xi}_{\checkmark,\,x}(t)}\\!=\\!0$, where
$\ket{\tilde{\xi}_{\checkmark,\,x}(t)}\\!\equiv\\![\ket{\partial_{x}\tilde{\psi}_{x|\checkmark}(t)}\\!+\\!\text{i}\partial_{x}\theta_{\checkmark,\,x}\ket{\tilde{\psi}_{x|\checkmark}(t)}]$
and $\partial_{x}\theta_{\checkmark,\,x}\\!\equiv\\!-\text{Re}\langle
F_{\checkmark,\,x}\rangle/\langle
E_{\checkmark,\,x}\rangle\\!=\\!\text{Im}\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}\rangle/\langle
E_{\checkmark,\,x}\rangle$.

A few comments in order. One special case is that if
$L_{j}\mathcal{T}e^{-\text{i}\int_{0}^{T}[H_{0x}(t)+H_{1}(t)]dt}\ket{\psi_{i}}\\!=\\!0$
for all $j$, which means that no jump occurs, then non-Hermitian dynamics is
effective unitary for the initial state $\ket{\psi_{i}}$. In this case,
$\langle E_{\checkmark,\,x}\rangle\\!=\\!0$ and
$L_{j}\ket{\tilde{\psi}_{x|\checkmark}(t)}\\!=\\!0$, which clearly satisfies
Theorem 2. Secondly, it can be calculated that [63]:

$\displaystyle\langle G_{\Omega}\rangle$ $\displaystyle=\langle
G_{\checkmark,\,x}\rangle+\int_{0}^{T}\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}(t)L^{2}(t)\partial_{x}M_{\checkmark,\,x}(t)\rangle,$
(16) $\displaystyle\langle F_{\Omega}\rangle$ $\displaystyle=\langle
F_{\checkmark,\,x}\rangle+\text{i}\int_{0}^{T}\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}(t)L^{2}(t)M_{\checkmark,\,x}(t)\rangle.$
(17)

As a result, even when the conditions in Theorem 2 is not satisfied or
partially satisfied, one can still estimate the loss of the average QFI
compared with the unitary quantum collision model, i.e.,
$\kappa\\!=\\!\left[1\\!-\\!p(\checkmark|x)I(\sigma_{x|\checkmark})/I^{Q}\right]$,
where $p(\checkmark|x)I(\sigma_{x|\checkmark})\\!=\\!4\left[\langle
G_{\checkmark,\,x}\rangle\\!-\\!\frac{|\langle
F_{\checkmark,\,x}\rangle|^{2}}{\langle E_{\checkmark,\,x}\rangle}\right]$
[49] and $I^{Q}$ can be calculated using Eqs. (7, 16, 17). This is our third
main result. We would like to emphasize the universality of the loss formula,
which does not depend on the microscopic environment and coupling details in
the quantum collision model.

To illustrate the application of the loss formula, we consider a simple yet
non-trivial case of pure dephasing channels with multiplicative estimation
Hamiltonian, where $H_{0x}=xH_{0}$ and $H_{0}$, $H_{1}(t)$, $L$ mutually
commute. Furthermore, we assume $\gamma_{j}(t)$ is time-independent for
simplicity. It can be readily calculated that
$M_{\checkmark,\,x}(T)\\!=\\!e^{-\text{i}[xH_{0}T+\int_{0}^{T}H_{1}(t)dt]}e^{-\frac{1}{2}L^{2}T}$,
$\langle E_{\checkmark,\,x}\rangle\\!=\\!\langle e^{-L^{2}T}\rangle$, $\langle
F_{\checkmark,\,x}\rangle\\!=\\!-T\langle e^{-L^{2}T}H_{0}\rangle$, and
$\langle G_{\checkmark,\,x}\rangle\\!=\\!T^{2}\langle
e^{-L^{2}T}H_{0}^{2}\rangle$. It is straightforward to check that the integral
completeness relation (15) is satisfied. Moreover, physically since the the
system-environment coupling commutes with $H_{0}$ and $H_{1}(t)$, it is
expected that QFI for closed-system must be $I^{Q}\\!=\\!4T^{2}\left(\langle
H_{0}^{2}\rangle-\langle H_{0}\rangle^{2}\right)$, which is the same as the
prediction using Eqs. (7, 16, 17). These observations demonstrate the validity
of the theory presented here. Finally, the loss for the dephasing model is
given by

$\kappa=1-\frac{\langle e^{-L^{2}T}H_{0}^{2}\rangle-(\langle
e^{-L^{2}T}H_{0}\rangle)^{2}/\langle e^{-L^{2}T}\rangle}{\langle
H_{0}^{2}\rangle-\langle H_{0}\rangle^{2}}.$ (18)

In the case where $H_{0}=\sigma_{z}$, $\ket{\psi_{i}}=\ket{+x}$, and
$L_{1}=L_{1}^{\dagger}=\sigma_{z}$, we find
$I(\sigma_{x|\checkmark})=I^{Q}=4T^{2}$ with loss $\kappa=1-e^{-\gamma_{1}T}$,
which increases with the probe time.

Conclusion. —We provide a framework for quantum metrology using quantum
measurement channels and hence may open a new research line. We give
fundamental theorems on conditions of lossless measurement encoding compared
with the unitary encoding and a fundamental formula in estimating the loss in
post-selected non-Hermitian sensing. We hope our findings here can further
inspire further studies on QFI-preserving quantum channels. Future directions
include investigating the interplay between many-body and post-selection
effects, generalizing to mixed states, and exploring non-Markovian post-
selection encoding, etc.

Acknowledgement. — The author would like to thank Adolfo del Campo for useful
feedback on the manuscript and Hongzhe Zhou for hosting his visit to Tsung-Dao
Lee Institute at Shanghai Jiao Tong University, during which this work was
completed. This work was funded by the Wallenberg Initiative on Networks and
Quantum Information (WINQ) program.

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Supplemental Material

###### Contents

  1. I Refined convexity inequality of the QFI
  2. II Proof of Theorem 1
  3. III Theorem 1 in generic gauge
  4. IV Approximate completeness relation
  5. V Proof of Theorem 2
  6. VI Integral representations of $\langle G_{\Omega}\rangle$ and $\langle F_{\Omega}\rangle$ for the quantum collision model

##  I Refined convexity inequality of the QFI

The proof of the convexity property Eq. (2) can be found in Refs. [62, 60,
61]. Here we provide a refined convexity inequality. Let us note the classical
Fisher information associated with the POVM measurement $\\{E_{\mu}\\}$ on the
state is

$J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})=\frac{[\text{Tr}(\partial_{x}\rho_{x}E_{\mu})]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}=\frac{\sum_{\omega}[\text{Tr}(\partial_{x}\tilde{\sigma}_{x|\omega}E_{\mu})]^{2}}{\text{Tr}(\rho_{x}E_{\mu})},$
(S1)

where

$\tilde{\sigma}_{x|\omega}=p(\omega|x)\sigma_{x|\omega}.$ (S2)

According to Ref. [65], we know

$J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})\leq
J_{\mu}(\rho_{x})\equiv\text{Tr}(\rho_{x}L_{x}E_{\mu}L_{x})=\sum_{\omega}\text{Tr}(\tilde{\sigma}_{x|\omega}L_{x}E_{\mu}L_{x}).$
(S3)

More specifically, one can show

$\displaystyle J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})$
$\displaystyle=\frac{\left[\text{Re}\text{Tr}(\rho_{x}E_{\mu}L_{x})\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}=\frac{\left[\text{Re}\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}L_{x}\right)\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
$\displaystyle\leq\frac{\left[\big{|}\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}L_{x}\right)\big{|}\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
(S4)
$\displaystyle\leq\frac{\left[\sum_{\omega}\big{|}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}L_{x}\right)\big{|}\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
(S5)
$\displaystyle\leq\frac{\left[\sum_{\omega}\sqrt{\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}\right)}\sqrt{\text{Tr}\left(\tilde{\sigma}_{x|\omega}L_{x}E_{\mu}L_{x}\right)}\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
(S6)
$\displaystyle\leq\frac{\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}\right)}{\text{Tr}(\rho_{x}E_{\mu})}\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}L_{x}E_{\mu}L_{x}\right)$
(S7) $\displaystyle=J_{\mu}(\rho_{x}),$ (S8)

where in the inequality (S5) we have used the triangle inequality and in
equalities (S6) and (S7), we have used the Cauchy-Schwarz inequality. We
define [60]

$\partial_{x}\tilde{\sigma}_{x|\omega}=\frac{\tilde{L}_{x|\omega}\tilde{\sigma}_{x|\omega}+\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}}{2}.$
(S9)

It is straightforward to verify that

$\tilde{L}_{x|\omega}=\partial_{x}\ln p(\omega|x)+L_{x|\omega},$ (S10)

where $L_{x|\omega}$ is defined as

$\partial_{x}\sigma_{x|\omega}=\frac{L_{x|\omega}\sigma_{x|\omega}+\sigma_{x|\omega}L_{x|\omega}}{2}.$
(S11)

Thus we find

$J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})=\frac{\left[\sum_{\omega}\text{Tr}(\partial_{x}\tilde{\sigma}_{x|\omega}E_{\mu})\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}=\frac{\left[\text{Re}\sum_{\omega}\text{Tr}(\tilde{\sigma}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega})\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}.$
(S12)

Following similar procedure in inequality (S7), it can be readily shown that

$\displaystyle J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})$
$\displaystyle\leq\frac{\left[\big{|}\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right)\big{|}\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
$\displaystyle\leq\frac{\left[\sum_{\omega}\big{|}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right)\big{|}\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
(S13)
$\displaystyle\leq\frac{\left[\sum_{\omega}\sqrt{\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}\right)}\sqrt{\text{Tr}\left(\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right)}\right]^{2}}{\text{Tr}(\rho_{x}E_{\mu})}$
(S14)
$\displaystyle\leq\frac{\left[\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}E_{\mu}\right)\right]\left[\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right)\right]}{\text{Tr}(\rho_{x}E_{\mu})}$
(S15)
$\displaystyle=\sum_{\omega}\text{Tr}\left(\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right).$
(S16)

We define

$L_{x}^{\text{SE}}=\sum_{\omega}\tilde{L}_{x|\omega}\otimes\ket{\pi_{\omega}^{\text{E}}}\bra{\pi_{\omega}^{\text{E}}},$
(S17)

which satisfies

$\partial_{x}\sigma_{x}^{\text{SE}}=\frac{L_{x}^{\text{SE}}\sigma_{x}^{\text{SE}}+\sigma_{x}^{\text{SE}}L_{x}^{\text{SE}}}{2}.$
(S18)

Then it is straightforward to calculate that

$\displaystyle\text{Tr}\left(\sum_{\omega}\tilde{\sigma}_{x|\omega}\tilde{L}_{x|\omega}E_{\mu}\tilde{L}_{x|\omega}\right)$
$\displaystyle=\text{Tr}_{\text{SE}}\left(\sum_{\omega}\tilde{\sigma}_{x|\omega}\otimes\ket{\pi_{\omega}^{\text{E}}}\bra{\pi_{\omega}^{\text{E}}}L_{x}^{\text{SE}}(E_{\mu}\otimes\mathbb{I}^{\text{E}})L_{x}^{\text{SE}}\right)$
$\displaystyle=\text{Tr}_{\text{SE}}\left(\sigma_{x}^{\text{SE}}L_{x}^{\text{SE}}(E_{\mu}\otimes\mathbb{I}^{\text{E}})L_{x}^{\text{SE}}\right)\equiv
J_{\mu}(\sigma_{x}^{\text{SE}}).$ (S19)

Thus, we find

$J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})\leq J_{\mu}(\sigma_{x}^{\text{SE}}).$
(S20)

Since $\sigma_{x|\omega}$ is a pure state,

$L_{x|\omega}=2\partial_{x}\sigma_{x|\omega},\,\tilde{L}_{x|\omega}=\partial_{x}\ln
p(\omega|x)+2\partial_{x}\sigma_{x|\omega},$ (S21)

and $J_{\mu}(\sigma_{x}^{\text{SE}})$ can be therefore expressed as

$\displaystyle J_{\mu}(\sigma_{x}^{\text{SE}})$
$\displaystyle=\sum_{\omega}\left[K^{\text{cl}}\left(p(\omega|x)\right)\text{Tr}\left(\sigma_{x|\omega}E_{\mu}\right)+4\partial_{x}p(\omega|x)\text{ReTr}\left(\partial_{x}\sigma_{x|\omega}\sigma_{x|\omega}E_{\mu}\right)+4p(\omega|x)\text{Tr}\left(\partial_{x}\sigma_{x|\omega}\sigma_{x|\omega}\partial_{x}\sigma_{x|\omega}E_{\mu}\right)\right],$
(S22)

where

$\text{ReTr}\left(\partial_{x}\sigma_{x|\omega}\sigma_{x|\omega}E_{\mu}\right)=\text{Re}\braket{\psi_{x}{\omega}\big{missing}}{E_{\mu}\big{|}\partial_{x}\psi_{x|\omega}}{,}$
(S23)
$\text{Tr}\left(\partial_{x}\sigma_{x|\omega}\sigma_{x|\omega}\partial_{x}\sigma_{x|\omega}E_{\mu}\right)=\braket{\partial_{x}\psi_{x}{\omega}\big{missing}}{E_{\mu}\big{|}\partial_{x}\psi_{x|\omega}}{+}\big{|}\braket{\partial_{x}\psi_{x}{\omega}\big{missing}}{\psi_{x|\omega}}{\big{missing}}|^{2}\braket{\psi_{x}{\omega}\big{missing}}{E_{\mu}\big{|}\psi_{x|\omega}}{-}2\text{Re}\braket{\psi_{x}{\omega}\big{missing}}{\partial_{x}\psi_{x|\omega}}{\braket{missing}}{\partial_{x}\psi_{x|\omega}\big{|}E_{\mu}\big{|}\psi_{x|\omega}}.$
(S24)

On the other hand, the qCRB for single-parameter is always saturable [64, 65],
i.e, there always exists a POVM measurement $\\{E_{\mu}\\}$ such that Eq. (S3)
is saturated. As such, we conclude

$J_{\mu}^{\text{cl}}(\rho_{x};E_{\mu})\leq J_{\mu}(\rho_{x})\leq
J_{\mu}(\sigma_{x}^{\text{SE}}),$ (S25)

which we dub as refined convexity inequality.

##  II Proof of Theorem 1

We consider a pure parameter-dependent state $\ket{\Psi_{x}}$ and a parameter-
independent quantum channel $\\{Q_{\omega}\\}_{\omega\in\Omega}$. In the
perpendicular gauge, where

$\braket{\partial_{x}\Psi_{x}\big{missing}}{\Psi_{x}}=0,$ (S26)

so that $\ket{\partial_{x}^{\perp}\Psi_{x}}=\ket{\partial_{x}\Psi_{x}}$. In
this gauge, the two conditions in Theorem 1 in Ref. [49] reduce to for
$\omega\in\checkmark$,

$\braket{\partial_{x}\Psi_{x}\big{missing}}{Q_{\omega}\big{missing}}{\Psi_{x}}=0,$
(S27)

and

$\braket{\partial_{x}\Psi_{x}\big{missing}}{Q_{\times}\big{missing}}{\partial_{x}\Psi_{x}}=0,$
(S28)

where

$Q_{\times}=\sum_{\omega\in\times}Q_{\omega}.$ (S29)

Eq. (S27) and (S28) is the condition for lossless compression. Furthermore,
when Eq. (S27) is saturated, the average QFI in the post-selected states is,
according to Ref. [49]

$\sum_{\omega\in\checkmark}p(\omega|\checkmark)I(\sigma_{x|\omega})=4\sum_{\omega\in\checkmark}\braket{\partial_{x}^{\perp}\Psi_{x}}{Q_{\omega}\big{missing}}{\partial_{x}^{\perp}\Psi_{x}}=4\braket{\partial_{x}\Psi_{x}\big{missing}}{Q_{\checkmark}\big{missing}}{\partial_{x}\Psi_{x}}.$
(S30)

Now we take $Q_{\omega}=\mathbb{I}\otimes\Pi_{\omega}^{\text{E}}$ with
$\Pi_{\omega}^{\text{E}}\equiv\ket{\pi_{\omega}^{\text{E}}}\bra{\pi_{\omega}^{\text{E}}}$
and define

$E_{\omega,\,x}\equiv
M_{\omega,\,x}^{\dagger}M_{\omega,\,x},\,F_{\omega,\,x}\equiv\text{i}\partial_{x}M_{\omega,\,x}^{\dagger}M_{\omega,\,x},\,G_{\omega,\,x}\equiv\partial_{x}M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}.$
(S31)

It can be readily observed that

$F_{\Omega,\,x}^{\dagger}=\sum_{\omega}F_{\omega,\,x}^{\dagger}=-\text{i}\sum_{\omega}M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}=\sum_{\omega}F_{\omega,x}=F_{\Omega,\,x}.$
(S32)

It is straightforward to calculate [16]

$\displaystyle\braket{\partial_{x}\Psi_{x}^{\text{SE}}\big{missing}}{\partial_{x}\Psi_{x}^{\text{SE}}}$
$\displaystyle=\bra{\psi_{i}}\otimes\bra{\phi_{i}^{\text{E}}}\partial_{x}U_{x}^{\text{SE}\dagger}\partial_{x}U_{x}^{\text{SE}}\ket{\psi_{i}}\otimes\ket{\phi_{i}^{\text{E}}}$
$\displaystyle=\sum_{\omega}\braket{\psi_{i}\big{missing}}{\partial_{x}M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}\big{missing}}{\psi_{i}}=\langle
G_{\Omega,\,x}\rangle,$ (S33)
$\displaystyle\braket{\partial_{x}\Psi_{x}^{\text{SE}}\big{missing}}{\Psi_{x}^{\text{SE}}}$
$\displaystyle=\bra{\psi_{i}}\otimes\bra{\phi_{i}^{\text{E}}}\partial_{x}U_{x}^{\text{SE}\dagger}U_{x}^{\text{SE}}\ket{\psi_{i}}\otimes\ket{\phi_{i}^{\text{E}}}$
$\displaystyle=\sum_{\omega}\braket{\psi_{i}\big{missing}}{\partial_{x}M_{\omega,\,x}^{\dagger}M_{\omega,\,x}\big{missing}}{\psi_{i}}=-\text{i}\langle
F_{\Omega,\,x}\rangle.$ (S34)

Thus, regardless of whether $M_{\omega}$ is lossless of not,

$I^{Q}=\langle G_{\Omega,\,x}\rangle-\left(\langle
F_{\Omega,\,x}\rangle\right)^{2}.$ (S35)

In particular, in the perpendicular gauge, we find

$\langle
F_{\Omega,\,x}\rangle=\text{i}\langle\\!\langle\partial_{x}U_{x}^{\text{SE}\dagger}U_{x}^{\text{SE}}\rangle\\!\rangle=0,$
(S36)

and

$I^{Q}=\langle G_{\Omega,\,x}\rangle.$ (S37)

Furthermore, Eq. (S27) and Eq. (S28) become

$\langle F_{\omega,\,x}\rangle=0,\,\omega\in\checkmark,$ (S38) $\langle
G_{\times,\,x}\rangle=0,$ (S39)

Eq. (S38) implies

$\displaystyle\text{Re}\langle F_{\omega,\,x}\rangle$
$\displaystyle=\frac{\text{i}}{2}\left(\langle\partial_{x}M_{\omega,\,x}^{\dagger}M_{\omega,\,x}\rangle-\langle
M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}\rangle\right)=0,$ (S40)
$\displaystyle\text{Im}\langle F_{\omega,\,x}\rangle$
$\displaystyle=\frac{1}{2}\left(\langle\partial_{x}M_{\omega,\,x}^{\dagger}M_{\omega,\,x}\rangle+\langle
M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}\rangle\right)=0,$ (S41)

which implies

$\langle\partial_{x}M_{\omega,\,x}^{\dagger}M_{\omega,\,x}\rangle=\langle
M_{\omega,\,x}^{\dagger}\partial_{x}M_{\omega,\,x}\rangle=0,\omega\in\checkmark,$
(S42)

i.e., Eq. (5) in the main text. Eq. (S39) implies Eq. (6) in the main text
since $G_{\times,\,x}$ is semi-positive definite.

When only Eq. (S38) is satisfied, the QFI for the average post-selected state
is, according to Eq. (S30),

$\displaystyle\sum_{\omega\in\checkmark}p(\omega|x)I(\sigma_{x|\omega})$
$\displaystyle=4\langle G_{\checkmark,\,x}\rangle,$ (S43)

and the loss of the QFI becomes

$\kappa=\frac{\langle G_{\times,\,x}\rangle}{\langle G_{\Omega,\,x}\rangle},$
(S44)

which are the equations in Eq. (8) in the main text.

##  III Theorem 1 in generic gauge

For arbitrary given state $\ket{\Psi_{x}^{\text{SE}}}$ that does not satisfy
the perpendicular gauge condition, one can always make the $U(1)$ gauge
transformation

$\ket{\Psi_{x}^{\text{SE}}}\to\ket{\tilde{\Psi}_{x}^{\text{SE}}}=e^{\text{i}\theta_{x}}\ket{\Psi_{x}^{\text{SE}}},$
(S45)

such that $\ket{\tilde{\Psi}_{x}^{\text{SE}}}$ satisfies the perpendicular
gauge condition, i.e.,

$\braket{\partial_{x}\tilde{\Psi}_{x}^{\text{SE}}\big{missing}}{\tilde{\Psi}_{x}^{\text{SE}}}=\braket{\partial_{x}\tilde{\Psi}_{x}^{\text{SE}}\big{missing}}{\tilde{\Psi}_{x}^{\text{SE}}}-\text{i}\partial_{x}\theta_{x}=0.$
(S46)

Then applying Theorem 1 to $\ket{\tilde{\Psi}_{x}^{\text{SE}}}$, we obtain the
following theorem for $\ket{\Psi_{x}^{\text{SE}}}$:

######  Theorem 3.

The measurement encoding for a quantum state $\ket{\Psi_{x}^{\text{SE}}}$,
which does not necessarily satisfy the perpendicular gauge, is lossless if and
only if

$\langle F_{\omega,\,x}\rangle=\langle F_{\Omega,\,x}\rangle\langle
E_{\omega,\,x}\rangle,\,\omega\in\checkmark,$ (S47) $\langle
G_{\times,\,x}\rangle=\langle F_{\Omega,\,x}\rangle\langle
F_{\times,\,x}^{\dagger}\rangle,$ (S48)

A few comments in order. Note that $\langle F_{\Omega,\,x}\rangle$, $\langle
E_{\omega,\,x}\rangle$ and $\langle G_{\times,\,x}\rangle$ are all real
numbers. Thus, to satisfy Eq. (S47) and Eq. (S48), it is necessary to have

$\displaystyle\text{Im}\langle F_{\omega,\,x}\rangle$
$\displaystyle=\frac{1}{2}\langle\partial_{x}E_{\omega,\,x}\rangle=0,\>\omega\in\checkmark,$
(S49) $\displaystyle\text{Im}\langle F_{\times,\,x}\rangle$
$\displaystyle=\frac{1}{2}\langle\partial_{x}E_{\times,\,x}\rangle=0.$ (S50)

Apparently, Eq. (S49) implies Eq. (S50) since
$\sum_{\omega\in\Omega}E_{\omega,\,x}=\mathbb{I}$.

When only Eq. (S47) is satisfied but Eq. (S48) may be violated, the QFI for
the average post-selected state is, according to Eq. (S30),

$\displaystyle\sum_{\omega\in\checkmark}p(\omega|x)I(\sigma_{x|\omega})$
$\displaystyle=4\left(\langle G_{\checkmark,\,x}\rangle-\langle
F_{\Omega,\,x}\rangle\langle F_{\checkmark,\,x}\rangle\right).$ (S51)

Thus the loss of the QFI becomes

$\kappa=1-\frac{\langle G_{\checkmark,\,x}\rangle-\langle
F_{\Omega,\,x}\rangle\langle F_{\checkmark,\,x}\rangle}{\langle
G_{\Omega,\,x}\rangle-\langle F_{\Omega,\,x}\rangle^{2}}=\frac{\langle
G_{\times,\,x}\rangle-\langle F_{\Omega,\,x}\rangle\langle
F_{\times,\,x}\rangle}{\langle G_{\Omega,\,x}\rangle-\langle
F_{\Omega,\,x}\rangle^{2}},$ (S52)

which vanishes when Eq. (S48) is satisfied. This is a further generalization
of the loss formula Eq. (S44) in generic gauge.

##  IV Approximate completeness relation

We show that the measurement operators for the discrete non-Hermitian
evolution satisfy the following property:

######  Proposition 1.

The measurement operators satisfy the approximate completeness relation, i.e.,

$\sum_{n=1}^{N}\sum_{j}M_{jn,\,x}^{\dagger}M_{jn,\,x}+M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}=\mathbb{I}+O(N\Delta
t^{2}).$ (S53)

###### Proof.

We prove by mathematical induction. It is straightforward to see that for
$N=2$, we have

$M_{j1,\,x}=M_{j}^{(1)},\,,M_{j2,\,x}=M_{j}^{(2)}M_{\checkmark,\,x}^{(1)},\,M_{\checkmark,\,x}=M_{\checkmark,\,x}^{(2)}M_{\checkmark,\,x}^{(1)}.$
(S54)

Thus it is straightforward to check

$\displaystyle\sum_{n=1}^{2}\sum_{j}M_{jn,\,x}^{\dagger}M_{jn,\,x}+M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}$
$\displaystyle=\sum_{j}M_{j}^{(1)\dagger}M_{j}^{(1)}+\sum_{j}M_{\checkmark,\,x}^{(1)\dagger}M_{j}^{(2)\dagger}M_{j}^{(2)}M_{\checkmark,\,x}^{(1)}+M_{\checkmark,\,x}^{(1)\dagger}M_{\checkmark,\,x}^{(2)\dagger}M_{\checkmark,\,x}^{(2)}M_{\checkmark,\,x}^{(1)}$
$\displaystyle=\sum_{j}M_{j}^{(1)\dagger}M_{j}^{(1)}+M_{\checkmark,\,x}^{(1)\dagger}\left(\sum_{j}M_{j}^{(2)\dagger}M_{j}^{(2)}+M_{\checkmark,\,x}^{(2)\dagger}M_{\checkmark,\,x}^{(2)}\right)M_{\checkmark,\,x}^{(1)}$
$\displaystyle=\sum_{j}M_{j}^{(1)\dagger}M_{j}^{(1)}+M_{\checkmark,\,x}^{(1)\dagger}M_{\checkmark,\,x}^{(1)}\left(\mathbb{I}+O(\Delta
t^{2})\right)$ $\displaystyle=\mathbb{I}+O(2\Delta t^{2}).$ (S55)

Now assume Eq. (S53) holds for $N$. Then for $N+1$, we define

$\displaystyle\tilde{M}_{\checkmark,\,x}$ $\displaystyle\equiv
M_{\checkmark,\,x}^{(N+1)}M_{\checkmark,\,x},\;\tilde{M}_{jn,\,x}\equiv
M_{jn,\,x},\;n\in[1,\,N],\;\tilde{M}_{jN+1,\,x}\equiv
M_{j}^{(N+1)}M_{\checkmark,\,x},$ (S56)

where the variables with and without tildes correspond to the case of $N+1$
and $N$ respectively. Then as before, it can be easily checked that

$\displaystyle\sum_{n=1}^{N+1}\sum_{j}\tilde{M}_{jn,\,x}^{\dagger}\tilde{M}_{jn,\,x}+\tilde{M}_{\checkmark,\,x}^{\dagger}\tilde{M}_{\checkmark,\,x}$
$\displaystyle=M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}^{(N+1)\dagger}M_{\checkmark,\,x}^{(N+1)}M_{\checkmark,\,x}+\sum_{j}M_{\checkmark,\,x}^{\dagger}M_{j}^{(N+1)\dagger}M_{j}^{(N+1)}M_{\checkmark,\,x}+\sum_{j}\sum_{n=1}^{N}M_{jn,\,x}^{\dagger}M_{jn,\,x}$
$\displaystyle=M_{\checkmark,\,x}^{\dagger}\left(\mathbb{I}+O(\Delta
t^{2})\right)M_{\checkmark,\,x}+\sum_{j}\sum_{n=1}^{N}M_{jn,\,x}^{\dagger}M_{jn,\,x}$
$\displaystyle=\mathbb{I}+O((N+1)\Delta t^{2}),$ (S57)

which completes the proof. ∎

In the continuum limit where $\Delta t\to 0$, $N\to\infty$ and $N\Delta t\to
T$ , the approximate completeness relation becomes the integral completeness
relation:

$\displaystyle\sum_{n=1}^{N}\sum_{j}M_{jn,\,x}^{\dagger}M_{jn,\,x}+M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}$
$\displaystyle=\sum_{j}\sum_{n=1}^{N}\gamma_{j}(t_{n})M_{\checkmark,\,x}^{(1)\dagger}M_{\checkmark,\,x}^{(n-1)\dagger}L_{j}^{\dagger}L_{j}M_{\checkmark,\,x}^{(n-1)}\cdots
M_{\checkmark,\,x}^{(1)}\Delta
t+M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}$
$\displaystyle=\sum_{j}\int_{0}^{T}\gamma_{j}(t)M_{\checkmark,\,x}^{\dagger}(t)L_{j}^{\dagger}L_{j}M_{\checkmark}(t)dt+M_{\checkmark,\,x}^{\dagger}(T)M_{\checkmark,\,x}(T)=\mathbb{I}.$
(S58)

##  V Proof of Theorem 2

We consider a sequence of unitary encoding interspersed by post-selections

$U_{x}^{\text{SE}}=U_{x}^{\text{SE}(N)}\cdots
U{}_{x}^{\text{SE}(2)}U_{x}^{\text{SE}(1)},$ (S59)

as shown in Fig. 2(b) in the main text. In the discrete case where $T=N\Delta
t$, we note that the operators corresponding to the retained state

$M_{\checkmark,\,x}=M_{\checkmark,\,x}^{(N)}\cdots
M{}_{\checkmark,\,x}^{(2)}M_{\checkmark,\,x}^{(1)},$ (S60)

and operators correspond to the discarded states are

$M_{jn,\,x}=M_{j}^{(n)}M_{\checkmark,\,x}^{(n-1)}\cdots
M_{\checkmark,\,x}^{(2)}M_{\checkmark,\,x}^{(1)},$ (S61)

where $n=1,\,2,\cdots N$.

Now we are in a position to prove Theorem 2. Let us note that under the gauge
transformation,

$\displaystyle U_{x}^{\text{SE}}$
$\displaystyle\to\bar{U}_{x}^{\text{SE}}=e^{\text{i}\theta_{x}}U_{x}^{\text{SE}},$
(S62) $\displaystyle M_{\omega,\,x}$
$\displaystyle\to\bar{M}_{\omega,\,x}=e^{\text{i}\theta_{x}}M_{\omega,\,x},$
(S63)

one can readily check

$\displaystyle\partial_{x}U_{x}^{\text{SE}\dagger}U_{x}^{\text{SE}}$
$\displaystyle\to\partial_{x}\bar{U}_{x}^{\text{SE}\dagger}\bar{U}_{x}^{\text{SE}}=\partial_{x}U_{x}^{\text{SE}\dagger}U_{x}^{\text{SE}}-\text{i}\partial_{x}\theta_{x},$
(S64) $\displaystyle\partial_{x}M_{\omega,\,x}$
$\displaystyle\to\partial_{x}\bar{M}_{\omega,\,x}=e^{\text{i}\theta_{x}}\left(\partial_{x}M_{\omega,\,x}+\text{i}\partial_{x}\theta_{x}M_{\omega,\,x}\right),$
(S65) $\displaystyle F_{\omega,\,x}$
$\displaystyle\to\bar{F}_{\omega,\,x}=F_{\omega,\,x}+\partial_{x}\theta_{x}E_{\omega,\,x}.$
(S66)

Clearly, $E_{\omega,\,x}$ is gauge-invariant. Then one can easily check for
$\omega\in\checkmark$

$\displaystyle\langle\bar{F}_{\omega,\,x}\rangle$ $\displaystyle=\langle
F_{\omega,\,x}\rangle+\partial_{x}\theta_{x}\langle E_{\omega,\,x}\rangle,$
(S67)

For non-Hermitian sensing, we have only one retained state. Without causing
any ambiguity, we can simply use $\checkmark$ instead of
$\omega\in\checkmark$. Based on the above observation, we can gauge

$\displaystyle M_{\checkmark,\,x}$
$\displaystyle\to\bar{M}_{\checkmark,\,x}=M_{\checkmark,\,x}e^{\text{i}\theta_{\checkmark,\,x}},$
(S68) $\displaystyle M_{jn,\,x}$
$\displaystyle\to\bar{M}_{jn,\,x}=M_{jn,\,x}e^{\text{i}\theta_{\checkmark,\,x}},$
(S69)

such that

$\text{Re}\langle\bar{F}_{\checkmark,\,x}\rangle=0,$ (S70)

where $\theta_{\checkmark,\,x}$ satisfies

$\partial_{x}\theta_{\checkmark,\,x}\langle
E_{\checkmark,\,x}\rangle=-\text{Re}\langle F_{\checkmark,\,x}\rangle.$ (S71)

Since lossless measurement encoding must satisfy Theorem 3, which implies Eq.
(S49) must satisfied, i.e..

$\text{Im}\langle\bar{F}_{\checkmark,\,x}\rangle=\frac{1}{2}\partial_{x}\langle
E_{\checkmark,\,x}\rangle=0,$ (S72)

and therefore

$\langle\bar{F}_{\checkmark,\,x}\rangle=0.$ (S73)

Furthermore, since $\langle E_{\checkmark,\,x}\rangle$ is strictly positively,
then Eq. (S47) forces

$\langle\bar{F}_{\Omega,\,x}\rangle=0,$ (S74)

which implies
$\ket{\bar{\Psi}_{x}^{\text{SE}}}=\ket{\Psi_{x}^{\text{SE}}}e^{\text{i}\theta_{\checkmark,\,x}}$
satisfies the perpendicular gauge condition. This condition in turn leads to,
according to Eq. (S48),

$\langle\bar{G}_{\times,\,x}\rangle=0,$ (S75)

which is equivalent to

$\partial_{x}\bar{M}_{jn,\,x}\ket{\psi_{i}}=0.$ (S76)

As a result, we know

$\langle\bar{F}_{jn,\,x}\rangle=0,$ (S77)

and thus $\langle\bar{F}_{\Omega,\,x}\rangle=0$. Then it straightforward to
calculate

$\partial_{x}\bar{M}_{jn,\,x}=\left(\partial_{x}M_{jn,\,x}+\text{i}\partial_{x}\theta_{\checkmark,\,x}M_{jn,\,x}\right)e^{\text{i}\theta_{\checkmark,\,x}},$
(S78)
$\partial_{x}\bar{M}_{jn,\,x}\ket{\psi_{i}}=\sqrt{\gamma_{j}(t_{n})\Delta
t}L_{j}\left(\ket{\partial_{x}\tilde{\psi}_{x|\checkmark}(t_{n-1})}+\text{i}\partial_{x}\theta_{\checkmark,\,x}\ket{\tilde{\psi}_{x|\checkmark}(t_{n-1})}\right)=\sqrt{\gamma_{j}(t_{n})\Delta
t}L_{j}\ket{\tilde{\xi}_{\checkmark,\,x}(t_{n-1})},$ (S79)

where

$\ket{\tilde{\psi}_{x|\checkmark}(t_{n-1})}\equiv
M_{\checkmark,\,x}^{(n-1)}\cdots
M_{\checkmark,\,x}^{(2)}M_{\checkmark,\,x}^{(1)}\ket{\psi_{i}}.$ (S80)

Thus

$\langle\bar{G}_{\times,\,x}\rangle=\sum_{j}\sum_{n=1}^{N}\langle\partial_{x}\bar{M}_{jn,\,x}^{\dagger}\partial_{x}\bar{M}_{jn,\,x}\rangle=\sum_{j}\sum_{n=1}^{N}\gamma_{j}(t_{n})\braket{\tilde{\xi}_{\checkmark,\,x}(t_{n-1})\big{missing}}{L_{j}^{\dagger}L_{j}\big{missing}}{\tilde{\xi}_{\checkmark,\,x}(t_{n-1})}\Delta
t.$ (S81)

In the continuum limit,

$\displaystyle\langle\bar{G}_{\times,\,x}\rangle$
$\displaystyle=\sum_{j}\int_{0}^{T}\gamma_{j}(t)\braket{\tilde{\xi}_{\checkmark,\,x}(t)\big{missing}}{L_{j}^{\dagger}L_{j}\big{missing}}{\tilde{\xi}_{\checkmark,\,x}(t)}dt$
$\displaystyle=\sum_{j}\int_{0}^{T}\gamma_{j}(t)\left[]dt,$ (S82)

Similarly, we find

$\langle\bar{F}_{\times,\,x}\rangle=\text{i}\sum_{j}\sum_{n=1}^{N}\langle\partial_{x}\bar{M}_{jn,\,x}^{\dagger}\bar{M}_{jn,\,x}\rangle=\text{i}\sum_{j}\int_{0}^{T}\gamma_{j}(t)\braket{\tilde{\xi}_{\checkmark,\,x}(t)\big{missing}}{L_{j}^{\dagger}L_{j}\big{missing}}{\tilde{\psi}_{\checkmark,\,x}(t)}dt.$
(S83)

##  VI Integral representations of $\langle G_{\Omega}\rangle$ and $\langle
F_{\Omega}\rangle$ for the quantum collision model

Similar to the integral representations above, it can be readily found,

$\displaystyle\langle G_{\Omega}\rangle$
$\displaystyle=\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}\partial_{x}M_{\checkmark,\,x}\rangle+\sum_{j}\sum_{n=1}^{N}\langle\partial_{x}M_{jn,\,x}^{\dagger}\partial_{x}M_{jn,\,x}\rangle$
$\displaystyle=\langle
G_{\checkmark,\,x}\rangle+\sum_{j}\int_{0}^{T}\gamma_{j}(t)\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}(t)L_{j}^{\dagger}L_{j}\partial_{x}M_{\checkmark,\,x}(t)\rangle,$
(S84)

and

$\displaystyle\langle F_{\Omega}\rangle$
$\displaystyle=\text{i}\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}M_{\checkmark,\,x}\rangle+\text{i}\sum_{j}\sum_{n=1}^{N}\langle\partial_{x}M_{jn,\,x}^{\dagger}M_{jn,\,x}\rangle$
$\displaystyle=\langle
F_{\checkmark,\,x}\rangle+\text{i}\sum_{j}\int_{0}^{T}\gamma_{j}(t)\langle\partial_{x}M_{\checkmark,\,x}^{\dagger}(t)L_{j}^{\dagger}L_{j}M_{\checkmark,\,x}(t)\rangle.$
(S85)

According to Eq. (S35), one can compute the $I^{Q}$ for the quantum collision
model and hence the loss of the average QFI $\kappa$.