# Quantum advantage of time-reversed ancilla-based metrology of absorption
parameters

Jiaxuan Wang  1,2 Ruynet. L. de Matos Filho  3 Girish S. Agarwal  1,2,4,* and
Luiz Davidovich1,2,3  1Institute for Quantum Science and Engineering, Texas
A&M University, College Station, Texas 77843, USA
2Department of Physics and Astronomy, Texas A&M University, College Station,
Texas 77843, USA
3Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro,
RJ 21941-972, Brazil
4Department of Biological and Agricultural Engineering, Texas A&M University,
College Station, Texas 77843, USA

††journal: opticajournal††articletype: Research Article

Quantum estimation of parameters defining open-system dynamics may be enhanced
by using ancillas that are entangled with the probe but are not submitted to
the dynamics. Here we consider the important problem of estimation of
transmission of light by a sample, with losses due to absorption and
scattering. We show, through the determination of the quantum Fisher
information, that the ancilla strategy leads to the best possible precision in
single-mode estimation – the one obtained for a Fock state input –, through
joint photon-counting of probe and ancilla, which are modes of a bimodal
squeezed state produced by an optical parametric amplifier. This proposal
overcomes the challenge of producing and detecting high photon-number Fock
states, and it is quite robust against additional noise: we show that it is
immune to phase noise and the precision does not change if the incoming state
gets disentangled. Furthermore, the quantum gain is still present under
moderate photon losses of the input beams. We also discuss an alternative to
joint photon counting, which is readily implementable with present technology,
and approaches the quantum Fisher information result for weak absorption, even
with moderate photons losses of the input beams before the sample is probed: a
time-reversal procedure, placing the sample between two optical parametric
amplifiers, with the second undoing the squeezing produced by the first one.
The precision of estimation of the loss parameter is obtained from the average
outgoing total photon number and its variance. In both procedures, the state
of the probe and the detection procedure are independent of the value of the
parameter.

##  1 Introduction

Quantum sensing involves the use of quantum resources, like entanglement and
squeezing, for the estimation of parameters characteristic of a physical
process, through its action on a probe that, upon a proper measurement, allows
the estimation of the value of the parameters [1, 2]. Entanglement of the
probe with an ancilla that is not submitted to the physical process may
increase the precision of estimation [3, 4]. This is true, however, only to
open-system dynamics. Here we apply the ancilla protocol to the estimation of
the photon-loss coefficient of a sample, due to absorption and scattering of
light. The probe and the ancilla correspond to two modes of a bimodal squeezed
state, produced by an optical parametric amplifier (OPA). Relevant aspects of
the ancilla protocol for estimating absorption were studied theoretically [5,
6, 7] and experimentally [8, 9]. Here we show, through a novel and clarifying
analytical procedure, that, for a given input intensity through the sample,
this scheme leads to precision of estimation identical to the best possible
one for single-mode estimation, obtained by using Fock states [10, 11, 12].
This has the advantage of avoiding the preparation of Fock states with high
photon numbers. Our derivation allows us to determine the corresponding best
measurement: a joint photon-counting of the outgoing probe and the ancilla.
Since this measurement could be challenging with current technologies, except
for joint Fock-state spaces of very small size, we present a time-reversal
detection alternative, consisting in placing the sample between two bimodal
squeezing transformations (two OPAs), such that the second squeezing is the
inverse of the first one. The precision in the estimation of the loss
parameter is obtained from the averaged total photon number and its variance
after the second transformation.

Time-reversal has been shown to increase the precision of estimation of
parameters characterizing unitary processes, beyond the classical limit, like
displacements [13, 14, 15, 16] and phases [16, 17, 18, 19, 20]. For absorption
estimation, time-reversal must be complemented by the use of ancilla. We show
that, for small absorption, the estimation obtained with this approach is very
close to the best possible precision, obtained from single-mode probes
prepared in a Fock state [11, 12], and it is superior to proposals based on a
single probe (no ancilla), prepared in a squeezed state [10]. It has the
further advantage that neither the input state nor the detection procedure
depends on the value of the parameter, which simplifies the experimental
realization and should motivate useful applications. We show that quantum
advantage is still present under moderate photon losses of the input beam.

Measuring the probe, after it undergoes the parameter-dependent dynamics,
leads to an estimation of the parameter through a function – an estimator –
that maps an experimental data set to a possible value of the parameter. There
are four basic questions that one would like to answer: (i) How to define the
precision of the estimate?; (ii) How to get the precision from the
experimental results?; (iii) What is the best initial state of the probe, in
order to get the best precision?; and (iv) What is the best measurement
procedure?

For unbiased estimations, the average of the estimator over a large number of
realizations of the measurement coincides with the true value of the
parameter. In this case, the precision of the estimation may be quantified by
the standard deviation of the measured values of the parameter with respect to
the average: $\Delta X=\sqrt{\langle X^{2}\rangle-\langle X\rangle^{2}}$.
Within the classical framework, a lower bound for the variance was obtained by
Cramér and Rao [21], and shown by Fisher [22] to be attainable when the
distribution of the possible values of the parameter is Gaussian, or when the
number of repetitions of the measurement is much larger than one. The Cramér-
Rao bound is expressed in terms of the Fisher information,

$F(X)=\sum_{j}\frac{1}{P_{j}(X)}\left[\frac{dP_{j}(X)}{dX}\right]^{2}\,,$ (1)

where $P_{j}(X)$ is the probability of getting an experimental result $j$ if
the value of the parameter is $X$. One has then $\Delta X\geq 1/\sqrt{{\cal
N}F(X)}$, where ${\cal N}$ is the number of independent measurements.

Generalization of this early work to quantum mechanics, through maximization
of $F(X)$ over all possible quantum measurements, leads to the inequality

$\Delta X\geq 1/\sqrt{{\cal N}{\cal F}_{Q}(X)}\,,$ (2)

where ${\cal F}_{Q}(X)$ is the quantum Fisher information (QFI). This relation
implies that the precision in the estimation of parameters can be increased
beyond the minimum uncertainty obtained by classical means, usually referred
to as the standard limit [1, 2, 23]. Quantum advantage has been proven for
estimations of displacements or rotations [24, 25, 26, 27, 15, 28, 16], phases
[29, 30, 31, 32, 33, 34, 35], electromagnetic fields [27, 28, 36, 14, 37],
damping and temperature [10, 11, 12], the gravitational field [38, 39, 40], or
yet the amount of squeezing of electromagnetic fields [41, 42], among many
other examples. More recently, interesting applications have been
demonstrated, among them gravimeters [43], accelerometers [44], gyroscopes
[45], magnetometers [46], high-resolution spectroscopy [47], detection of
gravitational waves [25, 48], and ultra-precise atomic clocks [49]. Quantum
metrology also concerns conceptual questions related to foundations of quantum
mechanics, as, for instance, the meaning of number-phase and energy-time
uncertainty relations [50], this last one being related to the quantum speed
limit [51, 52].

For noiseless quantum processes, with probe dynamics governed by unitary
evolution, and unbiased estimators, simple expressions are obtained for the
quantum Cramér-Rao bound. This is not so, however, for open systems, that is,
systems in the presence of an environment. Exact solutions can be found for
one or two qubits [3, 4], but for higher dimensions it is not possible, in
general, to find analytical solutions. Lower bounds for the variance can be
found through purification of the non–unitary dynamics, by adding an ad hoc
environment, such that the dynamics of the enlarged system is unitary and the
reduced dynamics, obtained by tracing out the added environment, coincides
with the original dynamics of the system [34, 36, 52, 53]. Lower bounds for
the precision have also been obtained via tools based on the geometry of
quantum channels and semi-definite programming [54]. Also, exact solutions for
the Cramér-Rao bound can be found for Gaussian systems [10, 55, 56, 57].

Parameter estimation is closely related to quantum channel identification,
that is, the distinguishability of quantum channels upon a change of one or
more of the parameters defining the channel [1]. In quantum information, a
quantum channel is a completely positive trace-preserving map between spaces
of operators, where a map $\Gamma$ acting on operators in a Hilbert space
${\cal H}_{1}$ is completely positive if the map $\Gamma\otimes{\bf 1}$ is
positive when acting on all possible extensions ${\cal H}_{1}\otimes{\cal
H}_{2}$ of ${\cal H}_{1}$. It is known that entanglement of the probe with an
ancilla, with the channel acting only on the probe, may improve parameter
estimation and the discrimination of quantum channels [3, 4, 58, 59, 60, 61,
62, 63, 64, 65]. The quantum advantage of this strategy is not universal, but
it was demonstrated in some specific examples. In particular, it does not hold
for unitary channels. Error correction, through the addition of multiple
ancillas, has also been shown to increase the precision of estimation [66, 67,
68].

Here we consider the quantum sensing of photon loss, due to absorption and
scattering by a material [10, 5, 7, 11, 12, 8, 9, 69]. It has direct
application to the estimation of the transmissivity of light by a sample,
especially for weak losses, and when low intensities are desirable, which may
be the case for fragile materials. Quantum metrology of absorption is also
important in absorption imaging. Reference [70] demonstrated sub-shot noise
quantum imaging using entangled photons produced by a down converter.
Reference [71] demonstrated that the use of an ancilla in quantum illumination
increases the signal-to-noise ratio beyond the classical value, in an
entanglement-breaking environment.

The absorption constant $\alpha$, to be estimated, is defined so that, if
$I_{0}$ and $I_{1}$ are the intensities of light before and after the
absorbing sample, then $I_{1}=(1-\alpha)I_{0}$.

While we concentrate here on the estimation of the absorption from a single-
mode probe, this strategy is of broader application. For instance, it can be
extended to the important spectroscopic technique that determines differential
absorption of two orthogonal polarized modes, which has recently been
investigated [73, 72].

Figure 1: Experimental setup for attaining a precision of estimation of an
absorption coefficient $\alpha$ equivalent to the one obtained by a Fock
state, using however a bimodal squeezed state produced by an optical
parametric amplifier as input. The estimation of $\alpha$ is obtained from the
joint photon counting of the two outgoing modes – probe plus ancilla.

##  2 Reaching the ultimate precision limit

The ultimate precision limit for the estimation of absorption of a sample in
the setup considered here is obtained through the quantum Fisher information
for the system probe + ancilla, which corresponds to the signal and idler
beams of a bimodal squeezed state, produced by an optical parametric amplifier
(OPA). Since the input state is Gaussian, the techniques used in [10, 55, 56,
57] can be applied to this case. This is done in the Supplementary Material
Sec. 1. Here, we adopt, however, a different procedure, which clarifies the
physical meaning of our results, and leads to the QFI derived in the
Supplementary Material.

The system to be considered is pictured in Fig. 1. The OPA implements a
bimodal squeezing transformation $\hat{S}(\xi)$ on a vacuum field [74],

$\hat{S}(\xi)=\exp(\xi\hat{a}^{\dagger}\hat{b}^{\dagger}-\xi^{*}\hat{a}\hat{b})\,$
(3)

leading to the squeezed state

$|\xi\rangle=\hat{S}(\xi)|0,0\rangle=\sum_{n=0}^{\infty}c_{n}|n,n\rangle\,,$
(4)

where $c_{n}=e^{in\varphi}(\tanh r)^{n}/\cosh r$. The average number of
photons in either of the two modes is $\langle n\rangle=\sinh^{2}r$.

The probability $p_{nm}$ of finding $n$ photons in the signal mode and $m$
photons in the idler mode, before the sample, is thus
$p_{nm}=\delta_{nm}|c_{n}|^{2}$, while the probability of finding $m$ photons
in the ancilla is $|c_{m}|^{2}$. After the sample is tested, the joint
probability is $p_{nm}=|c_{m}|^{2}p_{n}^{(m)}$, where $p_{n}^{(m)}$ is the
probability of counting $n$ photons in the output probe beam for the input of
$m$ photons. The corresponding Fisher information for the estimation of the
absorption $\alpha$ is then, according to Eq. (1), since only $p_{n}^{(m)}$
depends on $\alpha$,

$\displaystyle F(\alpha)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{m}\frac{1}{|c_{m}|^{2}p_{n}^{(m)}}\left[\frac{\partial|c_{m}|^{2}p_{n}^{(m)}}{\partial\alpha}\right]^{2}$
(5) $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}|c_{m}|^{2}\sum_{n=0}^{m}\frac{1}{p_{n}^{(m)}}\left[\frac{\partial
p_{n}^{(m)}}{\partial\alpha}\right]^{2}=\sum_{m=0}^{\infty}|c_{m}|^{2}F^{(m)}(\alpha)\,,$

where $F^{(m)}(\alpha)$ is the Fisher information for a $m$-photons Fock state
probing the absorbing sample, obtained through photon counting. One knows,
however, that photon counting actually leads to the quantum Fisher information
for a Fock state input [11, 12], so $F^{(m)}(\alpha)={\cal
F}_{Q}^{(m)}(\alpha)=m/[\alpha(1-\alpha)]$. Therefore, $F(\alpha)$ is a
weighted average of QFIs corresponding to Fock states:

$F(\alpha)={\sum_{m=0}^{\infty}|c_{m}|^{2}m\over\alpha(1-\alpha)}={\bar{n}\over\alpha(1-\alpha)}\,,$
(6)

since the sum in the first term on the right-hand side of the above equation
is the average number of photons $\bar{n}$ in either the probe or the ancilla,
before probing the sample. For the incoming two-mode squeezed state considered
here, $\bar{n}=\sinh^{2}r$. In Supplementary Material Eq. (S9), it is shown
that the Fisher information above coincides with the QFI of the probe +
ancilla output state. Therefore,

${\cal F}_{Q}(\alpha)={{\bar{n}\over\alpha(1-\alpha)}}\,.$ (7)

The Fisher information Eq. (7) coincides with the upper bound, derived in
[10], on the QFI for the estimation of absorption, for any single-mode quantum
state with mean photon number $\bar{n}$. This was done by replacing the
absorption medium by a beam splitter, with transmissivity equal to the
absorption coefficient, thus turning the open system dynamics into a unitary
one, involving the two modes of the beam splitter. The corresponding quantum
Fisher information should be an upper bound on the corresponding quantity for
the open system, since having access to the environment should result in
better precision of estimation of the absorption parameter [34]. Consequently,
no single mode quantum state with a mean photon number $\bar{n}$ can beat the
precision reached with a bimodal squeezed vacuum state with the same mean
photon number in the probe beam. This can be generalized to the system probe
plus ancilla considered here. If one replaces the absorbing medium by a beam
splitter, as done for the single-mode case, the system will have a unitary
evolution, and therefore the ancilla would not play any role: the upper bound
is the same as in the single-mode case! And it is reached by the the two-mode
squeezed state considered here, when the probe plus ancilla output is detected
through joint photon counting.

Eq. (7) leads to two important conclusions:

  * (i)

Joint photon counting on probe and ancilla is an optimal measurement, leading
to the QFI corresponding to the parameter $\alpha$;

  * (ii)

The QFI related to the bimodal squeezed input state coincides with a Fock
state QFI for which the photon number is replaced by $\bar{n}$.

The resulting bound for the precision $\Delta\alpha$ in the estimation is
given by Eq. (S2):

$\Delta\alpha=\sqrt{{\alpha(1-\alpha)\over\bar{n}}}\,,$ (8)

setting ${\cal N}=1$ in Eq. (S2).

As shown in [11, 12], Fock states lead to the best precision in the estimation
of the absorption, for a fixed photon number. This implies that, through the
use of an ancillary system, and for a given average photon number of photons
probing the sample, it is possible to achieve the best precision in the
estimation of the parameter $\alpha$, outperforming not only the Gaussian
states. This generalizes the result in [7], that a two-mode squeezed vacuum
maximizes QFI among all two-mode Gaussian states.

Figure 2: Uncertainty in the estimation of the absorption constant, for
$\alpha=0.05$. Comparison between the bound for the uncertainty $\Delta\alpha$
obtained from the quantum Fisher information for probe plus ancilla
corresponding to two modes of an incoming bimodal squeezed state (red line)
and the bound from the best single-mode Gaussian state [10] (pink dots), for
the same average number of photons testing the absorption. The green line
represents the standard limit, obtained for a single-mode coherent state
testing the sample. For the probe plus ancilla setup with $\bar{n}=10$ the
increase in precision from the standard limit is about 5 dB.

This interesting result stems from the perfect correlation of photon numbers
of probe and ancilla, as shown in Eq. (4). Although the bimodal squeezed state
describing the ancilla+probe system does not have a well-defined number of
photons, photon counting on the ancilla tells us that a Fock state with the
same number of photons probed the sample. The result of photon counting in the
probe then allows one to get information about the parameter $\alpha$ as if
the input state of the probe was a Fock state, leading to the best possible
precision in the estimation of $\alpha$. Several runs of joint photon counting
lead to quantum Fisher information of a Fock state for which the photon number
is replaced by $\bar{n}$.

###  2.1 Resilience of the joint photon counting procedure

The discussion above, stemming from the derivation of the Fisher information
in Eq. (5), leads to important consequences: Any state, pure or mixed, with
photon-number correlation between probe and ancilla could lead to Eq. (8). Not
even entanglement is needed. Indeed, the following mixed product state of
probe and ancilla,

$\hat{\rho}=\sum_{n}p_{n}|n,n\rangle\langle n,n|$ (9)

has the same QFI as a two-mode vacuum squeezed state with the same average
number of photons. This implies that the preparation of the input state via
OPA is resilient to phase noise. Furthermore, it also allows one to understand
and generalize numerical results published in [5], where it was shown that the
state $|\psi\rangle_{d}=(1/d)\sum_{k=1}^{d}|k,k\rangle$ with $3\leq d\leq 6$
has the same QFI as a two-mode vacuum squeezed state with the same average
number of photons, and that there are states with less entanglement than
$|\psi\rangle_{d}$ with similar performance.

###  2.2 Quantum advantage of the probe+ancilla setup

Fig. 2 compares the result for $\Delta\alpha$ obtained from the ancilla-based
QFI Eq. (7) with the one corresponding to the best single-mode Gaussian state
[10], which is a parameter-dependent squeezed and displaced vacuum state, for
the same average number of photons probing the sample. The bound obtained from
Eq. (6) prevails, as expected from a QFI for a Fock-state expression with
photon number equal to $\bar{n}=\sinh^{2}r$. This result is in conformity with
[5], which pointed out the non-optimal nature of single-mode Gaussian states.
One should note that both the input state and the detection procedure in the
probe plus ancilla setup do not depend on the (unknown) parameter to be
estimated, which is not the case of the procedure in [10].

In Fig. 2, these results are compared with the standard limit, which
corresponds to probing the sample with an incoming coherent state, with the
same average number of photons as in the previous setups, and measuring the
intensity of the field after its interaction with the sample. It can be
derived from the corresponding single-mode quantum Fisher information [10]:

$\Delta\alpha=\sqrt{{1-\alpha\over\bar{n}_{1}}}\,,$ (10)

where $\bar{n}_{1}$ is the incoming average number of photons probing the
sample. One should note that, for $\alpha\rightarrow 1$, that is, for strong
absorption, one has $\Delta\alpha\rightarrow 0$. This is also the limit of
vanishing outgoing intensity. Fig. 2 shows that, for $\alpha=0.05$ and
$\bar{n}=10$, the increase in precision from the standard limit is about 5 dB.
In the limit of strong absorption, $\alpha\rightarrow 1$, both quantum Fisher
information, for the single mode and the probe plus ancilla setups, converge
to the standard limit, expressing the environment-induced emergence of
classicality [75, 76]. This can be verified by comparing (8) and (10) when
$\alpha\rightarrow 1$.

##  3 Time-reversal strategy

One should note, however, that joint photon counting is challenging, with
present technologies. We show now that, for weak absorption, there is an
interesting and useful alternative, which does not rely on joint photon
counting, and involves a time-reversal detection scheme, illustrated in Fig.
3. It is based on a SU(1,1) interferometer [77, 78, 79, 80, 81, 82, 83, 84,
85, 86, 87, 88, 89, 90, 91], consisting of two OPAs, with the probed sample
between them. The first one generates a two-mode squeezed state from a vacuum
input, with the signal mode probing the sample, and the idler beam playing the
role of an ancilla. The second OPA reverses the transformation implemented by
the first one, so that, in the absence of photon losses, there is no outgoing
field. The time-reversed operation can be carried out in many different ways
[83, 84, 85]. The simplest is to have a $\pi$ phase difference between the
beams pumping the first and the second OPA. We assume that proper calibration
compensates for the difference in optical paths of both arms due to the
presence of the sample. Detection of the total number of outgoing photons
leads to the estimation of the absorption, defined, as before, by the constant
$\alpha$, so that, if $I_{1}$ and $I_{2}$ are the intensities of light in the
upper arm of the interferometer, before and after the absorbing medium, then
$I_{2}=(1-\alpha)I_{1}$. Fig. 3 displays the annihilation operators
corresponding to the electromagnetic fields in several regions of the device.
The relations between them are obtained from the squeezing transformations and
the absorption. Thus, from the first OPA, one has [74]

$\displaystyle\hat{a}_{1}=\hat{a}_{\rm in}\cosh\\!r+\hat{b}_{\rm
in}^{\dagger}e^{i\phi}\sinh\\!r\,,$ $\displaystyle\hat{b}_{1}=\hat{b}_{\rm
in}\cosh\\!r+\hat{a}_{\rm in}^{\dagger}e^{i\phi}\sinh\\!r\,,$ (11)

where $[\hat{a}_{\rm in},\hat{a}_{\rm in}^{\dagger}]=1$, $[\hat{b}_{\rm
in},\hat{b}_{\rm in}^{\dagger}]=1$ and the squeezing transformation is

$\hat{S}(\xi)=\exp(\xi\hat{a}^{\dagger}\hat{b}^{\dagger}-\xi^{*}\hat{a}\hat{b})\,,$
(12)

where $\xi=re^{i\phi}$ is the squeezing parameter, with
$\hat{a}_{1}=\hat{S}^{-1}\hat{a}_{\rm in}\hat{S}$,
$\hat{b}_{1}=\hat{S}^{-1}\hat{b}_{\rm in}\hat{S}$.

The second OPA applies the time-reversed transformation
$(\xi\rightarrow-\xi)$, resulting in the output operators (see Fig. 3):

$\displaystyle\hat{a}_{\rm
out}=\hat{a}_{2}\cosh\\!r-\hat{b}_{1}^{\dagger}e^{i\phi}\sinh\\!r\,,$
$\displaystyle\hat{b}_{\rm
out}=\hat{b}_{1}\cosh\\!r-\hat{a}_{2}^{\dagger}e^{i\phi}\sinh\\!r\,.$ (13)

Figure 3: Experimental SU(1,1) setup for squeezed vacuum time-reversal
metrology. The absorption medium is placed between two optical parametric
amplifiers (OPAs), on the upper arm of the interferometer. The first one, with
vacuum input, produces a two-mode squeezed state, the signal beam probing the
medium and the idler playing the role of an ancilla. The second OPA reverses
the squeezing transformation, so that in the absence of the absorption medium,
there is no outgoing field. Detection of the total number of outgoing photons
leads to the estimation of the photon-loss coefficient $\alpha$.

The photon loss, due to absorption and scattering, can be described by

$\hat{a}_{2}=\hat{a}_{1}\sqrt{1-\alpha}+\hat{c}\sqrt{\alpha}\,,$ (14)

where $\hat{c}$ stands for the annihilation operator corresponding to the
vacuum noise mode. The presence of $\hat{c}$ preserves the commutation
relation of the field operators:
$[\hat{a}_{2},\hat{a}^{\dagger}_{2}]=[\hat{a}_{1},\hat{a}^{\dagger}_{1}]=1$.

From Eq. (3), Eq. (3), and Eq. (14), it follows that

$\displaystyle\hat{a}_{\rm out}$ $\displaystyle=$ $\displaystyle\hat{a}_{\rm
in}\left(\cosh^{2}r\sqrt{1-\alpha}-\sinh^{2}r\right)$ $\displaystyle-$
$\displaystyle\hat{b}_{\rm
in}^{\dagger}e^{i\phi}\\!\sinh\\!r\cosh\\!r\\!\left(1-\sqrt{1-\alpha}\right)+\hat{c}\cosh\\!r\sqrt{\alpha}\,,$
$\displaystyle\hat{b}_{\rm out}$ $\displaystyle=$ $\displaystyle\hat{b}_{\rm
in}\left(\cosh^{2}r-\sinh^{2}r\sqrt{1-\alpha}\right)$ (15) $\displaystyle+$
$\displaystyle\hat{a}_{\rm
in}^{\dagger}e^{i\phi}\\!\sinh\\!r\cosh\\!r\\!\left(1-\sqrt{1-\alpha}\right)-\hat{c}^{\dagger}e^{i\phi}\sinh\\!r\sqrt{\alpha}\,.$

In the absence of the sample, it is easy to check that $\hat{a}_{\rm
out}=\hat{a}_{\rm in}$, $\hat{b}_{\rm out}=\hat{b}_{\rm in}$.

From Eq. (3), one gets the average total number of output photons:

$\displaystyle\bar{N}_{\rm out}$ $\displaystyle=$
$\displaystyle\langle\hat{a}_{\rm out}^{\dagger}\hat{a}_{\rm out}+\hat{b}_{\rm
out}^{\dagger}\hat{b}_{\rm
out}\rangle=2\sinh^{2}\\!r\cosh^{2}\\!r(1-\sqrt{1-\alpha})^{2}+\alpha\sinh^{2}r\,.$
(16)

The variance $\Delta^{2}N_{\rm out}$ is displayed in the Supplementary
Material Eq. (S16) [supmat].

We calculate $\Delta\alpha$ through the sensitivity, which can be related in
this case to photon number fluctuations:

$\Delta\alpha={\Delta N_{\rm out}\over\left|d\bar{N}_{\rm
out}/d\alpha\right|}\,,$ (17)

where

$\Delta^{2}N_{\rm out}=\langle(N_{\rm out}-\bar{N}_{\rm
out}\rangle)^{2}\rangle$ (18)

is the variance of the total number of photons.

Figure 4: Uncertainty bounds for probe plus ancilla QFI and SU(1,1) result.
The uncertainty bounds from the quantum Fisher information corresponding to
the probe and ancilla associated with the two modes of a bimodal squeezed
state (red curve), given by Eq. (6), and the one resulting from a sensitive
calculation for the total number of outgoing photons in the SU(1,1) setup
illustrated in Fig. 3 (dashed blue line). For weak absorption ($\alpha\ll 1$),
results are practically indistinguishable for the range of $\bar{n}$
considered here.

From these expressions, the sensitivity can be calculated. Details are given
in the Supplementary Material Eqs. (S17-S19). The corresponding uncertainty is
plotted in Fig. 4, and compared with the one obtained from the QFI in Eq. (7).
For weak absorption, the result obtained from the time-reversal procedure is
practically indistinguishable from the probe plus ancilla quantum Fisher
information bound.

Since only the measurements of the total output photon number and its variance
are needed here, they can be obtained through measurement of the intensity of
photocurrents produced by the output fields and their cross-correlations,
which does not require reconstructing the photon-number distribution [92].

In the next section, we demonstrate another advantage of this method: the
resilience to moderate photon losses of the incoming probe plus the ancilla
beam.

##  4 Resilience of time-reversal to extra photon losses

Figure 5: (a). The impurity of the two-mode squeezed state generated by
general optical parametric amplifiers is modeled by adding additional loss to
the two-mode squeezed vacuum before passing through the sample $\alpha$. (b).
For comparison, the corresponding sensing protocol with a coherent beam is
shown. Figure 6: The ratio between the quantum Fisher Informations
corresponding to a two-mode squeezed vacuum (2MSV) and to a coherent state,
$QFI_{2MSV}/QFI_{coh}$, in dB, for $\alpha=0.05$ and $\bar{n}=25$, as a
function of the impurity loss $\alpha_{0}$.

General optical parametric amplifiers (OPAs) can produce impure entanglement
due to various factors, leading to a reduction in the quality of the generated
entanglement. We have already shown that the joint photon counting procedure
is not affected if the input state becomes a mixture of photon-correlated
probe plus ancilla states. We consider now the effect of extra photon losses.

It is worth noting that while losses can introduce impurities in the
entanglement produced by general OPAs, it is still possible to have quantum
gain for the joint photon counting and the time-reversal setup. In Fig. 5,
additional loss is introduced to both modes of the two-mode squeezed vacuum
before it passes through the sample. For simplicity, we assume the degree of
loss $\alpha_{0}$ to be the same for both modes. The calculations are similar
to those in the Supplementary Material Sec. 2.

As shown in Fig. 6, where the QFI for estimation of $\alpha$ corresponding to
the two-mode squeezed state is compared to the QFI for a single-mode coherent
state input with the same additional loss $\alpha_{0}$, quantum advantage
persists even after undergoing significant loss (advantage of 3 dB with
$\alpha_{0}\sim 5\alpha$). This actually refers to the joint photon counting
procedure, since the second OPA does not change the QFI.

The resilience of the time-reversal procedure to noise is illustrated in Fig.
7. Even for extra noise equal to the sample absorption constant, there is
still significant advantage, as compared to the estimation corresponding to a
coherent state input, subject to the same extra noise. This is an important
and useful property of the time-reversal strategy proposed here.

Figure 7: Resilience of the time-reversal procedure: Precision $\Delta\alpha$
in the estimation of the sample absorption constant as a function of the extra
loss $\alpha_{0}$ in both probe and conjugate beams, as in Fig. 5(a). For
comparison, the standard quantum limit, corresponding to a single-mode
coherent state input, as in Fig. 5(b), is also shown. The top scale is the
number of photons thru absorber, modified by extra loss. The bottom scale is
the number of photons produced by the OPA in each beam.

##  5 Discussion

Optimal quantum sensing of open-system dynamics may require strategies that
differ markedly from those applied to lossless systems. Entanglement of the
probe with an ancilla may enhance the precision of estimation, even though the
ancilla does not interact with the parameter-dependent system, a property that
is absent for unitary dynamics. Here we have considered the estimation of
photon loss for a light beam propagating in a sample or, more generally, of
the loss coefficient in a bosonic channel, due to absorption or scattering by
the sample.

The probe and ancilla are the modes of a bimodal squeezed state, produced by
an optical parametric amplifier. We calculated the corresponding quantum
Fisher information for estimation of the absorption coefficient and showed
that the respective uncertainty bound coincides with the one for a Fock state,
being saturated by joint photon counting for the outgoing probe and ancilla, a
detection procedure that does not depend on the value of the parameter. The
ancilla strategy benefits therefore from the extreme precision associated with
Fock states of the probe, surpassing the sensing obtained with the best
parameter-dependent single-mode Gaussian state [10], while overcoming the
challenge of producing high photon-number eigenstates.

Joint photon counting is however challenging, with present technologies [6].
We have therefore described an approach that does not require this procedure.
It is based on the conjunction of the ancilla strategy with a time-reversal
strategy, implemented with a SU(1,1) interferometer consisting of two optical
parametric amplifiers, so that the first one generates a two-mode squeezed
state, corresponding to the probe and the ancilla, and the second one undoes
the squeezing produced by the first one, after the probe has interacted with
the sample. The addition of the second optical parametric amplifier does not
change the quantum Fisher information, which is invariant under unitary
transformations. An estimation based on a sensitivity relation for the total
number of outgoing photons and its variance leads to an uncertainty in the
absorption coefficient with a precision that is, for weak absorption,
practically indistinguishable from the bound obtained from the quantum Fisher
information for the probe plus ancilla system. This detection setup is also
independent of the (unknown) parameter to be estimated, implying that phase
stabilization and mode-matching for the two optical parametric amplifiers can
be done once and for all: the device is then ready to be used, independently
of the value of the parameter. Interestingly, quantum gain is still achieved
for this protocol under moderate photon losses of the probe plus ancilla input
beam, even for losses compared to the estimated absorption parameter.

We note that previously several papers [6, 7, 8, 9, 69] have reported on the
quantum metrology of absorption. In particular, the ancilla strategy with
spontaneous down-converted photons was demonstrated in [8, 9], by using an
adjustable estimator, with parameters that must be experimentally determined
in order to minimize the uncertainty and make the estimator unbiased. This
procedure is circumvented by our approach, since the joint photon counting
leads to the best precision, given by the quantum Fisher information, without
any further adjustment, and the same is valid, in the regime of weak
absorption, for the time-reversal strategy. A general approach, related to the
analysis made here, was presented in [6], which considered the technical
challenges involved in joint photon counting, restricted however to pure
states and a noiseless setup. These contributions are not based on a SU(1, 1)
setup, which, as shown here, leads, for weak absorption, to the precision
obtained from the quantum Fisher information, is resilient to extra photon
losses, and avoids technical challenges associated to joint photon counting.

The quantum advantage in the precision of estimation obtained with the ancilla
and time-reversal strategy, demonstrated here, relies on available technology
and opens the way to the increase in precision of a diversity of metrological
tasks involving open systems.

## Supplementary Material 1. quantum Fisher information of absorption constant
for the time-reversed ancilla-based metrology

We note that the two-mode squeezed state has a Gaussian Wigner function. It
can be shown that if one of the modes goes through an absorber, the output is
still a Gaussian Wigner function, though it corresponds to a mixed state of
the output field [93]. The general method for calculating the quantum Fisher
information (QFI) for a Gaussian system has been developed by [94, 95, 96, 97,
98, 99, 100, 101, 102]. We apply specifically the method in [102] to obtain
the quantum Fisher information of the system described by Fig. 1. For a two-
mode bosonic system, we may define a vector of annihilation and creation
operators given by

$\displaystyle\hat{A}=\left(a_{1},a_{2},a_{1}^{\dagger},a_{2}^{\dagger}\right)^{T}\,.$
(S1)

Gaussian states can be fully characterized by their first moments (the
displacement vector) $d_{m}=\mathrm{tr}\big{[}\rho{\hat{A}}_{m}\big{]}$ and
the second moments (the covariance matrix)
$\sigma_{mn}=\mathrm{tr}\big{[}\hat{\rho}\,\\{{\Delta\hat{A}}_{m},{\Delta\hat{A}}_{n}^{\dagger}\\}\big{]}$,
where ${\Delta\hat{A}}:={\hat{A}}-d$. The subscripts $m$ and $n$ stand for the
components of the vector defined in Eq. (S1). The QFI is given by

$\displaystyle{\cal F}_{Q}(\alpha)=lim_{v\rightarrow
1}\frac{1}{2}vec(\frac{\partial\sigma}{\partial\alpha})^{\dagger}M^{-1}vec(\frac{\partial\sigma}{\partial\alpha})\,,$
(S2)

where the matrix $M$ can be expressed by

$\displaystyle M=v^{2}\bar{\sigma}\otimes\sigma-K\otimes K\,,$ (S3)

for the system considered here, with a zero displacement vector. Other
notations in Eq. (S2) include the symplectic form $K=diag(1,1,-1,-1)$, and the
operator $vec(\Lambda)$. Applying $vec(\Lambda)$ on a matrix
$\Lambda=\left(\Lambda_{1},\Lambda_{2}\right)$ will transform it to a vector
$vec(\Lambda)=\left(\Lambda_{1}^{T},\Lambda_{2}^{T}\right)^{T}$. Replacing the
operators $a_{1}$ and $b_{1}$ in Eq. (S1) by $\hat{a}_{\rm out}$ and
$\hat{b}_{\rm out}$, we obtain

$\displaystyle\sigma=\left(\begin{array}[]{cccc}\left\langle\hat{a}_{o}\hat{a}_{o}^{\dagger}+\hat{a}_{o}^{\dagger}\hat{a}_{o}\right\rangle&2\left\langle\hat{a}_{o}\hat{b}_{o}^{\dagger}\right\rangle&2\left\langle\hat{a}_{o}\hat{a}_{o}\right\rangle&2\left\langle\hat{a}_{o}\hat{b}_{o}\right\rangle\\\
2\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}\right\rangle&\left\langle\hat{b}_{o}\hat{b}_{o}^{\dagger}+\hat{b}_{o}^{\dagger}b_{o}\right\rangle&2\left\langle\hat{a}_{o}\hat{b}_{o}\right\rangle&2\left\langle\hat{b}_{o}\hat{b}_{o}\right\rangle\\\
2\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}^{\dagger}\right\rangle&2\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle&\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}+\hat{a}_{o}\hat{a}_{o}^{\dagger}\right\rangle&2\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}\right\rangle\\\
2\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle&2\left\langle\hat{b}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle&2\left\langle\hat{a}_{o}\hat{b}_{o}^{\dagger}\right\rangle&\left\langle\hat{b}_{o}^{\dagger}b_{o}+\hat{b}_{o}\hat{b}_{o}^{\dagger}\right\rangle\end{array}\right)\,,$
(S8)

where for brevity, we note $\hat{a}_{\rm out}$ and $\hat{b}_{\rm out}$ by
$\hat{a}_{\rm o}$ and $\hat{b}_{\rm o}$. These are given by

$\displaystyle\hat{a}_{o}$
$\displaystyle=\hat{a}_{1}\sqrt{1-\alpha}+\hat{c}\sqrt{\alpha}\,,$
$\displaystyle\hat{b}_{o}$ $\displaystyle=\hat{b}_{1},$ (S9)

where $\hat{a}_{1}$ and $\hat{b}_{1}$ are given by Eq. (11). In the system
considered here, many of the off-diagonal terms are zero

$\displaystyle\left\langle\hat{a}_{o}\hat{a}_{o}\right\rangle$
$\displaystyle=\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}^{\dagger}\right\rangle=\left\langle\hat{b}_{o}\hat{b}_{o}\right\rangle=\left\langle\hat{b}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle$
$\displaystyle=\left\langle\hat{a}_{o}\hat{b}_{o}^{\dagger}\right\rangle=\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}\right\rangle=0\,.$
(S10)

Thus, we obtain

$\displaystyle\sigma=\left(\begin{array}[]{cccc}2\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}\right\rangle+1&0&0&2\left\langle\hat{a}_{o}\hat{b}_{o}\right\rangle\\\
0&2\left\langle\hat{b}_{o}^{\dagger}b_{o}\right\rangle+1&2\left\langle\hat{a}_{o}\hat{b}_{o}\right\rangle&0\\\
0&2\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle&2\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}\right\rangle+1&0\\\
2\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle&0&0&2\left\langle\hat{b}_{o}^{\dagger}b_{o}\right\rangle+1\end{array}\right)\,,$
(S15)

where

$\displaystyle\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}\right\rangle$
$\displaystyle=(1-\alpha)\sinh^{2}r\,,$
$\displaystyle\left\langle\hat{b}_{o}^{\dagger}b_{o}\right\rangle$
$\displaystyle=\sinh^{2}r\,,$
$\displaystyle\left\langle\hat{a}_{o}\hat{b}_{o}\right\rangle$
$\displaystyle=\left\langle\hat{b}_{o}\hat{a}_{o}\right\rangle=e^{i\phi}\sinh
r\cosh r\sqrt{1-\alpha}\,,$
$\displaystyle\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle$
$\displaystyle=\left\langle\hat{b}_{o}^{\dagger}\hat{a}_{o}^{\dagger}\right\rangle=e^{-i\phi}\sinh
r\cosh r\sqrt{1-\alpha}\,.$

From Eq. (S2), Eq. (Supplementary Material 1. quantum Fisher information of
absorption constant for the time-reversed ancilla-based metrology), and Eq.
(S15), we obtain the quantum Fisher information corresponding to the
absorption constant $\alpha$, for the ancilla-based metrology,

$\displaystyle{\cal F}_{Q}(\alpha)=\frac{\sinh^{2}r}{\alpha(1-\alpha)}\,,$
(S17)

which coincides with Eq. (6) of the main text. We note that this result can be
derived from the quantum Fisher information obtained in [5], where it was
shown that input two-mode squeezed states outperform any other class of
Gaussian states, for the estimation of the time-dependent parameter
$\gamma=\Gamma t$, where $\Gamma$ is the coupling of the channel to a thermal
reservoir. The connection with Fock states, through the expression Eq. (S17),
was not discussed there, as well as the resilience of these states against
additional noise, which are direct consequences of our derivation method in
the main text. Since the time-reversal procedure is a unitary transformation,
which does not change the quantum Fisher information, Eq. (S17) also applies
to the time-reversed ancilla system. This can be checked explicitly for the
arrangement of Fig. 3 for which $\hat{a}_{\rm out}$ and $\hat{b}_{\rm out}$
are given by Eq. (15). We can derive

$\displaystyle\left\langle\hat{a}_{o}^{\dagger}\hat{a}_{o}\right\rangle$
$\displaystyle=\sinh^{2}r\cosh^{2}r\left(1-\sqrt{1-\alpha}\right)^{2}\,,$
$\displaystyle\left\langle\hat{b}_{o}^{\dagger}b_{o}\right\rangle$
$\displaystyle=\sinh^{2}r\cosh^{2}r\left(1-\sqrt{1-\alpha}\right)^{2}+\alpha\sinh^{2}r\,,$
$\displaystyle\left\langle\hat{a}_{o}\hat{b}_{o}\right\rangle$
$\displaystyle=\left\langle\hat{b}_{o}\hat{a}_{o}\right\rangle=-e^{i\phi}\sinh
r\cosh
r\left(1-\sqrt{1-\alpha}\right)\left(\cosh^{2}r-\sinh^{2}r\sqrt{1-\alpha}\right)\,,$
$\displaystyle\left\langle\hat{a}_{o}^{\dagger}\hat{b}_{o}^{\dagger}\right\rangle$
$\displaystyle=-e^{-i\phi}\sinh r\cosh
r\left(1-\sqrt{1-\alpha}\right)\left(\cosh^{2}r-\sinh^{2}r\sqrt{1-\alpha}\right)\,.$

On substituting Eq. (Supplementary Material 1. quantum Fisher information of
absorption constant for the time-reversed ancilla-based metrology) in Eq.
(S15) and using Eq. (S2) and Eq. (S3), we do obtain Eq. (S17).

## Supplementary Material 2. sensitivity for the time-reversed scheme

Here we present the expressions needed for the evaluation of $\Delta\alpha$ in
Eq. (17). For simplicity, we express Eq. (15) in the main text as

$\displaystyle\hat{a}_{\rm out}$ $\displaystyle=c_{11}\hat{a}_{\rm
in}+c_{12}\hat{b}_{\rm in}^{\dagger}+c_{13}\hat{c}\,,$
$\displaystyle\hat{b}_{\rm out}$ $\displaystyle=c_{21}\hat{a}_{\rm
in}^{\dagger}+c_{22}\hat{b}_{\rm in}+c_{23}\hat{c}^{\dagger}\,,$ (S19)

where $c_{11}$, $c_{12}$, $c_{13}$, $c_{21}$, $c_{22}$, and $c_{23}$ are given
by

$\displaystyle c_{11}$ $\displaystyle=\sqrt{1-\alpha}\cosh^{2}r-\sinh^{2}r\,,$
$\displaystyle c_{12}$ $\displaystyle=-(1-\sqrt{1-\alpha})e^{i\phi}\cosh
r\sinh r\,,$ $\displaystyle c_{13}$ $\displaystyle=\cosh r\sqrt{\alpha}\,,$
$\displaystyle c_{21}$ $\displaystyle=(1-\sqrt{1-\alpha})e^{i\phi}\cosh r\sinh
r\,,$ $\displaystyle c_{22}$
$\displaystyle=\cosh^{2}r-\sqrt{1-\alpha}\sinh^{2}r\,,$ $\displaystyle c_{23}$
$\displaystyle=-e^{i\phi}\sinh r\sqrt{\alpha}\,.$

We obtain the average total number of output photons

$\displaystyle\bar{N}_{\rm out}$ $\displaystyle=\langle\hat{a}_{\rm
out}^{\dagger}\hat{a}_{\rm out}+\hat{b}_{\rm out}^{\dagger}\hat{b}_{\rm
out}\rangle=|c_{12}|^{2}+|c_{21}|^{2}+|c_{23}|^{2}\,,$ (S21)

and its variance

$\displaystyle\Delta^{2}N_{\rm out}=$ $\displaystyle\langle(N_{\rm
out}-\bar{N}_{\rm out}\rangle)^{2}\rangle$ $\displaystyle=$
$\displaystyle|c_{12}|^{2}(|c_{11}|^{2}+|c_{13}|^{2}+|c_{22}|^{2})+|c_{22}|^{2}|c_{23}|^{2}+2|c_{12}c_{22}(c_{11}c_{21}+c_{13}c_{23})|\,.$
(S22)

Defining $\eta:=(1-\sqrt{1-\alpha})$, we obtain

$\frac{d\bar{N}_{\rm
out}}{d\alpha}=2\cosh^{2}r\sinh^{2}r\frac{\eta}{1-\eta}+\sinh^{2}r\,,$ (S23)

and

$\displaystyle\Delta^{2}N_{\rm out}$
$\displaystyle=\eta^{2}\cosh^{2}r(2\eta-1+3\eta^{2}\cosh^{2}r\sinh^{2}r)$
$\displaystyle+2\alpha\eta\cosh^{2}r(1+\eta\sinh^{2}r)+(1+\eta\sinh^{2}r)^{2}(2\eta\cosh^{2}r-\alpha\sinh^{2}r)\,.$
(S24)

From these expressions, the sensitivity can be calculated.

$\Delta\alpha={\sqrt{A}\over B}\,,$ (S25)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\eta^{2}\left(\bar{n}+1\right)\left[2\eta-1+3\eta^{2}(\bar{n}+1)\bar{n}\right]$
(S26) $\displaystyle+$ $\displaystyle
2\alpha\eta\left(\bar{n}+1\right)\left(1+\eta\bar{n}\right)+\left(1+\eta\bar{n}\right)^{2}\left[2\eta(\bar{n}+1)-\alpha\bar{n}\right],$

and

$B=\sqrt{\bar{n}}\left[1+\frac{2\eta}{\sqrt{1-\alpha}}\left(\bar{n}+1\right)\right],$
(S27)

in terms of the average number of photons interacting with the sample,
$\bar{n}=\langle\hat{a}_{1}^{\dagger}\hat{a}_{1}\rangle=\sinh^{2}r$, and
$\eta:=(1-\sqrt{1-\alpha})$. For small values of $\alpha$ such that $\alpha\ll
1$ and $\alpha\bar{n}\lesssim 1$, Eq. (S25) can be expanded to

$\Delta\alpha=\frac{\sqrt{\alpha}[1+\alpha
n+\frac{1}{8}\alpha^{3}n^{3}+\frac{1}{2}\alpha(1+\frac{1}{2}\alpha
n+\frac{1}{2}\alpha^{2}n^{2})]}{\sqrt{\bar{n}}\left[1+\alpha\bar{n}+\alpha\left(1+\frac{3}{4}\alpha\bar{n}\right)\right]},$
(S28)

where the leading term is $\Delta\alpha\sim\sqrt{\alpha/\bar{n}}$, showing
that the SU(1, 1) sensitivity estimate goes over the result in Eq. (8), for
$\alpha\ll 1$.

Acknowledgments G.S.A and J.W. are grateful for the support of Air Force
Office of Scientific Research (Award No. FA-9550-20-1-0366) and the Robert A.
Welch Foundation (A-1943-20210327). R.L.M.F. and L.D. acknowledge the support
of the Brazilian agencies CNPq, CAPES, and the Rio de Janeiro State Foundation
for Research Support (FAPERJ). R. L. M. F. acknowledges the support of the
John Templeton Foundation (Grant 62424).

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