# Quantum-Enhanced Transmittance Sensing

Zihao Gong, , Nathaniel Rodriguez, , Christos N. Gagatsos, Saikat Guha, , and
Boulat A. Bash This paper was presented in part at the 2021 International
Symposium on Information Theory (ISIT 2021) [1].ZG and BAB were supported by
the National Science Foundation (NSF) under Grant No. CCF-2045530. NR was
supported by the NSF under Grant Nos. EEC-1941583 and CCF-2006679. CNG and SG
acknowledge funding support from the Office of Naval Research, grant number
N00014-19-1-2189.Zihao Gong and Nathaniel Rodriguez are with Electrical and
Computer Engineering Department, The University of Arizona, Tucson, AZ 85721
USA (email<EMAIL_ADDRESS>and
nrod968@email.arizona.edu).Christos N. Gagatsos is with the College of Optical
Sciences, The University of Arizona, Tucson, AZ 85721 USA (email:
cgagatsos@arizona.edu).Saikat Guha and Boulat A. Bash are with the Electrical
and Computer Engineering Department, The University of Arizona, Tucson, AZ
85721 USA, and also with the College of Optical Sciences, The University of
Arizona, Tucson, AZ 85721 USA (email<EMAIL_ADDRESS>and
boulat@arizona.edu).

###### Abstract

We consider the problem of estimating unknown transmittance $\theta$ of a
target bathed in thermal background light. As quantum estimation theory yields
the fundamental limits, we employ the lossy thermal-noise bosonic channel
model, which describes sensor-target interaction quantum mechanically in many
practical active-illumination systems (e.g., using emissions at optical,
microwave, or radio frequencies). We prove that quantum illumination using
two-mode squeezed vacuum (TMSV) states asymptotically achieves minimal quantum
Cramér-Rao bound (CRB) over all quantum states (not necessarily Gaussian) in
the limit of low transmitted power. We characterize the optimal receiver
structure for TMSV input, and show its advantage over other receivers using
both analysis and Monte Carlo simulation.

##  I Introduction

A precise measurement of power transmittance is a fundamental task in
engineering. It translates to measuring both target reflectance in active
sensing systems, such as RADAR and LIDAR, and signal distortion from
attenuation in communications systems. Transmittance is also critical to
quantum-system design. It determines the precision of quantum methods for
phase estimation [2, 3], the point-to-point quantum-communication capacity
[4], and whether a quantum channel preserves the entanglement [5].

The importance of measuring transmittance led to the development of classical
signal processing methods covering many practical scenarios [6, 7]. However,
the fundamental precision limits for all sensing tasks as well as the
approaches to achieve these limits are governed by quantum information theory
[8, 9, 10, 11]. As we briefly discuss in Section II-C, this is because quantum
information methods optimize the underlying physical measurement process that
generates the data for the estimator, as well as the estimator itself. Indeed,
quantum-enhanced sensing systems can significantly outperform those limited by
classical methodology [12, 13].

Consider active sensing of target reflectance, using optical, microwave, or
radio-frequency emissions in the presence of background Gaussian noise. This
task is modeled quantum mechanically by the measurement of power transmittance
of a lossy thermal-noise bosonic channel. Despite the progress in quantum
transmittance sensing [14, 15, 16, 17, 18, 19, 20], outlined briefly in
Section II-D, a design of a sensor transceiver that attains the quantum limit
in the presence of environmental thermal noise, has been elusive. In fact, the
authors of [18], upon establishing the fundamental lower bound on the mean
squared error (MSE) of quantum transmittance estimation, question whether a
thermal-noise scenario even exists where this bound is saturated. Here, we
answer this question affirmatively: the lower bound is achievable for probes
with low transmitted power. Furthermore, we characterize the corresponding
transceiver and provide analysis, and simulations supporting its near-term
physical implementation.

We begin by describing in Section II our notation and the lossy thermal-noise
bosonic-channel model. We then cover the basics of quantum estimation theory.
This allows us to introduce the quantum perspective on the transmittance-
sensing problem and to consider the use of _quantum illumination_ (QI) [21,
22, 23, 24, 25, 26, 18], which, in general, improves precision by using
entanglement between the transmitted probe and a reference state retained in
the transceiver. In Section III we prove that probes constructed from two-mode
squeezed vacuum (TMSV) states can achieve the ultimate bound in the limit of
low transmitted power. As was done previously for quantum-enhanced target
detection [21, 22, 23, 24, 26, 18], our transmitter generates $n$ TMSV states,
and probes the target’s transmittance with one mode of each TMSV state, while
retaining the other mode as a reference. In Section IV, we characterize a
matching quantum receiver, that measures the $n$ returned probes and
corresponding entangled reference signals, and applies maximum likelihood
estimation (MLE) on the resulting classical measurements. In the limit of low
transmitted power and large $n$ this transceiver achieves the fundamental
lower bound on MSE from [18]. Although they are not classical, the components
used in our receiver are well-known to the optics community: a two-mode
squeezer followed by a photon-number-resolving (PNR) measurement. However,
despite this convenience, our receiver’s existence is limited to certain
ranges of system parameters: transmittance, signal power, and thermal noise
power. Thus, in Section V, we compare its theoretical limits to those of other
well-known receivers, and show significant advantage derived from using TMSV
input and our receiver.

The MSE of our sensing system converges to the quantum lower bound as the
number of probes $n\to\infty$. However, practical sensing is limited to a
finite number of probes: $n<\infty$. This motivates evaluating the speed of
convergence to the limit. Further complicating the analysis is the dependence
of our receiver’s structure on the parameter of interest, transmittance. This
is common in quantum estimation, and is addressed using the two-stage method
from [11, Ch. 6.4], [27, 28]. Unfortunately, this complicates the analysis. In
Section VI we use Monte Carlo simulation to study the performance of
transmittance sensing using our sensor, and two other receivers, when the
number of probes is finite. We show that the MSEs for these systems converge
rapidly to their corresponding quantum limits, even when the two-stage method
is used.

The transceiver derived here is optimal in the low-transmitted-power regime.
This corresponds to previous QI results, which demonstrated quantum advantage
in this regime [21, 22, 23, 25, 26, 18]. Nevertheless, this has significant
practical applications to sensors operating under total transmitted power
constraints. These can be imposed by, e.g., covertness/low probability of
detection (LPD) [29, 30, 31, 32, 33, 34, 35, 36], battery size, or the need to
limit exposure of a biological sample to light. Although technically
challenging, an experimental validation of our design is feasible in the near
term, as the required squeezing and PNR measurement have been demonstrated. We
conclude with a discussion of future work in Section VII.

_Note:_ The fundamental limits of transmittance estimation using TMSV in
Section III and the preliminary comparison to other systems in Section V were
presented at the 2021 International Symposium on Information Theory (ISIT
2021) and published in its proceedings [1]. Receiver design and simulations in
Sections IV and VI are new, while Section V is significantly expanded from
[1].

##  II Prerequisites

###  II-A Notation

As is customary in quantum information theory, we indicate operators with
hats, e.g., $\hat{a}$ and $\hat{\rho}$. Conjugate transpose and trace of
$\hat{a}$ are indicated by $\hat{a}^{\dagger}$ and
$\operatorname{Tr}\left\\{\hat{a}\right\\}$. Density operators, which describe
quantum states, are denoted using Greek letters, e.g., $\hat{\rho}$ and
$\hat{\sigma}$. We employ ket $\left|x\right>$ and bra
$\left<x\right|\equiv\left|x\right>^{\dagger}$ notation for pure quantum
states. A particularly useful pure state is a Fock, or photon number, state
$\left|n\right>$ which represents exactly $n$ photons. We employ caret to
indicate classical estimators that input measured data and output an estimate,
e.g., $\check{\theta}$ is an estimator for $\theta$. We denote mean quantities
with an overbar, e.g., $\bar{n}$ often indicates mean photon number.

###  II-B Channel Model

A classical linear channel relates the complex-valued input and output
amplitude mode functions $a$ and $b$ by $b=\sqrt{\theta}a+z$, where $\theta$
is power transmittance and $z$ is additive noise. Using an independent
Gaussian random variable to represent noise, $z$, yields an additive white
Gaussian noise (AWGN) channel, and simplifies estimation of transmittance
$\theta$ [6, 7]. However, optical, microwave, and radio-frequency (RF) light
used for communications and sensing is fundamentally an electromagnetic wave
described quantum mechanically by a boson field. Noises of quantum-mechanical
origin limit the performance of advanced high-sensitivity photodetection
systems [37, 38, 39]. Therefore, achieving the ultimate limits of estimation
generally requires tools from quantum-information processing [8, 9, 10, 11]
and quantum optics [40, 41, 42].

Formally, a single-mode lossy thermal-noise bosonic channel
$\mathcal{E}^{\bar{n}_{\rm T},\theta}_{S\rightarrow R}$ in Fig. 2 describes
quantum mechanically the transmission of a single (spatio-temporal-
polarization) mode of the electromagnetic field at a given transmission
wavelength (such as optical, microwave, RF) over linear loss and additive
Gaussian noise (such as noise stemming from blackbody radiation). A
beamsplitter with transmittance $\theta$ models power loss. In contrast to the
classical linear model, here the input-output relationship
$\hat{b}=\sqrt{\theta}\hat{a}+\sqrt{1-\theta}\hat{e}$ between the bosonic
modal annihilation operators of the single-mode channel requires the
environment mode $\hat{e}$ to preserve the unitarity by ensuring that
commutator $\left[\hat{b},\hat{b}^{\dagger}\right]=\hat{I}$, where $\hat{I}$
is the identity operator. Excess noise is modeled by mode $\hat{e}$ being in a
zero-mean thermal state $\hat{\tau}_{\bar{n}_{\rm T}}$, where $\bar{n}_{\rm
T}$ is the mean photon number per mode injected by the environment. Thermal
state is represented in Fock (photon number) basis as
$\hat{\tau}_{\bar{n}_{\rm T}}=\sum_{k=0}^{\infty}q\left(k;\bar{n}_{\rm
T}\right)\left|k\right>\left<k\right|$, where the Bose-Einstein probability
mass function

$\displaystyle q(k;\bar{n})$
$\displaystyle=\frac{\bar{n}^{k}}{(1+\bar{n})^{k+1}}$ (1)

is a variant of a geometric distribution. In this paper, we are interested in
estimating unknown transmittance $\theta$. Before we state our problem
formally in Section II-D, we review the concepts from quantum estimation
theory that we require.

Figure 1: Single-mode thermal-noise lossy bosonic channel
$\mathcal{E}^{(\bar{n}_{\rm T},\theta)}$ modeled by a beamsplitter with
transmittance $\theta$. The sensor injects a probe in one of the input ports,
while an environment injects a thermal state $\hat{\tau}_{\bar{n}_{\rm T}}$
with mean photon number $\bar{n}_{\rm T}$ in the other input port. The sensor
has access to one of the output ports while the other output is absorbed by
the environment. $\hat{a}$, $\hat{e}$, $\hat{b}$, and $\hat{c}$ label the
corresponding input/output modal annihilation operators.

###  II-C Introduction to Quantum Estimation

Suppose a quantum state $\hat{\sigma}(\theta)$ physically encodes information
about an unknown parameter $\theta$ (transmittance in this paper). We are
interested in estimating $\theta$ from $\hat{\sigma}(\theta)$. A physical
device that extracts information about $\theta$ from $\hat{\sigma}(\theta)$ is
described by a positive operator-valued measure (POVM)
$\left\\{\hat{\Gamma}_{x}\right\\}$ that satisfies the non-negativity and
completeness conditions:

$\displaystyle\forall x:\hat{\Gamma}_{x}\geq 0,\ \
\sum_{x}\hat{\Gamma}_{x}=\hat{I},$ (2)

where $\hat{I}$ is the identity operator. For example, an ideal photon-number-
resolving (PNR) measurement is described by
$\left\\{\left|x\right>\left<x\right|\right\\}$, where $\left|x\right>$ is the
Fock (photon number) state. Classical statistics of an output of a device
characterized by POVM $\left\\{\hat{\Gamma}_{x}\right\\}$ are described by a
random variable $X$ with probability mass function
$p_{X}(x;\theta)=\operatorname{Tr}\left\\{\hat{\Gamma}_{x}\hat{\sigma}(\theta)\right\\}$
[8, Ch. III].

An estimator $\check{\theta}(x)$, is a function of the observation $x$, which
is an instance of $X$. We desire an unbiased $\check{\theta}(x)$, i.e.,
$E_{X}\left[\check{\theta}(X)\right]=\theta_{0}$, that minimizes the mean
square error (MSE)
$V_{\theta_{0}}\left(\check{\theta}\right)=E_{X}\left[\left(\check{\theta}(X)-\theta_{0}\right)^{2}\right]$,
where $\theta_{0}$ is the true value of $\theta$ and $E_{X}[f(X)]$ is the
expected value of $f(X)$. The classical Cramér-Rao bound (CRB) lower bounds
the MSE:

$\displaystyle V_{\theta_{0}}\left(\check{\theta}\right)$
$\displaystyle\geq\frac{1}{\mathcal{I}_{\theta}(X)},$ (3)

where $\mathcal{I}_{\theta}(X)=E_{X}\left[\left(\partial\log
p_{X}(X;\theta)/\partial\theta\right)^{2}\right]$ is the classical Fisher
information (FI) associated with $\theta$ for random variable $X$. By
additivity of classical FI, for a sequence of $n$ independent and identically
distributed (i.i.d.) random variables $\left\\{X_{k}\right\\}_{k=1}^{n}$,
$\mathcal{I}_{\theta}\left(\left\\{X_{k}\right\\}_{k=1}^{n}\right)=n\mathcal{I}_{\theta}\left(X_{1}\right)$.
Estimators that achieve the classical CRB in (3) with equality are called
_efficient_. For observations described by an i.i.d. sequence of random
variables $\left\\{X_{k}\right\\}_{k=1}^{n}$, maximum likelihood estimator
(MLE) is asymptotically unbiased and efficient as $n\to\infty$, up to mild
regularity conditions [6, 7].

However, classical estimation theory implicitly assumes that the measurement
device (described by POVM $\left\\{\hat{\Gamma}_{x}\right\\}$) is fixed.
Quantum methodology enriches estimation theory by allowing analysis and
optimization of $\left\\{\hat{\Gamma}_{x}\right\\}$ [8, Ch. VIII]. The quantum
CRB lower bounds the classical CRB since it assumes an optimal measurement:

$\displaystyle V_{\theta_{0}}\left(\check{\theta}\right)$
$\displaystyle\geq\frac{1}{\mathcal{J}_{\theta}\left(\hat{\sigma}(\theta)\right)},$
(4)

where
$\mathcal{J}_{\theta}\left(\hat{\sigma}(\theta)\right)=\operatorname{Tr}\left\\{\hat{\Lambda}_{\theta}^{2}\hat{\sigma}(\theta)\right\\}$
is the quantum FI associated with $\theta$ for state $\hat{\sigma}(\theta)$.
$\hat{\Lambda}_{\theta}$ is the symmetric logarithm derivative (SLD) operator
that is Hermitian but not necessarily positive. It is defined implicitly by
[8, Ch. VIII.4(b)]:

$\displaystyle\frac{\partial\hat{\sigma}(\theta)}{\partial{\theta}}=\frac{1}{2}\left(\hat{\Lambda}_{\theta}\hat{\sigma}(\theta)+\hat{\sigma}(\theta)\hat{\Lambda}_{\theta}\right).$
(5)

Analogous to classical FI, by the additivity of quantum FI, for a tensor
product of $n$ states $\hat{\sigma}^{\otimes n}(\theta)$,
$\mathcal{J}_{\theta}\left(\hat{\sigma}^{\otimes
n}(\theta)\right)=n\mathcal{J}_{\theta}\left(\hat{\sigma}(\theta)\right)$. We
call a combination of a quantum measurement and a classical estimator on the
corresponding output _quantum efficient_ , if it achieves the quantum CRB in
(4) with equality.

Consider an eigendecomposition of SLD
$\hat{\Lambda}_{\theta}=\sum_{x}\lambda_{x}(\theta)\left|\lambda_{x}(\theta)\right>\left<\lambda_{x}(\theta)\right|$,
where $\left\\{\left|\lambda_{x}(\theta)\right>\right\\}$ is a set of
orthonormal pure eigen-states of $\hat{\Lambda}_{\theta}$ and
$\left\\{\lambda_{x}(\theta)\right\\}$ are the corresponding eigenvalues. For
a tensor-product state $\hat{\sigma}^{\otimes n}(\theta)$, a tensor-product
measurement
$\left\\{\left|\lambda_{x}(\theta)\right>\left<\lambda_{x}(\theta)\right|\right\\}^{\otimes
n}$ constructed from eigendecomposition of SLD $\hat{\Lambda}_{\theta}$ and
followed by an MLE on the corresponding classical i.i.d. random output
sequence $\left\\{X_{k}\right\\}_{k=1}^{n}$ is asymptotically unbiased and
quantum efficient. Thus, unlike in optimal decoders for general classical-
quantum channels [43], a complicated joint-detection measurement that
entangles the output of $n$ probes is unnecessary:
$\left\\{\left|\lambda_{x}(\theta)\right>\left<\lambda_{x}(\theta)\right|\right\\}$
is applied separately to each of $n$ states $\hat{\sigma}(\theta)$. However,
it is important to note that there is an infinite number of
eigendecompositions of $\hat{\Lambda}_{\theta}$, and that their mathematical
expressions are typically unavailable in closed form. Even when they are
found, translating these expressions to physical devices can be extremely
challenging.

Furthermore, the structure of a quantum CRB-achieving measurement
$\left\\{\left|\lambda_{x}(\theta)\right>\left<\lambda_{x}(\theta)\right|\right\\}$
often depends on the parameter of interest $\theta$. This seeming paradox of
needing to know $\theta$ to build a device for its measurement is addressed by
the following two-stage approach [11, Ch. 6.4], [27, 28]. First, one obtains a
rough pre-estimate $\check{\theta}_{0}$ using $n^{\beta}$,
$\beta\in\left(\frac{1}{2},1\right)$, probes and a sub-optimal measurement
that does not depend in $\theta$. Then, one employs $\check{\theta}_{0}$ to
construct
$\left\\{\left|\lambda_{x}(\check{\theta}_{0})\right>\left<\lambda_{x}(\check{\theta}_{0})\right|\right\\}$
and refine $\check{\theta}$ using the remaining $n-n^{\beta}$ probes. This
achieves quantum CRB asymptotically as $n\to\infty$, under conditions outlined
in [11, Ch. 6.4], [28].

We encourage the reader, interested in learning more about the foundations of
quantum sensing, to consult the classic texts on quantum detection and
estimation [8] and quantum information theory [9], as well as more recent
texts [10, 11] covering these subjects in greater depth. We are now ready for
the formal treatment of quantum transmittance estimation.

Figure 2: Sensing of unknown transmittance $\theta$. Sensor transmits $n$-mode
signal systems $S$ of bipartite state $\hat{\rho}_{I^{n}S^{n}}$ by using $n$
times the single-mode thermal noise lossy bosonic channel
$\mathcal{E}^{(\bar{n}_{\rm T},\theta)}$ described in Fig. 1, where mean
thermal photon number $\bar{n}_{\rm T}\equiv\frac{\bar{n}_{\rm B}}{1-\theta}$.
Idler systems $I$ are retained as a reference for the measurement of output
state $\hat{\sigma}_{I^{n}R^{n}}(\theta)$, which yields the estimator
$\check{\theta}_{n}$.

###  II-D Quantum Transmittance Estimation

Fig. 2 depicts our system setup. Sensor’s goal is to estimate unknown power
transmittance $\theta$. It prepares a bipartite quantum state
$\hat{\rho}_{I^{n}S^{n}}$ which occupies $n$ signal systems $S$ and $n$ idler
systems $I$. Signal systems $S$ interrogate the target over $n$ available
modes of channel $\mathcal{E}^{\bar{n}_{\rm T},\theta}_{S\rightarrow R}$.
Idler systems $I$ are retained losslessly and noiselessly as a reference. The
output state
$\hat{\sigma}_{I^{n}R^{n}}(\theta)=\left(\mathcal{I}_{I}\otimes\mathcal{E}^{\bar{n}_{\rm
T},\theta}_{S\rightarrow R}\right)^{\otimes
n}\left[\hat{\rho}_{I^{n}S^{n}}\right]$ carries information about
transmittance $\theta$ in the returned systems $R$, where $\mathcal{I}$ is the
identity channel. As described in Section II-C, we seek an unbiased estimator
$\check{\theta}_{n}$ on a measurement of $\hat{\sigma}_{I^{n}R^{n}}(\theta)$
that minimizes the mean squared error
$V_{\theta_{0}}\left(\check{\theta}_{n}\right)$. Early work focused on the
transmittance sensing in a pure-loss bosonic channel
$\mathcal{E}^{0,\theta}_{S\rightarrow R}$, with $\bar{n}_{\rm T}=0$ [14, 15,
16]. Notably, the author of [16] proved that Fock (photon number) states are
optimal for transmittance sensing in this environment.

Unfortunately, the pure-loss bosonic channel model has limited applications,
due to the omnipresence of thermal noise in practical scenarios. Quantum
illumination (QI) improves detection and estimation in thermal noise using
entanglement between $S$ and $I$. The quantum CRB for joint estimation of
unknown $\theta$ and $\bar{n}_{\rm T}$ using Gaussian subset of quantum states
[44] was derived in [17]. Although the two-mode squeezed vacuum (TMSV) state
was proved an optimal Gaussian state in [17], the sensor was allowed to
estimate $\theta$ from just the thermal background – called “shadow effect” in
[19]. QI literature [18, 21, 22, 23, 26, 25] addresses arguably more practical
settings where the thermal background cannot aid the estimation. The “shadow
effect” is removed by setting:

$\displaystyle\bar{n}_{\rm T}$ $\displaystyle\equiv\frac{\bar{n}_{\rm
B}}{1-\theta},$ (6)

where $\bar{n}_{\rm B}$ is the mean number of thermal background photons per
mode that corrupt the sensor’s probes. When the background light does not help
the estimation of $\theta$, quantum CRB is quantitatively different from
results in [17]111Reparameterizing [17, Eqs. (B8a)-(B8d)] to
$\theta=e^{-\gamma}$ shows that, even in the absence of probes ($\bar{n}_{\rm
S}=0$), quantum FI associated with $\theta$ is positive due to the thermal
background., and the quantum FI is upper-bounded by [18, Eq. (21)]:

$\displaystyle\mathcal{J}_{\theta}\left(\hat{\sigma}_{I^{n}R^{n}}(\theta)\right)$
$\displaystyle\leq\mathcal{J}_{\theta,\mathrm{ub}}\left(\bar{n}_{\rm
S,tot}\right)\equiv\frac{\bar{n}_{\rm S,tot}}{\theta(\bar{n}_{\rm
B}+1-\theta)},$ (7)

where $\bar{n}_{\rm S,tot}=\sum_{i=1}^{n}\bar{n}_{\mathrm{S},i}$ is the total
mean photon number transmitted over $n$ modes, and the individual mode mean
photon numbers $\bar{n}_{\mathrm{S},i}$ may be unequal. However, [18] leaves
open the structure of $\hat{\rho}_{I^{n}S^{n}}$ that saturates (7), and the
design of the corresponding quantum-CRB-achieving measurement.

Excited by the gap identified in [18] and using the prescription for
$\bar{n}_{\rm T}$ in (6), we analyze transmittance sensing with the TMSV
states. In the next section, we find that the quantum FI of TMSV states
saturates the ultimate bound in (7) as per-mode transmitted photon number
$\bar{n}_{\rm S}\to 0$. In Section V, we report that, at $\bar{n}_{\rm S}>0$,
the quantum FI of TMSV states significantly exceeds the FI of other well-known
transmittance estimation schemes. This motivates the derivation of the
quantum-CRB-achieving receiver structure for TMSV probes in Section IV, and
the numerical analysis of the convergence of its MSE to the optimal in Section
VI.

Following the conference presentation [1] of our preliminary results, [19]
investigated transmittance sensing with and without noise aiding the
estimation. Notably, [19] proved that TMSV maximized the quantum FI within the
class of Gaussian quantum states [44] with and without the “shadow effect.”
While the treatment of quantum FI in [19] is comprehensive, the receiver
structures that achieve it are not considered. Even more recently, [20]
presented an experimental study of estimating multiple transmittance
parameters in Xanadu’s X8 integrated-photonic-quantum computer [45].
Unfortunately, the limitations of Xanadu’s platform limit the study in [20] to
photon-number-resolving (PNR) measurements of TMSV states. The same authors
follow up by analyzing the limits of transmittance sensing using coherent and
Fock states in the presence of detector dark counts [46], and report results
that are qualitatively similar to those in Section V.

##  III TMSV is Optimal for Transmittance Sensing

The TMSV state is represented in the Fock (photon number) basis as follows:

$\displaystyle\left|\psi\right>_{IS}$
$\displaystyle=\sum_{k=0}^{\infty}\sqrt{q(k;\bar{n}_{\rm
S})}\left|k\right>_{I}\left|k\right>_{S},$ (8)

where $q\left(k;\bar{n}\right)$ is defined in (1). TMSV is a zero-displacement
pure Gaussian state, which among all two-mode-Gaussian states with mean photon
number $\bar{n}_{\rm S}$ is maximally entangled [44]. It is critical in
quantum-information processing. Generating TMSV is a well-known (bordering on
routine) process in quantum optics. We show that TMSV becomes optimal for
transmittance estimation in thermal noise as transmitted photon number per
mode $\bar{n}_{\rm S}$ decays to zero:

######  Theorem 1.

The following limit holds for the quantum Fisher information
$\mathcal{J}_{\theta}\left(\hat{\sigma}_{I^{n}R^{n}}(\theta)\right)$:

$\displaystyle\lim_{\bar{n}_{\rm S}\rightarrow
0}\frac{\mathcal{J}_{\theta}\left(\hat{\sigma}_{I^{n}R^{n}}(\theta)\right)}{\bar{n}_{\rm
S}}$ $\displaystyle=\frac{n}{\theta(\bar{n}_{\rm B}+1-\theta)}$ (9)

when $n$ TMSV probes described by tensor-product state
$\hat{\rho}_{I^{n}S^{n}}=\left|\psi\right>\left<\psi\right|_{IS}^{\otimes n}$
are used and $\hat{\sigma}_{I^{n}R^{n}}(\theta)$ is the quantum state
describing the returned probes and retained references.

We first note that the lossy thermal-noise bosonic channel
$\mathcal{E}^{\bar{n}_{\rm T},\theta}_{S\rightarrow R}$ acts independently on
each transmitted mode. Therefore, for input tensor product of TMSV states
$\left|\psi\right>\left<\psi\right|_{IS}^{\otimes n}$, the output state
$\hat{\sigma}_{I^{n}R^{n}}(\theta)=\hat{\sigma}_{IR}^{\otimes n}(\theta)$ is a
tensor product of states
$\hat{\sigma}_{IR}(\theta)=\left(\mathcal{I}_{I}\otimes\mathcal{E}^{\bar{n}_{\rm
T},\theta}_{S\rightarrow
R}\right)\left[\left|\psi\right>\left<\psi\right|_{IS}\right]$. By the
additivity of the quantum FI for tensor product states,
$\mathcal{J}_{\theta}\left(\hat{\sigma}_{I^{n}R^{n}}(\theta)\right)=n\mathcal{J}_{\theta}\left(\hat{\sigma}_{IR}(\theta)\right)$.
In Appendix A we employ the method from [47, 48] to derive the quantum FI

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\mathcal{J}_{\theta,\text{TMSV}}\left(\bar{n}_{\rm
S}\right)\equiv\mathcal{J}_{\theta}\left(\hat{\sigma}_{IR}(\theta)\right)}$
(10) $\displaystyle=$ $\displaystyle\frac{\bar{n}_{\rm S}\left(\bar{n}_{\rm
B}+1+(1-\theta)\bar{n}_{\rm S}+\bar{n}_{\rm B}\bar{n}_{\rm
S}\right)}{\theta\left(\bar{n}_{\rm B}+1-\theta\right)\left(\bar{n}_{\rm
B}+1+\bar{n}_{\rm S}(2\bar{n}_{\rm B}+1-\theta)\right)}$

associated with the quantum state $\hat{\sigma}_{IR}(\theta)$ that describes
the returned probe and retained reference when a single TMSV probe is used.
Multiplying (10) by $\frac{n}{\bar{n}_{\rm S}}$ and taking the limit in (9)
yields the proof.222Our expression for the quantum FI of TMSV in (10) is
exact, unlike [25, Eq. (6)] and [18, Eq. (23)]. In fact, all of the quantum FI
expressions in [25] are approximations at $\theta_{\rm a}=0$, where
$\theta_{\rm a}=\sqrt{\theta}$ is the amplitude transmittance, and [18, Eq.
(23)] is [25, Eq. (6)] reparametrized from $\theta_{\rm a}$ to $\theta$.
Reparametrizing (10) to $\theta_{\rm a}$ and setting $\theta_{\rm a}=0$ yields
[25, Eq. (6)]. That a crude approximation with a zero-order Taylor series term
yields such a close result is striking.

Theorem 1 proves that TMSV is optimal over _all_ low-input-power states,
including the non-Gaussian ones. Although it has been shown to be an optimal
Gaussian state for all values of $\theta$, $\bar{n}_{\rm B}$, and
$\bar{n}_{\rm S}$ [19], characterization of a general quantum input state that
maximizes quantum FI associated with $\theta$ is an open problem. That being
said, as mentioned in Section I, the low transmitted-photon-number per mode
regime is important for the design of sensors operating under the total power
constraints. Thus, we characterize and analyze a receiver structure that
achieves asymptotically the quantum CRB for TMSV.

##  IV Transmittance Estimation Using TMSV Probes

###  IV-A Optimal Receiver Structure

We determine the eigendecomposition of the symmetric logarithmic derivative
(SLD) operator $\hat{\Lambda}_{\theta}$ defined in (5) for the quantum state
$\hat{\sigma}_{IR}(\theta)$ describing the returned probe and retained
reference. Since $\hat{\sigma}_{IR}(\theta)$ is Gaussian,
$\hat{\Lambda}_{\theta}$ is a degree-2 polynomial of creation and annihilation
operators $\hat{a}_{I}$, $\hat{a}_{R}$, $\hat{a}_{I}^{\dagger}$, and
$\hat{a}_{R}^{\dagger}$ [49, Sec. III]. Thus, there are infinitely-many
unitary transformations $\hat{U}_{\theta}$ that represent the application of a
finite sequence of squeezing, displacement, phase rotation, and beam splitter
operators, that diagonalize $\hat{\Lambda}_{\theta}$.

In Appendix B, we adapt the approach from [14] to show that
$\hat{\Lambda}_{\theta}=c_{1}\hat{S}(\omega)\left(\hat{a}^{\dagger}_{I}\hat{a}_{I}+\hat{a}^{\dagger}_{R}\hat{a}_{R}\right)\hat{S}^{\dagger}(\omega)+c_{0}\hat{I}$,
where
$\hat{S}(\omega)=\exp(\omega^{\ast}\hat{a}_{R}\hat{a}_{I}-\omega\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger})$
is a two-mode squeezing operator, $\hat{I}$ is the identity operator, and
$c_{0}$, $c_{1}$ are scalars. Unitary $\hat{S}(\omega)$ diagonalizes
$\hat{\Lambda}_{\theta}$ in the two-mode Fock (photon number) basis
$\left\\{\left|km\right>\right\\}\equiv\left\\{\left|k\right>\otimes\left|m\right>\right\\}$,
$k,m=0,1,\ldots$ since this is an eigenbasis for
$\hat{a}^{\dagger}_{I}\hat{a}_{I}+\hat{a}^{\dagger}_{R}\hat{a}_{R}$. Thus,
$\left\\{\hat{S}(\omega)\left|km\right>:k,m=0,1,\ldots\right\\}$ is an
eigendecomposition of $\hat{\Lambda}_{\theta}$. Hence, a receiver for TMSV
transmitter that achieves the quantum Cramér-Rao bound (CRB) asymptotically is
a two-mode squeezer followed by the photon-number-resolving (PNR) measurements
of each output mode. Fig. 3 depicts transmittance sensing using TMSV and this
receiver. Formally, it is the positive operator-valued measurement (POVM)
$\mathcal{M}=\left\\{\left|s_{km}\right>\left<s_{km}\right|\right\\}$, where
$\left|s_{km}\right>=\hat{S}(\omega)\left|km\right>$. POVM $\mathcal{M}$
followed by the maximum likelihood estimation (MLE) of $\theta$ from the
classical outcome of PNR measurements achieves the quantum CRB as
$n\to\infty$. The squeezing parameter is $\omega=\lambda-\zeta$, where $\zeta$
and $\lambda$ are functions of $\theta$, $\bar{n}_{\rm B}$, and $\bar{n}_{\rm
S}$ defined in (80) and (151), respectively. We emphasize that, although our
measurement is convenient, in that it can be physically implemented using
well-known optical elements (squeezer and PNR receivers), it is only _one of
the infinitely-many_ measurements that achieve the quantum CRB asymptotically.

Figure 3: Sensing the unknown transmittance $\theta$ using $n$ two-mode
squeezed vacuum (TMSV) states $\left|\psi\right>_{IS}$. When
$\left|\psi\right>_{IS}$ is transmitted, a bipartite output state
$\hat{\sigma}_{IR}$ occupies a retained idler system $I$ and a corresponding
returned probe system $R$. The receiver achieves the quantum CRB by applying a
two-mode squeezer (TMS) separately to each of the $n$ output states
$\hat{\sigma}_{IR}$, followed by independent photon-number-resolving (PNR)
measurement of each output mode. Maximum likelihood estimator (MLE) is applied
to the resulting classical output to obtain the estimate $\check{\theta}_{n}$.
As in Fig. 2, $\bar{n}_{\rm T}\equiv\frac{\bar{n}_{\rm B}}{1-\theta}$.

###  IV-B Remarks and Caveats

The structure of our receiver is remarkable in that it achieves the quantum
CRB using well-known optical components:

  * •

Optical elements implementing a two-mode squeezer required for the receiver
have been demonstrated in the laboratory [50].

  * •

Our Monte Carlo simulation described in Section VI shows that, at a moderate
amount of thermal noise ($\bar{n}_{\rm B}\approx 1$ photons per mode), and
with a low transmitted photon number ($\bar{n}_{\rm S}\approx 0.01$), nine
photons is the sufficient resolution for PNR measurements. Such measurements,
while technically complex, have been demonstrated [51].

However, two caveats are in order:

####  IV-B1 Value of $\omega$ depends on $\theta$

As mentioned in Section II-C, the dependence of the structure of a quantum
CRB-achieving measurement on the parameter of interest is a common occurrence
in quantum estimation theory. In our Monte Carlo simulation detailed in
Section VI we employ the two-stage approach [11, Ch. 6.4], [27, 28] described
in Section II-C with coherent transceiver from Section V-A used to construct a
rough pre-estimate, $\check{\theta}_{0}$.

Figure 4: Satisfaction of the condition in (11) for the existence of the
receiver employing two-mode squeezing, followed by photon counting at various
values of system parameters $\theta$, $\bar{n}_{\rm S}$, and $\bar{n}_{\rm
B}$. Red shading denotes the regions where (11) is not satisfied and white the
regions where (11) is satisfied.

####  IV-B2 Existence of $\omega$ depends on $\theta$, $\bar{n}_{\rm S}$, and
$\bar{n}_{\rm B}$

The eigenbasis of the SLD $\hat{\Lambda}_{\theta}$ in the form
$\left\\{\hat{S}(\omega)\left|km\right>:k,m=0,1,\ldots\right\\}$ requires the
scalars $\lambda$, $F^{\prime}$, $T_{1}$, $T_{2}$ to be real (see Appendix
B-D). Thus, per (145)-(149), we must have:

$\displaystyle(C+D)^{2}<4E^{2},$ (11)

where $C$, $D$, and $E$ are defined in (130)-(132). As Fig. 4 illustrates,
(11) does not hold for a certain range of system parameters: $\theta$,
$\bar{n}_{\rm S}$, and $\bar{n}_{\rm B}$. Therefore, another, likely more
complicated, receiver structure is necessary to realize the full quantum
advantage that the TMSV states yield.333In Appendix B we adapt the approach in
[14], where the authors derive a quantum CRB-achieving receiver for
transmittance sensing when $\bar{n}_{\rm T}=0$. That receiver similarly does
not exist for certain ranges of $\bar{n}_{\rm S}$ and $\theta$.

Notwithstanding these caveats, the range of operating parameters where our
proposed receiver exists, corresponds to the low transmitted photon number
regime ($\bar{n}_{\rm S}\ll 1$) which has significant practical applications,
(e.g., in covert/low probability of detection (LPD) [29, 30, 31, 32, 33, 34,
35, 36], battery-constrained, or light-sensitive-sample scenarios). It is also
the regime where TMSV states are quantum optimal. Nevertheless, we analyze and
compare alternative transmittance-sensing approaches next.

##  V Comparison with Alternative Transmittance-Sensing Methods: Fundamental
Limits

We compare the classical and quantum Fisher information (FI) associated with
transmittance $\theta$ for several well-known receivers with the quantum FI
for TMSV state derived in Section III and the ultimate upper bound (7) from
[18].

###  V-A Coherent Transceiver

Single-mode coherent state $\left|\alpha\right>$ with complex-valued amplitude
$\alpha\in\mathbb{C}$ is an eigenstate of the annihilation operator $\hat{a}$
and a quantum-mechanical description of laser light. There are two types of
coherent measurement: an optical homodyne receiver implements a measurement
that uses the eigendecomposition of either
$\hat{p}=\frac{\hat{a}-\hat{a}^{\dagger}}{\sqrt{2}}$ or
$\hat{q}=\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}}$ quadrature operators and
yields a single Gaussian random variable, while a heterodyne receiver
implements measurement that uses the eigendecompositions of both $\hat{p}$ and
$\hat{q}$ and outputs a pair of independent Gaussian random variables. A
coherent transceiver employs both coherent states and a receiver, and is
called “classical” in the literature. The quantum FI associated with
transmittance $\theta$ for a coherent state $\left|\alpha\right>_{S}$ probe is
[18, Eq. (23)]:

$\displaystyle\mathcal{J}_{\theta,\text{Coh}}\left(\bar{n}_{\rm
S}\right)\equiv\mathcal{J}_{\theta}\left(\mathcal{E}^{\bar{n}_{\rm
T},\theta}_{S\rightarrow
R}\left[\left|\alpha\right>\left<\alpha\right|_{S}\right]\right)$
$\displaystyle=\frac{\bar{n}_{\rm S}}{\theta(2\bar{n}_{\rm B}+1)},$ (12)

where the transmitted mean photon number $\bar{n}_{\rm S}\equiv|\alpha|^{2}$.
Use of a homodyne receiver as depicted in Fig. 5 achieves the corresponding
quantum CRB.

Figure 5: Sensing the unknown transmittance $\theta$ using $n$ coherent states
$\left|\alpha\right>_{S}$ and a homodyne receiver. The output state
$\hat{\sigma}_{R}$ is called a displaced thermal state. A Gaussian random
variable describes the output of homodyne measurement [44],[52, Ch. 7.3.2]. A
maximum likelihood estimator (MLE) that uses the homodyne receiver’s output
and the value of $\alpha$ as a classical reference achieves the quantum CRB
for $\hat{\sigma}_{R}^{\otimes n}$ as $n\to\infty$, and is derived in Section
VI-A1.

###  V-B TMSV and Optical Parametric Amplifier (OPA) Receiver

Let’s use the TMSV probes as in Section III, but apply an optical parametric
amplifier (OPA) to the output state $\hat{\sigma}_{IR}(\theta)$ instead of a
two-mode squeezer, and discard one of the OPA outputs. The remaining output of
the OPA is then in a zero-mean thermal state with average photon number [23,
Sec. A]:

$\displaystyle\bar{n}_{\rm OPA}$ $\displaystyle=G\bar{n}_{\rm
S}+(G-1)(\bar{n}_{\rm B}+1+\theta\bar{n}_{\rm S})$
$\displaystyle\phantom{=}+2\sqrt{G(G-1)\theta\bar{n}_{\rm S}(\bar{n}_{\rm
S}+1)},$ (13)

where the OPA gain $G>1$. PNR measurement then outputs a random photon count
$K$, distributed geometrically with mass function $q_{K}(k;\bar{n}_{\rm OPA})$
from (1). The OPA receiver was proposed for target detection with quantum
illumination [23, 24], however, it can also be used to estimate transmittance
$\theta$, as depicted in Fig. 6. Classical FI associated with $\theta$ in $K$
is:

$\displaystyle\mathcal{J}_{\theta,\mathrm{OPA}}\left(\bar{n}_{\rm S}\right)$
$\displaystyle=\max_{G>1}\frac{\left((G-1)\bar{n}_{\rm
S}+\sqrt{\frac{G(G-1)\bar{n}_{\rm S}(\bar{n}_{\rm
S}+1)}{\theta}}\right)^{2}}{\bar{n}_{\rm OPA}(\bar{n}_{\rm OPA}+1)}.$ (14)

For fixed $\bar{n}_{\rm S}>0$, the constrained maximization over gain $G$ is
challenging analytically. Hence, we use numerical approaches to plot classical
FI for an OPA receiver in Fig. 9. The dependence of optimal $G$ on $\theta$
can be addressed using the two-stage estimation approach described in Section
IV-B; we employ it in the Monte Carlo simulation described in Section VI.
Furthermore, when $\bar{n}_{\rm S}$ is small, we have:

$\displaystyle\left.\frac{\mathcal{J}_{\theta,\mathrm{OPA}}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}\right|_{\bar{n}_{\rm S}=0}$
$\displaystyle=\max_{G>1}\frac{G}{\theta(\bar{n}_{\rm B}+1)(G(1+\bar{n}_{\rm
B})-\bar{n}_{\rm B})}$ (15) $\displaystyle=\frac{1}{\theta(\bar{n}_{B}+1)},$
(16)

where (15) is maximized when $G\to 1$. This shows that the TMSV+OPA
combination can perform close to the limit in (7) in the low-transmitted-power
low-transmittance regime.

Figure 6: Sensing the unknown transmittance $\theta$ using $n$ two-mode
squeezed vacuum (TMSV) states $\left|\psi\right>_{IS}$ and an optical
parametric amplifier (OPA) receiver. Per [23], one of the outputs of the OPA
is discarded, resulting in the other being in a thermal state. A maximum
likelihood estimator (MLE) that uses the output of the photon-number-resolving
(PNR) measurement achieves the classical CRB asymptotically as $n\to\infty$,
and is derived in Section VI-A2.

###  V-C Fock States

It is well-known that Fock state $\left|m\right>$ is optimal for transmittance
sensing in a vacuum ($\bar{n}_{\rm B}=0$), achieving quantum FI
$\mathcal{J}_{\theta}\left(\mathcal{E}^{0,\theta}_{S\rightarrow
R}\left[\left|m\right>\left<m\right|_{S}\right]\right)=\frac{m}{\theta(1-\theta)}$
[15]. Here we show that this breaks down in thermal noise. The output state
$\hat{\nu}_{R}(\theta,m)=\mathcal{E}^{\bar{n}_{\rm T},\theta}_{S\rightarrow
R}\left[\left|m\right>\left<m\right|_{S}\right]$ is diagonal in the Fock
basis, making PNR measurement in the sensor as depicted in Fig. 7 optimal. The
mass function for random variable $K$ describing the photon count is derived
in [52, Sec. 7.3.1]. Transformation of [52, Eq. (7.37)] using [53, 9.131.1]
yields:

$\displaystyle p_{K}(k;m)$ $\displaystyle=\binom{m+k}{k}\frac{(\bar{n}_{\rm
B}+1-\theta)^{m-k}\theta^{k}}{(\bar{n}_{\rm B}+1)^{m+k+1}}$
$\displaystyle\phantom{=}\times{}_{2}F_{1}\left[\begin{array}[]{c}-k,-k\\\
-(m+k)\end{array};z(\theta,\bar{n}_{\rm B})\right],$ (19)

where $z(\theta,\bar{n}_{\rm B})=\frac{(\theta-\bar{n}_{\rm B})(\bar{n}_{\rm
B}+1)}{\theta}$ and

$\displaystyle{}_{2}F_{1}\left[\begin{array}[]{c}a,b\\\ c\end{array};z\right]$
$\displaystyle=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}z^{2}+\cdots$ (22)

is the hypergeometric series. The quantum FI associated with $\theta$ is:

$\displaystyle\mathcal{J}_{\theta,\text{Fock}}(\left|m\right>)$
$\displaystyle\equiv\mathcal{J}_{\theta}\left(\hat{\nu}_{R}(\theta,m)\right)$
$\displaystyle=m\left(\frac{1-\theta}{(\bar{n}_{\rm
B}+1-\theta)^{2}}+\frac{1}{\theta}\right)$
$\displaystyle\phantom{=}-\bar{n}_{\rm B}\left(\frac{1}{(\bar{n}_{\rm
B}+1-\theta)^{2}}-\frac{1}{\theta^{2}}\right)$
$\displaystyle\phantom{=}-\frac{2\bar{n}_{\rm B}(\theta m+\bar{n}_{\rm
B}+1)}{(\bar{n}_{\rm B}+1-\theta)\theta^{2}}$
$\displaystyle\phantom{=}-\frac{2\bar{n}_{\rm B}^{2}\left(2(\theta
m+\bar{n}_{\rm B}+\theta m\bar{n}_{\rm B})+\bar{n}_{\rm
B}^{2}+1\right)}{(\bar{n}_{\rm B}+1-\theta)^{2}\theta^{2}}$
$\displaystyle\phantom{=}-\frac{\bar{n}_{\rm B}^{2}m(m-1)}{(\bar{n}_{\rm
B}+1-\theta)^{2}}+R_{m}(\theta,\bar{n}_{\rm B}),$ (23)

where

$\displaystyle R_{m}(\theta,\bar{n}_{\rm B})$
$\displaystyle=\sum_{k=0}^{\infty}\binom{m+k}{k}\frac{k^{4}\bar{n}_{\rm
B}^{2}(\bar{n}_{\rm B}+1-\theta)^{m-k}\theta^{k-4}}{(m+k)^{2}(\bar{n}_{\rm
B}+1)^{m+k-1}}$
$\displaystyle\phantom{=\sum_{k=0}^{\infty}}\times\frac{{}_{2}F_{1}\left[\begin{array}[]{c}-(k-1),-(k-1)\\\
-(m+k-1)\end{array};z(\theta,\bar{n}_{\rm
B})\right]^{2}}{{}_{2}F_{1}\left[\begin{array}[]{c}-k,-k\\\
-(m+k)\end{array};z(\theta,\bar{n}_{\rm B})\right]}.$ (28)

The ratio of the hypergeometric series in (V-C) makes both the analysis and
numerical evaluation of the quantum FI
$\mathcal{J}_{\theta,\text{Fock}}(\left|m\right>)$ for large $m$ challenging.
Furthermore, on-demand generation of Fock states $\left|m\right>$ for
arbitrary $m$ presents technical challenges that appear insurmountable in the
near term. However, such sources exist for single photons [54, 55, 56].
Setting $m=1$ in (19) yields

$\displaystyle p_{K}(k;1)$ $\displaystyle=\left(\bar{n}_{\rm B}(\bar{n}_{\rm
B}+1)+\theta(k-\bar{n}_{\rm B})\right)\bar{n}_{\rm B}^{k-1}(\bar{n}_{\rm
B}+1)^{-(k+2)}.$ (29)

The corresponding quantum FI is

$\displaystyle\mathcal{J}_{\theta,\text{Fock}}\left(\left|1\right>\right)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(k-\bar{n}_{\rm B})^{2}\bar{n}_{\rm
B}^{k-1}(\bar{n}_{\rm B}+1)^{-(k+2)}}{\left(\bar{n}_{\rm B}(\bar{n}_{\rm
B}+1)+\theta(k-\bar{n}_{\rm B})\right)}.$ (30)

We note that, for $\bar{n}_{\rm B}>0$, (30) is well approximated by the first
three terms of the sum. The contribution from terms corresponding to $k>2$
decreases rapidly due to the exponential decay of $\bar{n}_{\rm
B}^{k-1}(\bar{n}_{\rm B}+1)^{-(k+2)}$ with increasing $k$.

Figure 7: Sensing the unknown transmittance $\theta$ using $m$-photon Fock
state $\left|m\right>_{S}$. Photon-number-resolving (PNR) measurement is
optimal. We can employ a maximum likelihood estimator (MLE) that uses the
output of the PNR measurement and the value of $m$ as a classical reference.

###  V-D TMSV and Heralded PNR Measurement

TMSV states and PNR measurement can be used for probabilistic generation of
Fock states. Detection of $m$ photons on the idler mode heralds
$\left|m\right>_{S}$ state on the signal mode. Thus, we consider transmittance
sensing with TMSV using two PNR measurements: one for the idler mode, and the
other for the returned signal port. This is depicted in Fig. 8. The classical
FI of this system is

$\displaystyle\mathcal{J}_{\theta,\mathrm{Her}}\left(\bar{n}_{\rm S}\right)$
$\displaystyle=E_{M}\left[\mathcal{J}_{\theta,\text{Fock}}(\left|M\right>)\right],$
(31)

where the output photon number $M$ mass function is in (1).

Figure 8: Sensing the unknown transmittance $\theta$ using $n$ two-mode
squeezed vacuum (TMSV) states $\left|\psi\right>_{IS}$. The photon-number-
resolving (PNR) measurement of the idler mode heralds the photon number of the
transmitted Fock state probe. This is used by a maximum likelihood estimator
(MLE) along with the output of the PNR measurement of the returned probe to
achieve the classical CRB asymptotically as $n\to\infty$.

###  V-E Comparison

We ensure a fair comparison of the fundamental limits for various
transmittance-sensing methods by analyzing their Fisher information attained
per photon per (transmitted) mode. That is, _photon Fisher information
efficiency_ (PFIE) $\jmath\left(\bar{n}_{\rm
S}\right)\equiv\frac{\mathcal{J}_{\theta}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}$ is our figure of merit, where
$\mathcal{J}_{\theta}\left(\bar{n}_{\rm S}\right)$ is classical or quantum FI
attainable with $\bar{n}_{\rm S}$ input mean signal photons per mode. The
ultimate upper bound for PFIE is derived from (7):

$\displaystyle\jmath_{\rm ub}\left(\bar{n}_{\rm S}\right)$
$\displaystyle\equiv\frac{\mathcal{J}_{\theta,\rm{ub}}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}=\frac{1}{\theta(\bar{n}_{\rm B}+1-\theta)}.$ (32)

The PFIE for TMSV source uses the quantum FI in (10):

$\displaystyle\jmath_{\rm TMSV}\left(\bar{n}_{\rm S}\right)$
$\displaystyle\equiv\frac{\mathcal{J}_{\theta,\text{TMSV}}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}$ $\displaystyle=\frac{\bar{n}_{\rm
B}+1+(1-\theta)\bar{n}_{\rm S}+\bar{n}_{\rm B}\bar{n}_{\rm
S}}{\theta\left(\bar{n}_{\rm B}+1-\theta\right)\left(\bar{n}_{\rm
B}+1+\bar{n}_{\rm S}(2\bar{n}_{\rm B}+1-\theta)\right)}.$ (33)

The PFIE for coherent and TMSV+OPA transceivers use classical FI expressions
(12) and (14), respectively:

$\displaystyle\jmath_{\rm Coh}\left(\bar{n}_{\rm S}\right)$
$\displaystyle\equiv\frac{\mathcal{J}_{\theta,\text{Coh}}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}=\frac{1}{2\theta(\bar{n}_{\rm B}+1)}$ (34)
$\displaystyle\jmath_{\rm OPA}\left(\bar{n}_{\rm S}\right)$
$\displaystyle\equiv\frac{\mathcal{J}_{\theta,\mathrm{OPA}}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}$
$\displaystyle=\max_{G>1}\frac{\left((G-1)\bar{n}_{\rm
S}+\sqrt{\frac{G(G-1)\bar{n}_{\rm S}(\bar{n}_{\rm
S}+1)}{\theta}}\right)^{2}}{\bar{n}_{\rm S}\bar{n}_{\rm OPA}(\bar{n}_{\rm
OPA}+1)}.$ (35)

The PFIE for Fock state and TMSV+Heralded PNR measurement use quantum and
classical FI expressions in (23) and (31), respectively:

$\displaystyle\jmath_{\rm Fock}\left(\left|m\right>\right)$
$\displaystyle\equiv\frac{\mathcal{J}_{\theta,\text{Fock}}\left(\left|m\right>\right)}{m}$
(36) $\displaystyle\jmath_{\rm Her}\left(\bar{n}_{\rm S}\right)$
$\displaystyle\equiv\frac{\mathcal{J}_{\theta,\mathrm{Her}}\left(\bar{n}_{\rm
S}\right)}{\bar{n}_{\rm S}}.$ (37)

We evaluate (32)-(37) and plot the results versus thermal-noise mean photon
number $\bar{n}_{\rm B}$ in Fig. 9. While $\jmath_{\rm ub}\left(\bar{n}_{\rm
S}\right)$ and $\jmath_{\rm Coh}\left(\bar{n}_{\rm S}\right)$ are constant
relative to transmitted mean photon number $\bar{n}_{\rm S}$, $\jmath_{\rm
TMSV}\left(\bar{n}_{\rm S}\right)$, $\jmath_{\rm OPA}\left(\bar{n}_{\rm
S}\right)$, and $\jmath_{\rm Her}\left(\bar{n}_{\rm S}\right)$ are not. Thus,
we include plots for various values of $\bar{n}_{\rm S}$. We set $\theta=0.5$
and note that, for other values $\theta$, the plots are qualitatively similar.
In plotting $\jmath_{\rm OPA}\left(\bar{n}_{\rm S}\right)$ we maximize (35)
over $G>1$ numerically. We include the plot of $\jmath_{\rm
Fock}\left(\left|m\right>\right)$ for single-photon Fock state
$\left|1\right>$ and a thirty-photon Fock state $\left|30\right>$ on all
plots. The challenges associated with computing the hypergeometric series
${}_{2}F_{1}$ [57] precluded evaluating $\jmath_{\rm
Fock}\left(\left|m\right>\right)$ for $m>30$. When evaluating $\jmath_{\rm
Her}\left(\bar{n}_{\rm S}\right)$, we had to truncate the sum in the
expectation in (31) at $m=30$. This accurately approximates $\jmath_{\rm
Her}\left(\bar{n}_{\rm S}\right)$ only up to $\bar{n}_{\rm S}\approx 4$. Thus,
we did not evaluate $\left.\jmath_{\rm Her}\left(\bar{n}_{\rm
S}\right)\right|_{\bar{n}_{\rm S}=10}$ and $\left.\jmath_{\rm
Her}\left(\bar{n}_{\rm S}\right)\right|_{\bar{n}_{\rm S}=100}$.

(a) $\bar{n}_{\rm S}=0.001$ (b) $\bar{n}_{\rm S}=0.01$

(c) $\bar{n}_{\rm S}=0.1$ (d) $\bar{n}_{\rm S}=1$

(e) $\bar{n}_{\rm S}=10$ (f) $\bar{n}_{\rm S}=100$

Figure 9: Photon Fisher information efficiency vs. thermal-noise mean number
of photons per mode $\bar{n}_{\rm B}$ for various transmittance estimation
methods and values of transmitted mean number of photons per mode
$\bar{n}_{\rm S}$. We set $\theta=0.5$, results for other values of $\theta$
are qualitatively similar.

It is evident from Fig. 9 that PFIE of the TMSV input combined with optimal
measurement exceeds that of other receivers we consider. While Fock state
transmitters’ performance rapidly decays with noise, they outperform the
TMSV+OPA transceiver when signal-to-noise ratio (SNR) is high. Indeed, the
TMSV+OPA transceiver performs very poorly when the transmitted mean photon
number is high.

On the other hand, the high PFIE of single-photon Fock state $\left|1\right>$
in low noise shows the promise of using the on-demand single-photon sources
and PNR measurement for transmittance sensing. Furthermore, since the first
three terms of the summation in (30) approximate
$\mathcal{J}_{\theta}\left(\hat{\nu}_{R}(\theta,1)\right)$ well for
$\bar{n}_{\rm B}<1$, a measurement that distinguishes zero, one, or more
photons suffices for accurate estimation of $\theta$ for these systems. Such
measurement is less complex than the full PNR one.

We also note that TMSV+heralded PNR performs as well as the single-photon Fock
state source in low-noise setting. In fact, our calculations suggest that, for
$\bar{n}_{\rm S}\ll 1$, TMSV+heralded PNR measurement matches PFIE of a
single-photon Fock state source while using a less-complex single-photon
detector (SPD) that distinguishes zero or more than zero photons instead of a
PNR measurement.

Nevertheless, many practical scenarios demand low-transmitted-power operation.
Even for moderate noise power, this results in low SNR. Transmittance sensors
that employ TMSV+receiver derived in Section IV and TMSV+OPA receiver behave
well in this setting. Thus, next we use Monte Carlo simulation to evaluate
these sensors and to compare their performance to a simpler one based of a
coherent transceiver.

##  VI Comparison with Alternative Transmittance-Sensing Methods: Simulations

Maximum likelihood estimators (MLEs) have a number of desirable properties,
the first and foremost being the availability of “turn-the-crank”
implementation in most practical settings. Furthermore, MLEs are usually
asymptotically consistent and efficient, as the number of observations
$n\to\infty$ [7, 6]. Here we employ MLEs to estimate transmittance $\theta$
from the outputs of coherent homodyne transceiver, TMSV+OPA receiver, and
TMSV+receiver derived in Section IV, analyzing their convergence to CRB for
increasing $n$.

###  VI-A Construction of MLEs

####  VI-A1 Coherent Homodyne Transceiver

Consider a transmittance sensing scheme described in Section V-A that uses a
tensor product $\left|\alpha\right>^{\otimes n}_{S}$ of $n$ coherent states
$\left|\alpha\right>_{S}$ with $\alpha=\sqrt{\bar{n}_{\rm S}}$ as probes and a
homodyne receiver. The corresponding output is a sequence of $n$ independent
and identically distributed (i.i.d.) Gaussian random variables
$\left\\{X_{k}\right\\}_{k=1}^{n}$, each with mean $\sqrt{\theta\bar{n}_{\rm
S}}$ and variance $\bar{n}_{\rm B}+1$. The MLE for $\theta$ is:

$\displaystyle\check{\theta}_{\text{Coh}}(n)$
$\displaystyle=\frac{1}{\bar{n}_{\rm
S}}\left(\frac{1}{n}\sum_{k=1}^{n}x_{k}\right)^{2},$ (38)

where $x_{k}$ is an observed instance of $X_{k}$.

####  VI-A2 TMSV Input and OPA Receiver

Now consider a scheme from Section V-B that uses a tensor product
$\hat{\rho}_{I^{n}S^{n}}=\left|\psi\right>\left<\psi\right|_{IS}^{\otimes n}$
of $n$ TMSV states defined in (8) as probes and an OPA receiver. The
corresponding output is a sequence of $n$ i.i.d. geometric random variables
$\left\\{Y_{k}\right\\}_{k=1}^{n}$, each with mass function
$q_{Y}(y;\bar{n}_{\rm OPA})$ defined in (1), where the $\bar{n}_{\rm OPA}$ is
in (13). The MLE for $\theta$ is:

$\displaystyle\check{\theta}_{\rm OPA}(n)$
$\displaystyle=\frac{\bar{y}(G^{\ast}-1)-1-\bar{n}_{\rm
B}}{(G^{\ast}-1)^{2}\bar{n}_{\rm S}}$
$\displaystyle\phantom{=}-\frac{2\sqrt{G^{\ast}(1+\bar{y}+\bar{n}_{\rm
B}-G^{\ast}\bar{n}_{\rm S})(1+\bar{n}_{\rm S})}}{(G^{\ast}-1)\bar{n}_{\rm S}}$
$\displaystyle\phantom{=}+\frac{G^{\ast}(G^{\ast}+2\bar{n}_{\rm
B}-G^{\ast}\bar{n}_{\rm B}+(G^{\ast}-1)\bar{n}_{\rm
S})}{(G^{\ast}-1)^{2}\bar{n}_{\rm S}},$ (39)

where $\bar{y}=\frac{1}{n}\sum_{k=1}^{n}y_{k}$ is the mean of the observed
instances of $Y_{k}$, and $G^{\ast}$ is the OPA gain that maximizes classical
FI in (14). Thus, $G^{\ast}$ minimizes the CRB for the receiver, and hence,
the asymptotic MSE of $\check{\theta}_{\rm OPA}(n)$. However, $G^{\ast}$
depends on the parameter of interest, $\theta$. Thus, we follow a two-stage
approach [11, Ch. 6.4], [27, 28]. First $\sqrt{n}$ probes are coherent states
from which we obtain a preliminary estimate
$\check{\theta}_{0}=\check{\theta}_{\rm Coh}\left(\sqrt{n}\right)$, using the
homodyne receiver described above. We compute $G^{\ast}_{0}$ by maximizing
(14) with $\check{\theta}_{0}$ substituted for $\theta$. The remaining
$n-\sqrt{n}$ probes are TMSV states, with output processed by the OPA
receiver, with gain set to $G^{\ast}_{0}$, and the corresponding MLE
$\check{\theta}_{\rm OPA}\left(n-\sqrt{n}\right)$ in (39), with $G^{\ast}$ set
to $G^{\ast}_{0}$.

####  VI-A3 TMSV Input and Receiver Derived in Section IV

Finally, we use a tensor product
$\hat{\rho}_{I^{n}S^{n}}=\left|\psi\right>\left<\psi\right|_{IS}^{\otimes n}$
of $n$ TMSV states defined in (8) as probes, but we employ the receiver
derived in Section IV, that uses the two-mode squeezer $\hat{S}(\omega)$,
followed by the PNR measurement of each mode. As noted in Section IV-B, the
squeezing parameter $\omega$ depends on the parameter of interest $\theta$. We
follow a two-stage approach as in the OPA-based scheme, with an identical
first stage that uses a coherent homodyne transceiver. We calculate the
squeezing parameter $\omega_{0}$ using the preliminary estimate
$\check{\theta}_{0}$ that employs the first $\sqrt{n}$ probes. The remaining
$n-\sqrt{n}$ probes are TMSV states, with each output state
$\hat{\sigma}_{IR}(\theta)$. These are processed by
$\hat{S}\left(\omega_{0}\right)$ followed by a two-mode PNR measurement. Thus,
the corresponding output is a sequence of $n$ i.i.d. pairs of random variables
$\left\\{\left(Z_{0,k},Z_{1,k}\right)\right\\}_{k=1}^{n}$, each with mass
function:

$\displaystyle p_{Z_{0},Z_{1}}(z_{0},z_{1})$
$\displaystyle=\left<z_{0}z_{1}\right|\hat{S}^{\dagger}\left(\omega_{0}\right)\hat{\sigma}_{IR}(\theta)\hat{S}\left(\omega_{0}\right)\left|z_{0}z_{1}\right>$
(40)
$\displaystyle=\sum_{s=0}^{\infty}\sum_{t=0}^{\infty}r_{st}\left|\left<st\right|\hat{S}\left(\omega_{0}+\zeta\right)\left|z_{0}z_{1}\right>\right|^{2},$
(41)

where, in Appendix C, we derive

$\left<st\right|S\left(\omega_{0}+\zeta\right)\left|km\right>\\\
=(-\tau_{0})^{s-k}\sqrt{s!t!k!m!}\delta(s-k,t-m)\qquad\\\
\phantom{=}\times\sum_{u=u_{\rm l}}^{u_{\rm
m}}\frac{(-|\tau_{0}|^{2})^{u}\nu_{0}^{-k-m+2u-1}}{(s-k+u)!u!(k-u)!(m-u)!}$
(42)

with $u_{\rm l}=\max(k-s,0)$, $u_{\rm m}=\min(k,m)$,
$\tau_{0}=\frac{\omega_{0}+\zeta}{|\omega_{0}+\zeta|}\tanh|\omega_{0}+\zeta|$,
$\nu_{0}=\cosh|\omega_{0}+\zeta|$, the Kronecker delta function
$\delta(x,y)=\\{1\text{~{}if~{}}x=y;0\text{~{}if~{}}x\neq y\\}$, and $\zeta$
and $r_{st}$ defined in (80) and (90), respectively. Another form of (42) is
in [58, Eq. (22)]. Since we multiply by $\delta(s-k,t-m)$, (42) is not zero
only if $s-k=t-m$. Thus, (41) simplifies to:

$\displaystyle p_{Z_{0},Z_{1}}(z_{0},z_{1})$
$\displaystyle=\sum_{s=0}^{\infty}r_{st^{\prime}}\left|\left<st^{\prime}\right|\hat{S}\left(\omega_{0}+\zeta\right)\left|z_{0}z_{1}\right>\right|^{2},$
(43)

where $t^{\prime}=s-k+m$. As there is no known closed-form solution for MLE,
we construct it using numerical optimization of (43) recast as a likelihood
function for $\theta$.

###  VI-B Results

While it is well-known that MLE is asymptotically consistent and efficient, as
number of observations $n\to\infty$ [7, 6], numerical approaches are needed
for its _finite-sample_ performance analysis at fixed $n<\infty$. Furthermore,
although the two-stage approach [11, Ch. 6.4], [27, 28] is consistent and
quantum efficient as $n\to\infty$, the convergence conditions in [11, Ch.
6.4], [28] are onerous to prove mathematically for many estimators (including
ours) and, to our knowledge, its finite sample analysis is missing from the
literature.

Thus, we compare the simulated MSEs for the MLEs developed in Section VI-A to
their corresponding CRBs. Our settings for $\bar{n}_{\rm S}=0.01$,
$\bar{n}_{\rm B}=1$, and $\theta=0.5$ are chosen because 1) they allow the
simulations to complete in reasonable time while ensuring the existence of the
receiver derived in Section IV; and, 2) $\bar{n}_{\rm S}\ll\bar{n}_{\rm B}$
models transmittance sensing in a low-SNR environment. The MSE is inversely
proportional to the number of probes $n$:
$\text{MSE}_{\theta}=\frac{c_{\theta}}{n}$. For $\bar{n}_{\rm S}$,
$\bar{n}_{\rm B}$, and $\theta$ fixed, the scaling factor $c_{\theta}\equiv
n\times\text{MSE}_{\theta}$ is a function of $n$, however, the asymptotic
efficiency of MLE suggests that, for estimators described in Section VI-A,
$\lim_{n\to\infty}c_{\theta}=\frac{1}{\mathcal{J}_{\theta}}$, where
$\frac{1}{\mathcal{J}_{\theta}}$ is a single-observation CRB.444As
$n\to\infty$, MLE $\check{\theta}_{\rm MLE}(n)$ for $\theta$ is usually
asymptotically efficient: $\check{\theta}_{\rm MLE}(n)$ converges in
probability to the true value $\theta_{0}$. It is also typically
asymptotically efficient as follows: $\sqrt{n}\left(\check{\theta}_{\rm
MLE}(n)-\theta_{0}\right)$ converges in law to a zero-mean Gaussian random
variable with variance equal to CRB for a single observation [7, 6]. However,
proving $\lim_{n\to\infty}c_{\theta}=\frac{1}{\mathcal{J}_{\theta}}$
mathematically requires showing uniform integrability, which is challenging
for MLEs (see, e.g., remarks following Proposition IV.D.2 in [59, Sec. IV.D]).
We are interested in the speed of $c_{\theta}$’s convergence to
$\frac{1}{\mathcal{J}_{\theta}}$ as $n$ increases, and the penalty (if any) of
the two-stage approach.

Figure 10: Scaling factor $c_{\theta}=n\times\text{MSE}_{\theta}$ vs. the
number of probes $n$ used to estimate $\theta$. We set $\bar{n}_{\rm S}=0.01$,
$\bar{n}_{\rm B}=1$, and $\theta=0.5$. We evaluate $c_{\theta}$ at 96
logarithmically-spaced values for $n\in[1,65536]$. The limitations of machine
precision, computational run-time, and memory precluded evaluation of
$c_{\theta}$ for $n>65536$. Each data point is an average of $5\times 10^{4}$
results from Monte Carlo simulations. Approximate points when scaling factors
converge to the corresponding CRBs are marked with $\otimes$. The 95%
confidence intervals are negligibly small, and the fluctuations in the plots
are due to machine-precision limits rather than stochastic variations between
experiments.

On the ordinate in Fig. 10 we plot the scaling factors
$c_{\theta,\text{Coh}}$, $c_{\theta,\text{OPA}}$, and $c_{\theta,\text{TMSV}}$
for the MLEs that use the output of a coherent homodyne transceiver, TMSV+OPA
receiver, and TMSV+receiver derived in Section IV, respectively. We also plot
the scaling factors $c_{\theta,\text{OPA}}^{\ast}$ and
$c_{\theta,\text{TMSV}}^{\ast}$ for the corresponding receivers, constructed
with knowledge of $\theta$. While such receivers cannot physically exist, this
enables us to isolate the impact of the two-stage method on the convergence of
the scaling factor.

Fig. 10 shows that $c_{\theta,\text{TMSV}}$ converges to
$\frac{1+e_{\text{two-
stage}}}{\mathcal{J}_{\theta,\text{TMSV}}\left(\bar{n}_{\rm S}\right)}$ at
approximately $n=3000$ modes, $c_{\theta,\text{OPA}}$ converges to
$\frac{1+e_{\text{two-
stage}}}{\mathcal{J}_{\theta,\text{OPA}}\left(\bar{n}_{\rm S}\right)}$ at
approximately $n=5000$ modes, and $c_{\theta,\text{Coh}}$ converges to
$\frac{1}{\mathcal{J}_{\theta,\text{Coh}}\left(\bar{n}_{\rm S}\right)}$ at
approximately $n=8000$ modes. The penalty from the use of the two-stage method
is $e_{\text{two-stage}}\approx 0.05$, and appears to decay with $n$. We note
that $\frac{1}{\mathcal{J}_{\theta,\text{TMSV}}\left(\bar{n}_{\rm S}\right)}$
is negligibly close to the ultimate lower bound
$\frac{1}{\mathcal{J}_{\theta,\text{ub}}\left(\bar{n}_{\rm S}\right)}$.

###  VI-C Towards Experimental Validation

Our results support the feasibility of near-future experimental validation of
quantum-enhanced transmittance sensing. Idler-mode storage and synchronization
are technical challenges that are similar to those the previous quantum
illumination experiments overcame, e.g., [24]. For $\bar{n}_{\rm S}=0.01$,
$\bar{n}_{\rm B}=1$, and $\theta=0.5$, the required squeezing parameter in the
receiver, that we derived in Section IV, is $\omega\approx 0.1428$. This
corresponds to the squeezing factor $10\log_{10}e^{2\omega}\approx 1.24$ dB.
Fig. 11 further explores the squeezing factor for our receiver. We note that
$\omega$ is a very-slowly-increasing function of thermal noise photon number
per mode $\bar{n}_{\rm B}$.555Numerical experiments suggest that $\omega$
grows sub-logarithmically with $\bar{n}_{\rm B}$. Hence, we set $\bar{n}_{\rm
B}=1$. We show that, unless transmittance $\theta$ is very small, for low
transmitted mean photon number per mode $\bar{n}_{\rm S}$ (which is the regime
where our sensor is optimal), substantially less than 10 dB of squeezing is
needed – a figure that has been demonstrated at 1550 nm “telecom” wavelength
[50]. Fig. 11 also illustrates how the existence of $\omega$ depends on the
parameters $\theta$, $\bar{n}_{\rm S}$, and $\bar{n}_{\rm B}$: when
$\bar{n}_{\rm S}=0.1$ and $\bar{n}_{\rm B}=1$, the condition in (11) is not
satisfied for $\theta<0.2882$. However, (11) is satisfied for $\theta>0.0385$
and $\theta>0.0040$ when $\bar{n}_{\rm S}=0.01$ and $\bar{n}_{\rm S}=0.001$,
respectively. Thus, at low transmitted power, our sensor can estimate almost
the entire range of $\theta$.

Figure 11: Squeezing factor $10\log_{10}e^{2\omega}$ vs. transmittance
$\theta$ for various values of transmitted mean photon number per mode
$\bar{n}_{\rm S}$. Thermal noise mean photon number per mode $\bar{n}_{\rm
B}=1$, results for other values are very similar, as $\omega$ grows very
slowly with $\bar{n}_{\rm B}$. The squeezing factor is shown only over the
regions of $\theta$ where the condition (11) holds and a solution for $\omega$
exists. These are $0.2882<\theta<1$ for $\bar{n}_{\rm S}=0.1$,
$0.0385<\theta<1$ for $\bar{n}_{\rm S}=0.01$, and $0.0040<\theta<1$ for
$\bar{n}_{\rm S}=0.001$. The squeezing factor required when $\bar{n}_{\rm
S}=0.01$, $\bar{n}_{\rm B}=1$, and $\theta=0.5$ (Section VI-B settings) is
marked with $\otimes$.

In our simulations, we approximate the ideal PNR by one that resolves up to 9
photons, which captures $>99\%$ of the probability mass in (40). Such
resolution has been demonstrated at 1550 nm using superconducting transition-
edge sensors (TES) [51]. Fig. 10 shows convergence to optimal MSE requiring
measurement of $<10^{5}$ probes. The output of a continuous-wave (cw)
spontaneous parametric down-conversion (SPDC) source of entangled photons
typically has an optical bandwidth of $W\sim 1$ THz [60]. The quantum
description of a $T$-second cw SPDC output is a tensor product of $n\approx
WT$ two-mode squeezed vacuum (TMSV) states. Due to the high modes-per-second
output of the SPDC, the duration of an experimental run will be governed by
the electronic bandwidth of the detector, which for TES is dominated by its
dead, or, reset time. The count rate of the TES reported in [51] is $\approx
20\times 10^{3}$ counts/s, which is far slower than commercially-available
(non-photon-number-resolving) superconducting nanowire single-photon detectors
(e.g., [61]). However, even the TES rate allows a less-than-a-second duration
for each experimental run that we simulated.

##  VII Conclusion

We showed that the TMSV state asymptotically minimizes the quantum CRB for
transmittance estimation in thermal noise and low-transmitted-power regime
over all states (not necessarily Gaussian). We derived a quantum CRB-achieving
receiver structure for TMSV source: a two-mode squeezer followed by the PNR
measurement. Although our design is restricted to a certain range of
parameters, the range corresponds to the low transmitted photon number regime
where TMSV source achieves the ultimate quantum FI bound in (7). Nevertheless,
alternate receiver structures for TMSV input that are not restricted to this
range of parameters should be explored. The structure of our receiver depends
on the parameter of interest, as is typical in quantum-enhanced sensing. This
necessitates a two-stage estimation approach [11, Ch. 6.4], [27, 28], however,
our simulations suggest that its impact on the MSE is negligible.

In our work, we assume thermal channel noise, availability of perfect PNR
measurement, error-free idler mode storage, and its perfect synchronization
with the returned probe mode. These are also standard assumptions in
theoretical investigations of quantum-enhanced sensing (e.g., [17, 18, 19, 21,
22, 23, 26, 25]). Strict enforcement of these assumptions in a controlled
laboratory environment allowed the experimental demonstration of quantum
advantage for target detection [24]. Our simulation results in Section VI
suggest a high likelihood of success for a similar experimental validation of
quantum advantage in transmittance sensing, with small transmitted photon
number ($\bar{n}_{\rm S}\approx 0.01$), and relatively large thermal noise
photon number ($\bar{n}_{\rm B}\approx 1$). However, our future work will
focus on extending both our analytic and numeric framework to account for the
limitations of practical systems. This will allow exploration of trade-offs
between their complexity and performance and hasten the integration of
quantum-enhanced sensing protocols into practical systems.

##  Appendix A Proof of Theorem 1

The proof of Theorem 1 was presented at the International Symposium on
Information Theory (ISIT) 2021 and included in its proceedings [1].

As explained in Section III, for input tensor product of TMSV states
$\left|\psi\right>\left<\psi\right|_{IS}^{\otimes n}$, the output is a tensor-
product state $\hat{\sigma}_{I^{n}R^{n}}(\theta)=\hat{\sigma}_{IR}^{\otimes
n}(\theta)$ and quantum FI
$\mathcal{J}_{\theta}\left(\hat{\sigma}_{I^{n}R^{n}}(\theta)\right)=n\mathcal{J}_{\theta}\left(\hat{\sigma}_{IR}(\theta)\right)$.
Since the TMSV state and the bosonic channel are Gaussian, the output state
$\hat{\sigma}_{IR}(\theta)$ is also Gaussian. This allows the use of the
symplectic formalism [44]. We use the $\hat{q}\hat{q}\hat{p}\hat{p}$ form for
representing and evolving the covariance matrices of Gaussian states in phase-
space, where $\hat{q}=\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}}$ and
$\hat{p}=\frac{\hat{a}-\hat{a}^{\dagger}}{\sqrt{2}}$ are the quadrature
operators. The input TMSV state’s covariance matrix is:

$\displaystyle\Sigma_{\hat{\rho}_{IS}}$
$\displaystyle=\left[\begin{array}[]{cccc}u_{1}&u_{2}&0&0\\\
u_{2}&u_{1}&0&0\\\ 0&0&u_{1}&-u_{2}\\\ 0&0&-u_{2}&u_{1}\end{array}\right],$
(48)

where $u_{1}=\bar{n}_{\rm S}+\frac{1}{2}$ and $u_{2}=\sqrt{\bar{n}_{\rm
S}(\bar{n}_{\rm S}+1)}$. The action of the lossy thermal-noise bosonic channel
$\mathcal{E}^{\bar{n}_{\rm T},\theta}_{S\rightarrow R}$ on the signal mode
does not displace the state and results in the covariance matrix of the
output:

$\displaystyle\Sigma_{\hat{\sigma}_{IR}(\theta)}$
$\displaystyle=X\Sigma_{\hat{\rho}_{IS}}X^{\rm T}+Y$ (49)
$\displaystyle=\left[\begin{array}[]{cccc}w_{11}&w_{12}&0&0\\\
w_{12}&w_{22}&0&0\\\ 0&0&w_{11}&-w_{12}\\\
0&0&-w_{12}&w_{22}\end{array}\right],$ (54)

where $Y=\operatorname{diag}\left(0,\bar{n}_{\rm
B}+\frac{1}{2}-\frac{\theta}{2},0,\bar{n}_{\rm
B}+\frac{1}{2}-\frac{\theta}{2}\right)$,
$X=\operatorname{diag}\left(1,\sqrt{\theta},1,\sqrt{\theta}\right)$, $A^{\rm
T}$ is a transpose of $A$, and

$\displaystyle w_{11}$ $\displaystyle=\bar{n}_{\rm S}+\frac{1}{2},$ (55)
$\displaystyle w_{22}$ $\displaystyle=\bar{n}_{\rm B}+\theta\bar{n}_{\rm
S}+\frac{1}{2},$ (56) $\displaystyle w_{12}$
$\displaystyle=\sqrt{\theta\bar{n}_{\rm S}(\bar{n}_{\rm S}+1)}.$ (57)

We now derive the quantum FI using the method in [47, 48]. First, the Uhlmann
fidelity between zero-displacement Gaussian states $\hat{\rho}_{1}$ and
$\hat{\rho}_{2}$ with covariance matrices $\Sigma_{1}$ and $\Sigma_{2}$ is
[47]:

$\displaystyle\mathcal{F}\left(\hat{\rho}_{1},\hat{\rho}_{2}\right)$
$\displaystyle=\frac{1}{\sqrt{\sqrt{\Gamma}+\sqrt{\Lambda}-\sqrt{(\sqrt{\Gamma}+\sqrt{\Lambda})^{2}-\Delta}}},$
(58)

where the symplectic invariants are:

$\displaystyle\Delta$
$\displaystyle=\det\left(\Sigma_{1}+\Sigma_{2}\right)\geq 1$ (59)
$\displaystyle\Gamma$
$\displaystyle=16\det\left(\Omega\Sigma_{1}\Omega\Sigma_{2}-\frac{I_{4\times
4}}{4}\right)\geq\Delta$ (60) $\displaystyle\Lambda$
$\displaystyle=16\det\left(\Sigma_{1}+\frac{j}{2}\Omega\right)\det\left(\Sigma_{2}+\frac{j}{2}\Omega\right)\geq
0,$ (61)

$\Omega=\left[\begin{array}[]{cc}0_{2\times 2}&I_{2\times 2}\\\ -I_{2\times
2}&0_{2\times 2}\end{array}\right]$ is the symplectic matrix, $I_{m\times m}$
is the $m\times m$ identity matrix, and $j=\sqrt{-1}$ is the imaginary unit.
The quantum FI is calculated using (58) as follows:

$\displaystyle\mathcal{J}_{\theta}\left(\hat{\sigma}_{IR}(\theta)\right)$
$\displaystyle=$
$\displaystyle-4\left.\frac{\partial^{2}\mathcal{F}\left(\hat{\sigma}_{IR}(\theta),\hat{\sigma}_{IR}(\theta+\delta_{\theta})\right)}{\partial\delta_{\theta}^{2}}\right|_{\delta_{\theta}=0},$
(62)

where (62) is derived in [48]. Evaluating (62) yields (10).

##  Appendix B Derivation of a Quantum CRB-achieving Receiver

We obtain a quantum CRB-achieving receiver for transmittance by adapting the
approach from [14]. We derive an eigendecomposion
$\left\\{\hat{S}(\omega)\left|km\right>:k,m=0,1,\ldots\right\\}$ of
$\hat{\Lambda}_{\theta}$ in three steps: 1) we find an orthonormal basis
$\left\\{\left|\psi_{km}\right>\right\\}$, $k,m=0,1,\ldots$ for the output
state $\hat{\sigma}_{IR}(\theta)$; 2) we use
$\left\\{\left|\psi_{km}\right>\right\\}$ to write $\hat{\Lambda}_{\theta}$ as
a linear combination of creation and annihilation operators $\hat{a}_{I}$,
$\hat{a}_{R}$, $\hat{a}_{I}^{\dagger}$, and $\hat{a}_{R}^{\dagger}$ of the
return and idler modes; and, 3) we recognize that the resulting linear
combination is produced by an action of a two-mode squeezing operator
$\hat{S}(\omega)$ on a number operator, yielding an expression for $\omega$.

###  B-A Orthonormal Basis for the Output State $\hat{\sigma}_{IR}(\theta)$

Squeezing the two modes of $\hat{\sigma}_{IR}(\theta)$ yields
$\hat{\sigma}_{IR}^{\rm
d}(\theta)=\hat{S}(\zeta)\hat{\sigma}_{IR}(\theta)\hat{S}^{\dagger}(\zeta)$
with the covariance matrix:

$\displaystyle\Sigma_{\hat{\sigma}_{IR}^{\rm d}(\theta)}$
$\displaystyle=Z\Sigma_{\hat{\sigma}_{IR}(\theta)}Z^{\rm T}$ (63)
$\displaystyle=\left[\begin{array}[]{cccc}w^{\rm d}_{11}&w^{\rm d}_{12}&0&0\\\
w^{\rm d}_{12}&w^{\rm d}_{22}&0&0\\\ 0&0&w^{\rm d}_{11}&-w^{\rm d}_{12}\\\
0&0&-w^{\rm d}_{12}&w^{\rm d}_{22}\end{array}\right],$ (68)

where

$\displaystyle Z$
$\displaystyle=\left[\begin{array}[]{cccc}\cosh(\zeta)&\sinh(\zeta)&0&0\\\
\sinh(\zeta)&\cosh(\zeta)&0&0\\\ 0&0&\cosh(\zeta)&-\sinh(\zeta)\\\
0&0&-\sinh(\zeta)&\cosh(\zeta)\end{array}\right]$ (73) $\displaystyle w^{\rm
d}_{11}$
$\displaystyle=w_{11}\cosh^{2}(\zeta)+w_{22}\sinh^{2}(\zeta)+w_{12}\sinh(2\zeta),$
(74) $\displaystyle w^{\rm d}_{22}$
$\displaystyle=w_{22}\cosh^{2}(\zeta)+w_{11}\sinh^{2}(\zeta)+w_{12}\sinh(2\zeta),$
(75) $\displaystyle w^{\rm d}_{12}$
$\displaystyle=w_{12}\cosh(2\zeta)+(w_{11}+w_{22})\cosh(\zeta)\sinh(\zeta).$
(76)

A value of $\zeta$ such that

$\displaystyle w^{\rm d}_{12}$ $\displaystyle=0$ (77)

makes $\hat{\sigma}_{IR}^{\rm d}(\theta)$ a thermal state that is diagonal in
the Fock basis. Note that in (76), $w_{11}>0$, $w_{22}>0$, $w_{12}>0$ and
$\cosh(\zeta)>0$. Thus, $\sinh(\zeta)<0$. Solution of (77) for $\zeta$ under
these constraints satisfies:

$\displaystyle\cosh(\zeta)$ $\displaystyle=\sqrt{\frac{1+\bar{n}_{\rm
B}+\bar{n}_{\rm S}+\theta\bar{n}_{\rm S}}{2\sqrt{a}}+\frac{1}{2}}\equiv\mu,$
(78) $\displaystyle\sinh(\zeta)$ $\displaystyle=-\sqrt{\frac{1+\bar{n}_{\rm
B}+\bar{n}_{\rm S}+\theta\bar{n}_{\rm S}}{2\sqrt{a}}-\frac{1}{2}}\equiv\nu.$
(79)

This yields, in turn,

$\displaystyle\zeta$ $\displaystyle=\log\left(\sqrt{\frac{1+\bar{n}_{\rm
B}+\bar{n}_{\rm S}+\theta\bar{n}_{\rm S}}{2\sqrt{a}}+\frac{1}{2}}\right.$
$\displaystyle\phantom{=\log\Bigg{(}}\left.-\sqrt{\frac{1+\bar{n}_{\rm
B}+\bar{n}_{\rm S}+\theta\bar{n}_{\rm S}}{2\sqrt{a}}-\frac{1}{2}}\right),$
(80)

where

$\displaystyle a$ $\displaystyle=\bar{n}_{\rm B}^{2}+(1+\bar{n}_{\rm
S}(1-\theta))^{2}+2\bar{n}_{\rm B}(1+\bar{n}_{\rm S}+\theta\bar{n}_{\rm S}).$
(81)

The covariance matrix of $\hat{\sigma}_{IR}^{\rm d}(\theta)$ can be expressed
as

$\displaystyle\Sigma_{\hat{\sigma}_{IR}^{\rm d}(\theta)}$
$\displaystyle=\left[\begin{array}[]{cccc}N_{1}+\frac{1}{2}&0&0&0\\\
0&N_{2}+\frac{1}{2}&0&0\\\ 0&0&N_{1}+\frac{1}{2}&0\\\
0&0&0&N_{2}+\frac{1}{2}\end{array}\right],$ (86)

where the mean thermal photon numbers in each mode are:

$\displaystyle N_{1}$ $\displaystyle=w^{\rm
d}_{11}-\frac{1}{2}=\frac{1}{2}\left(\sqrt{a}+\bar{n}_{\rm B}-1-\bar{n}_{\rm
S}(1-\theta)\right)$ (87) $\displaystyle N_{2}$ $\displaystyle=w^{\rm
d}_{22}-\frac{1}{2}=\frac{1}{2}\left(\sqrt{a}-\bar{n}_{\rm B}-1+\bar{n}_{\rm
S}(1-\theta)\right).$ (88)

Hence, using two-mode Fock basis, we have:

$\displaystyle\hat{\sigma}_{IR}^{\rm d}(\theta)$
$\displaystyle=\sum_{km}r_{km}\left|km\right>\left<km\right|,$ (89)

where, using the definition of $q(k,\bar{n})$ from (1),

$\displaystyle r_{km}$ $\displaystyle\equiv
q(k,N_{1})q(m,N_{2})=\frac{N_{1}^{k}N_{2}^{m}}{(1+N_{1})^{k+1}(1+N_{2})^{m+1}}.$
(90)

Therefore, the output state $\hat{\sigma}_{IR}(\theta)$ is diagonal in the
two-mode squeezed Fock basis $\\{\left|\psi_{km}\right>\\}$, $k,m=0,1,\ldots$:

$\displaystyle\hat{\sigma}_{IR}(\theta)=$
$\displaystyle\hat{S}^{\dagger}(\zeta)\hat{\sigma}_{IR}^{\rm
d}(\theta)\hat{S}(\zeta)=\sum_{km}r_{km}\left|\psi_{km}\right>\left<\psi_{km}\right|,$
(91)

where $\left|\psi_{km}\right>=\hat{S}^{\dagger}(\zeta)\left|km\right>$ defines
an orthonormal basis.

###  B-B Actions of Modal Creation and Annihilation Operators on Output State

Since the squeezing parameter $\zeta$ is real, we have [42, Eq. (5.35)]

$\displaystyle S(\zeta)\hat{a}_{R}S^{\dagger}(\zeta)$
$\displaystyle=\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger},$ (92) $\displaystyle
S(\zeta)\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)$
$\displaystyle=\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I},$ (93)

where $\mu$ and $\nu$ are defined in (78) and (79), respectively. These facts,
the diagonalization of the output state $\hat{\sigma}_{IR}(\theta)$ in (91),
and the photon-number raising and lowering properties of creation and
annihilation operators allow us to derive the following six expressions for
use in Appendix B-C:

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\left<\psi_{km}\right|2\hat{a}_{R}\hat{\sigma}_{IR}(\theta)\hat{a}_{R}^{\dagger}\left|\psi_{st}\right>}$
(94) $\displaystyle=$
$\displaystyle\left<km\right|S(\zeta)2\hat{a}_{R}S^{\dagger}(\zeta)\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|S(\zeta)\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|(\mu\sqrt{r+1}\left|s+1,t\right>$
$\displaystyle\phantom{\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|}+\nu\sqrt{t-1}\left|s,t-1\right>)$
$\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(\mu
r_{s+1,t}\sqrt{r+1}\left|s+1,t\right>$
$\displaystyle\phantom{\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(}+\nu
r_{s,t-1}\sqrt{t-1}\left|s,t-1\right>)$ $\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(\mu
r_{s+1,t}\hat{a}_{R}^{\dagger}+\nu r_{s,t-1}\hat{a}_{I})\left|st\right>$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\left<\psi_{km}\right|\hat{a}_{R}^{\dagger}\hat{a}_{R}\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}$
(95) $\displaystyle=$
$\displaystyle\left<km\right|S(\zeta)\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)S(\zeta)\hat{a}_{R}S^{\dagger}(\zeta)\sum_{ul}r_{ul}\left|ul\right>\langle{\psi_{ul}}|{\psi_{st}}\rangle$
$\displaystyle=$
$\displaystyle\left<km\right|(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})r_{st}\left|st\right>$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\left<\psi_{km}\right|\hat{\sigma}_{IR}(\theta)\hat{a}_{R}^{\dagger}\hat{a}_{R}\left|\psi_{st}\right>}$
(96) $\displaystyle=$
$\displaystyle\left<\psi_{km}\right|\sum_{ul}r_{ul}\left|\psi_{ul}\right>\left<ul\right|S(\zeta)\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)S(\zeta)\hat{a}_{R}S^{\dagger}(\zeta)\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|r_{km}(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})\left|st\right>$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\left<\psi_{km}\right|2\hat{a}_{R}^{\dagger}\hat{\sigma}_{IR}(\theta)\hat{a}_{R}\left|\psi_{st}\right>}$
(97) $\displaystyle=$
$\displaystyle\left<km\right|S(\zeta)2\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|S(\zeta)\hat{a}_{R}S^{\dagger}(\zeta)\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|(\mu\sqrt{k}\left|s-1,t\right>$
$\displaystyle\phantom{\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})\sum_{ul}r_{ul}\left|ul\right>\left<ul\right|}+\nu\sqrt{t+1}\left|s,t+1\right>)$
$\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(\mu
r_{s-1,t}\sqrt{k}\left|s-1,t\right>$
$\displaystyle\phantom{\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(}+\nu
r_{s,t+1}\sqrt{t+1}\left|s,t+1\right>)$ $\displaystyle=$
$\displaystyle\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(\mu
r_{s-1,t}\hat{a}_{R}+\nu r_{s,t+1}\hat{a}_{I}^{\dagger})\left|st\right>$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\left<\psi_{km}\right|\hat{a}_{R}\hat{a}_{R}^{\dagger}\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}$
(98) $\displaystyle=$
$\displaystyle\left<km\right|S(\zeta)\hat{a}_{R}S^{\dagger}(\zeta)S(\zeta)\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)\sum_{ul}r_{ul}\left|ul\right>\langle{\psi_{ul}}|{\psi_{st}}\rangle$
$\displaystyle=$
$\displaystyle\left<km\right|(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})r_{st}\left|st\right>$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\left<\psi_{km}\right|\hat{\sigma}_{IR}(\theta)\hat{a}_{R}\hat{a}_{R}^{\dagger}\left|\psi_{st}\right>}$
(99) $\displaystyle=$
$\displaystyle\left<\psi_{km}\right|\sum_{ul}r_{ul}\left|\psi_{ul}\right>\left<ul\right|S(\zeta)\hat{a}_{R}S^{\dagger}(\zeta)S(\zeta)\hat{a}_{R}^{\dagger}S^{\dagger}(\zeta)\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|r_{km}(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})\left|st\right>$

###  B-C Characterization of SLD $\hat{\Lambda}_{\theta}$

First, we use (5) to relate the $kmst$-th term of the SLD operator
$\hat{\Lambda}_{\theta}$ in the basis $\\{\left|\psi_{km}\right>\\}$ to the
corresponding term of the derivative
$\frac{d\hat{\sigma}_{IR}(\theta)}{d\theta}$:

$\displaystyle\left<\psi_{km}\right|\frac{d\hat{\sigma}_{IR}(\theta)}{d\theta}\left|\psi_{st}\right>$
$\displaystyle=\frac{1}{2}\left<\psi_{km}\right|\hat{\sigma}_{IR}(\theta)\hat{\Lambda}_{\theta}\left|\psi_{st}\right>$
$\displaystyle\phantom{=}+\frac{1}{2}\left<\psi_{km}\right|\hat{\Lambda}_{\theta}\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>$
(100)
$\displaystyle=\frac{r_{km}+r_{st}}{2}\left<\psi_{km}\right|\hat{\Lambda}_{\theta}\left|\psi_{st}\right>.$
(101)

Thus, the SLD operator is expressed as follows:

$\displaystyle\hat{\Lambda}_{\theta}$
$\displaystyle=\sum_{kmst}\left<\psi_{km}\right|\frac{2\frac{d\hat{\sigma}_{IR}(\theta)}{d\theta}}{r_{km}+r_{st}}\left|\psi_{st}\right>\left|\psi_{km}\right>\left<\psi_{st}\right|.$
(102)

The probe state evolving in a thermal bath is characterized by the Lindblad
Master equation [62, Ch. 4]:

$\displaystyle\frac{d\hat{\sigma}_{IR}(\theta)}{dt}=\frac{\gamma}{2}\left[(\bar{n}_{\rm
B}+1)\hat{\mathcal{L}}_{R}[\hat{a}]+\bar{n}_{\rm
B}\hat{\mathcal{L}}_{R}[\hat{a}^{\dagger}]\right]\hat{\sigma}_{IR}(\theta),$
(103)

where the superoperator $\hat{\mathcal{L}}_{R}[\cdot]$ is defined as follows:

$\displaystyle\hat{\mathcal{L}}_{R}[\hat{a}]\hat{\sigma}_{IR}(\theta)$
$\displaystyle=2\hat{a}_{R}\hat{\sigma}_{IR}(\theta)\hat{a}_{R}^{\dagger}-\hat{a}_{R}^{\dagger}\hat{a}_{R}\hat{\sigma}_{IR}(\theta)$
$\displaystyle\phantom{=}-\hat{\sigma}_{IR}(\theta)\hat{a}_{R}^{\dagger}\hat{a}_{R}$
(104)
$\displaystyle\hat{\mathcal{L}}_{R}[\hat{a}^{\dagger}]\hat{\sigma}_{IR}(\theta)$
$\displaystyle=2\hat{a}_{R}^{\dagger}\hat{\sigma}_{IR}(\theta)\hat{a}_{R}-\hat{a}_{R}\hat{a}_{R}^{\dagger}\hat{\sigma}_{IR}(\theta)$
$\displaystyle\phantom{=}-\hat{\sigma}_{IR}(\theta)\hat{a}_{R}\hat{a}_{R}^{\dagger}.$
(105)

The dissipation rate $\gamma$ satisfies
$\exp\left(-\frac{\gamma}{2}t\right)=\sqrt{\theta}$, which, in turn, implies:

$\displaystyle\frac{dt}{d\theta}=\frac{1}{\gamma\theta}.$ (106)

Employing the chain rule in (103) using (106), and substituting the result
into (102) yields:

$\displaystyle\hat{\Lambda}_{\theta}$
$\displaystyle=\sum_{kmst}\left<\psi_{km}\right|\frac{2\frac{d\hat{\sigma}_{IR}(\theta)}{dt}\frac{dt}{d\theta}}{r_{km}+r_{st}}\left|\psi_{st}\right>\left|\psi_{km}\right>\left<\psi_{st}\right|$
(107) $\displaystyle=\frac{(\bar{n}_{\rm
B}+1)}{\theta}\sum_{kmst}\frac{\left<\psi_{km}\right|\hat{\mathcal{L}}_{R}[\hat{a}]\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}\left|\psi_{km}\right>\left<\psi_{st}\right|$
$\displaystyle\phantom{=}+\frac{\bar{n}_{\rm
B}}{\theta}\sum_{kmst}\frac{\left<\psi_{km}\right|\hat{\mathcal{L}}_{R}[\hat{a}^{\dagger}]\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}\left|\psi_{km}\right>\left<\psi_{st}\right|$
(108)

We analyze the two summations in (108) separately. First,

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\frac{\left<\psi_{km}\right|\hat{\mathcal{L}}_{R}[\hat{a}]\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}}$
$\displaystyle=$
$\displaystyle\frac{\left<\psi_{km}\right|2\hat{a}_{R}\hat{\sigma}_{IR}(\theta)\hat{a}_{R}^{\dagger}\left|\psi_{st}\right>}{r_{km}+r_{st}}$
$\displaystyle-\frac{\left<\psi_{km}\right|\hat{a}_{R}^{\dagger}\hat{a}_{R}\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}$
$\displaystyle-\frac{\left<\psi_{km}\right|\hat{\sigma}_{IR}(\theta)\hat{a}_{R}^{\dagger}\hat{a}_{R}\left|\psi_{st}\right>}{r_{km}+r_{st}}$
$\displaystyle=$
$\displaystyle\frac{\left<km\right|2(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(\mu
r_{s+1,t}\hat{a}_{R}^{\dagger}+\nu
r_{s,t-1}\hat{a}_{I})\left|st\right>}{r_{km}+r_{st}}$
$\displaystyle-\left<km\right|(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|\left(\frac{2r_{s+1,t}}{r_{km}+r_{st}}-1\right)\mu^{2}\hat{a}_{R}^{\dagger}\hat{a}_{R}\left|st\right>$
$\displaystyle+\left<km\right|\left(\frac{2r_{s+1,t}}{r_{km}+r_{st}}-1\right)\nu\mu\hat{a}_{I}^{\dagger}\hat{a}_{R}^{\dagger}\left|st\right>$
$\displaystyle+\left<km\right|\left(\frac{2r_{s,t-1}}{r_{km}+r_{st}}-1\right)\mu\nu\hat{a}_{R}\hat{a}_{I}\left|st\right>$
$\displaystyle+\left<km\right|\left(\frac{2r_{s,t-1}}{r_{km}+r_{st}}-1\right)\nu^{2}\hat{a}_{I}^{\dagger}\hat{a}_{I}\left|st\right>$
$\displaystyle+\left(\frac{2\mu^{2}r_{s+1,t}}{r_{km}+r_{st}}-\nu^{2}\right)\left\langle
km\middle|st\right\rangle,$ (111)

where $\mu$ an $\nu$ are defined in (78) and (79), respectively, (B-C) is
derived using (94)-(99) in Appendix B-B, and (111) is due to the commutation
relation $\left[\hat{a},\hat{a}^{\dagger}\right]=\hat{I}$ (with $\hat{I}$
denoting the identity operator), and rearrangement of terms. Observe that the
first, fourth, and fifth terms in $\eqref{eq:second,hatK}$ are not zero only
when $\\{k=s,m=t\\}$, while the second and third terms are not zero when
$\\{k=s+1,m=t+1\\}$ and $\\{k=s-1,m=t-1\\}$, respectively. Since

$\displaystyle\left.\frac{2r_{s+1,t}}{r_{km}+r_{st}}\right|_{k=s,m=t}-1$
$\displaystyle=-\frac{1}{1+N_{1}}$ (112)
$\displaystyle\left.\frac{2r_{s+1,t}}{r_{km}+r_{st}}\right|_{k=s+1,m=t+1}-1$
$\displaystyle=\frac{N_{1}-N_{2}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}$ (113)
$\displaystyle\left.\frac{2r_{s,t-1}}{r_{km}+r_{st}}\right|_{k=s-1,m=t-1}-1$
$\displaystyle=\frac{N_{1}-N_{2}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}$ (114)
$\displaystyle\left.\frac{2r_{s,t-1}}{r_{km}+r_{st}}\right|_{k=s,m=t}-1$
$\displaystyle=\frac{1}{N_{2}},$ (115)

with $N_{1}$ and $N_{2}$ defined in (87) and (88), respectively, we have:

$\sum_{kmst}\frac{\left<\psi_{km}\right|\hat{\mathcal{L}}_{R}[\hat{a}]\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}\left|\psi_{km}\right>\left<\psi_{st}\right|\\\
=\hat{S}^{\dagger}(\zeta)\hat{K}_{1}\hat{S}(\zeta),$ (116)

where

$\displaystyle\hat{K}_{1}$
$\displaystyle=\left(\frac{N_{1}-N_{2}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}\right)\nu\mu\left(\hat{a}_{I}^{\dagger}\hat{a}_{R}^{\dagger}+\hat{a}_{R}\hat{a}_{I}\right)$
$\displaystyle\phantom{=}-\frac{\mu^{2}\hat{a}_{R}^{\dagger}\hat{a}_{R}}{1+N_{1}}+\frac{\nu^{2}}{N_{2}}\hat{a}_{I}^{\dagger}\hat{a}_{I}+\left(\mu^{2}\frac{N_{1}}{1+N_{1}}-\nu^{2}\right)\hat{I}.$
(117)

Now, the $kmst$-th term in the second summation in (108) is:

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\frac{\left<\psi_{km}\right|\hat{\mathcal{L}}_{R}[\hat{a}^{\dagger}]\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}}$
$\displaystyle=$
$\displaystyle\frac{\left<\psi_{km}\right|2\hat{a}_{R}^{\dagger}\hat{\sigma}_{IR}(\theta)\hat{a}_{R}\left|\psi_{st}\right>}{r_{km}+r_{st}}$
$\displaystyle-\frac{\left<\psi_{km}\right|\hat{a}_{R}\hat{a}_{R}^{\dagger}\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}$
$\displaystyle-\frac{\left<\psi_{km}\right|\hat{\sigma}_{IR}(\theta)\hat{a}_{R}\hat{a}_{R}^{\dagger}\left|\psi_{st}\right>}{r_{km}+r_{st}}$
$\displaystyle=$
$\displaystyle\frac{\left<km\right|2(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})(\mu
r_{s-1,t}\hat{a}_{R}+\nu
r_{s,t+1}\hat{a}_{I}^{\dagger})\left|st\right>}{r_{km}+r_{st}}$
$\displaystyle-\left<km\right|(\mu\hat{a}_{R}+\nu\hat{a}_{I}^{\dagger})(\mu\hat{a}_{R}^{\dagger}+\nu\hat{a}_{I})\left|st\right>$
$\displaystyle=$
$\displaystyle\left<km\right|\left(\frac{2r_{s-1,t}}{r_{km}+r_{st}}-1\right)\mu^{2}\hat{a}_{R}^{\dagger}\hat{a}_{R}\left|st\right>$
$\displaystyle+\left<km\right|\left(\frac{2r_{s-1,t}}{r_{km}+r_{st}}-1\right)\nu\mu\hat{a}_{I}\hat{a}_{R}\left|st\right>$
$\displaystyle+\left<km\right|\left(\frac{2r_{s,t+1}}{r_{km}+r_{st}}-1\right)\mu\nu\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}\left|st\right>$
$\displaystyle+\left<km\right|\left(\frac{2r_{s,t+1}}{r_{km}+r_{st}}-1\right)\nu^{2}\hat{a}_{I}^{\dagger}\hat{a}_{I}\left|st\right>$
$\displaystyle+\left(\frac{2\nu^{2}r_{s+1,t}}{r_{km}+r_{st}}-\mu^{2}\right)\left\langle
km\middle|st\right\rangle,$ (120)

where $\mu$ an $\nu$ are defined in (78) and (79), respectively, (B-C) is
derived using (94)-(99) in Appendix B-B, and (120) is due to the commutation
relation $\left[\hat{a},\hat{a}^{\dagger}\right]=\hat{I}$, and rearrangement
of terms. Observe that the first, fourth, and fifth terms in
$\eqref{eq:fourth,hatK}$ are not zero only when $\\{k=s,m=t\\}$, while the
second and third terms are not zero when $\\{k=s-1,m=t-1\\}$ and
$\\{k=s+1,m=t+1\\}$, respectively. Since

$\displaystyle\left.\frac{2r_{s-1,t}}{r_{km}+r_{st}}\right|_{k=s,m=t}-1$
$\displaystyle=\frac{1}{N_{1}}$ (121)
$\displaystyle\left.\frac{2r_{s-1,t}}{r_{km}+r_{st}}\right|_{k=s-1,m=t-1}-1$
$\displaystyle=\frac{N_{2}-N_{1}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}$ (122)
$\displaystyle\left.\frac{2r_{s,t+1}}{r_{km}+r_{st}}\right|_{k=s+1,m=t+1}-1$
$\displaystyle=\frac{N_{2}-N_{1}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}$ (123)
$\displaystyle\left.\frac{2r_{s,t+1}}{r_{km}+r_{st}}\right|_{k=s,m=t}-1$
$\displaystyle=-\frac{1}{1+N_{2}},$ (124)

we have:

$\sum_{kmst}\frac{\left<\psi_{km}\right|\hat{\mathcal{L}}_{R}[\hat{a}^{\dagger}]\hat{\sigma}_{IR}(\theta)\left|\psi_{st}\right>}{r_{km}+r_{st}}\left|\psi_{km}\right>\left<\psi_{st}\right|\\\
=\hat{S}^{\dagger}(\zeta)\hat{K}_{2}\hat{S}(\zeta),$ (125)

where

$\displaystyle\hat{K}_{2}$
$\displaystyle=\left(\frac{N_{2}-N_{1}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}\right)\nu\mu\left(\hat{a}_{I}^{\dagger}\hat{a}_{R}^{\dagger}+\hat{a}_{R}\hat{a}_{I}\right)$
$\displaystyle\phantom{=}+\frac{\mu^{2}\hat{a}_{R}^{\dagger}\hat{a}_{R}}{N_{1}}-\frac{\nu^{2}}{1+N_{2}}\hat{a}_{I}^{\dagger}\hat{a}_{I}+\left(\nu^{2}\frac{N_{2}}{1+N_{2}}-\mu^{2}\right)\hat{I}.$
(126)

Combining (116) and (125) yields:

$\displaystyle\hat{\Lambda}_{\theta}$
$\displaystyle=\frac{1}{\theta}\hat{S}^{\dagger}(\zeta)\hat{K}\hat{S}(\zeta),$
(127)

where

$\displaystyle\hat{K}$ $\displaystyle=(\bar{n}_{\rm
B}+1)\hat{K}_{1}+\bar{n}_{\rm B}\hat{K}_{2}$ (128)
$\displaystyle=C\hat{a}_{R}^{\dagger}\hat{a}_{R}+D\hat{a}_{I}^{\dagger}\hat{a}_{I}+E(\hat{a}_{I}^{\dagger}\hat{a}_{R}^{\dagger}+\hat{a}_{R}\hat{a}_{I})+F\hat{I},$
(129)

and real scalars

$\displaystyle C$ $\displaystyle=\mu^{2}\frac{\bar{n}_{\rm
B}-N_{1}}{N_{1}(1+N_{1})}$ (130) $\displaystyle D$
$\displaystyle=\nu^{2}\left(\frac{\bar{n}_{\rm
B}+1+N_{2}}{N_{2}(1+N_{2})}\right)$ (131) $\displaystyle E$
$\displaystyle=\mu\nu\left(\frac{N_{1}-N_{2}-2\bar{n}_{\rm
B}-1}{2N_{1}N_{2}+N_{1}+N_{2}+1}\right)$ (132) $\displaystyle F$
$\displaystyle=\mu^{2}\left(\frac{N_{1}-\bar{n}_{\rm
B}}{1+N_{1}}\right)-\nu^{2}\left(\frac{\bar{n}_{\rm
B}+1+N_{2}}{1+N_{2}}\right),$ (133)

with $\mu$, $\nu$, $N_{1}$, and $N_{2}$ defined in (78), (79), (87), and (88),
respectively.

###  B-D Eigenbasis of the SLD $\hat{\Lambda}_{\theta}$

Application of a two-mode squeezing operator $\hat{S}\left(\lambda
e^{j\theta_{\lambda}}\right)$ to a photon number operator
$\hat{a}_{I}^{\dagger}\hat{a}_{I}$ results in:

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\hat{S}\left(\lambda
e^{j\theta_{\lambda}}\right)\hat{a}_{R}^{\dagger}\hat{a}_{R}\hat{S}^{\dagger}\left(\lambda
e^{j\theta_{\lambda}}\right)}$ (134) $\displaystyle=$
$\displaystyle\hat{S}\left(\lambda
e^{j\theta_{\lambda}}\right)\hat{a}_{R}^{\dagger}\hat{S}^{\dagger}\left(\lambda
e^{j\theta_{\lambda}}\right)\hat{S}\left(\lambda
e^{j\theta_{\lambda}}\right)\hat{a}_{R}\hat{S}^{\dagger}\left(\lambda
e^{j\theta_{\lambda}}\right)$ $\displaystyle=$
$\displaystyle(\kappa\hat{a}_{R}^{\dagger}+\xi
e^{-j\theta_{\lambda}}\hat{a}_{I})(\kappa\hat{a}_{R}+\xi
e^{j\theta_{\lambda}}\hat{a}_{I}^{\dagger})$ $\displaystyle=$
$\displaystyle\kappa^{2}\hat{a}_{R}^{\dagger}\hat{a}_{R}+\kappa\xi(e^{j\theta_{\lambda}}\hat{a}_{R}\hat{a}_{I}+e^{-j\theta_{\lambda}}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger})+\xi^{2}\hat{a}_{I}\hat{a}_{I}^{\dagger},\qquad$

where $\kappa=\cosh\lambda$ and $\xi=\sinh\lambda$. Similarly,

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\hat{S}\left(\lambda
e^{j\theta_{\lambda}}\right)\hat{a}_{I}^{\dagger}\hat{a}_{I}\hat{S}^{\dagger}\left(\lambda
e^{j\theta_{\lambda}}\right)}$ (135) $\displaystyle=$
$\displaystyle\kappa^{2}\hat{a}_{I}^{\dagger}\hat{a}_{I}+\kappa\xi(e^{-j\theta_{\lambda}}\hat{a}_{I}\hat{a}_{R}+e^{j\theta_{\lambda}}\hat{a}_{I}^{\dagger}\hat{a}_{R}^{\dagger})+\xi^{2}\hat{a}_{R}\hat{a}_{R}^{\dagger}.\qquad$

Thus, provided scalars $\lambda$, $F^{\prime}$, $T_{1}$, $T_{2}$ exist, we can
write:

$\displaystyle\hat{K}$
$\displaystyle=F^{\prime}\hat{I}+\hat{S}(\lambda)(T_{1}\hat{a}_{R}^{\dagger}\hat{a}_{R}+T_{2}\hat{a}_{I}^{\dagger}\hat{a}_{I})\hat{S}^{\dagger}(\lambda)$
(136)
$\displaystyle=F^{\prime}\hat{I}+(T_{1}\kappa\xi+T_{2}\kappa\xi)(\hat{a}_{R}\hat{a}_{I}+\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger})+T_{1}\xi^{2}+T_{2}\xi^{2}$
$\displaystyle\phantom{=}+(T_{1}\kappa^{2}+T_{2}\xi^{2})\hat{a}_{R}^{\dagger}\hat{a}_{R}+(T_{1}\xi^{2}+T_{2}\kappa^{2})\hat{a}_{I}^{\dagger}\hat{a}_{I},$
(137)

where (137) is from substituting (134) and (135) in (136) and rearranging
terms. Note that it is necessary that $\theta_{\lambda}=0$, which means that
scalars $\lambda$, $F^{\prime}$, $T_{1}$, $T_{2}$ must be real and satisfy:

$\displaystyle C$ $\displaystyle=T_{1}\kappa^{2}+T_{2}\xi^{2}$ (138)
$\displaystyle D$ $\displaystyle=T_{1}\xi^{2}+T_{2}\kappa^{2}$ (139)
$\displaystyle E$
$\displaystyle=(T_{1}+T_{2})\kappa\xi=(T_{1}+T_{2})\frac{\sinh 2\lambda}{2},$
(140) $\displaystyle F$ $\displaystyle=F^{\prime}+T_{1}\xi^{2}+T_{2}\xi^{2}$
(141)

where scalars $C$, $D$, $E$, and $F$ are given in (130)-(133). Now,

$\displaystyle C-D$
$\displaystyle=T_{1}(\kappa^{2}-\xi^{2})+T_{2}(\xi^{2}-\kappa^{2})=T_{1}-T_{2},$
(142) $\displaystyle C+D$
$\displaystyle=T_{1}(\kappa^{2}+\xi^{2})+T_{2}(\xi^{2}+\kappa^{2})=(T_{1}+T_{2})\cosh
2\lambda,$ (143)

Furthermore,

$\displaystyle(C+D)^{2}-4E^{2}$
$\displaystyle=(T_{1}+T_{2})^{2}(\cosh^{2}2\lambda-\sinh^{2}2\lambda)$
$\displaystyle=(T_{1}+T_{2})^{2}.$ (144)

Using (142)-(144), we obtain:

$\displaystyle T_{1}$ $\displaystyle=\frac{1}{2}(\sqrt{(C+D)^{2}-4E^{2}}+C-D)$
(145) $\displaystyle T_{2}$
$\displaystyle=\frac{1}{2}(\sqrt{(C+D)^{2}-4E^{2}}-C+D)$ (146)
$\displaystyle\cosh 2\lambda$
$\displaystyle=\frac{C+D}{\sqrt{(C+D)^{2}-4E^{2}}}$ (147) $\displaystyle\sinh
2\lambda$ $\displaystyle=\frac{2E}{\sqrt{(C+D)^{2}-4E^{2}}}$ (148)
$\displaystyle F^{\prime}$ $\displaystyle=F-(T_{1}+T_{2})\sinh^{2}(\lambda).$
(149)

Finally, we can show that

$\displaystyle\IEEEeqnarraymulticol{3}{l}{\hat{\Lambda}_{\theta}S^{\dagger}(\zeta)S(\lambda)\left|km\right>}$
(150) $\displaystyle=$
$\displaystyle\frac{1}{\theta}S^{\dagger}(\zeta)\hat{K}S(\zeta)S^{\dagger}(\zeta)S(\lambda)\left|km\right>$
$\displaystyle=$
$\displaystyle\frac{1}{\theta}S^{\dagger}(\zeta)[F^{\prime}+S(\lambda)(T_{1}\hat{a}_{R}^{\dagger}\hat{a}_{R}+T_{2}\hat{a}_{I}^{\dagger}\hat{a}_{I})^{\dagger}S(\lambda)^{\dagger}]S(\lambda)\left|km\right>$
$\displaystyle=$
$\displaystyle\frac{1}{\theta}\left(F^{\prime}+T_{1}k+T_{2}m\right)S^{\dagger}(\zeta)S(\lambda)\left|km\right>,$

where

$\displaystyle\lambda$
$\displaystyle=\frac{1}{2}\log\left(\frac{2E+|C+D|}{\sqrt{(C+D)^{2}-4E^{2}}}\right).$
(151)

Thus, $\hat{S}(\omega)\left|km\right>$, $\omega=\lambda-\zeta$ is an
eigenvector of the SLD $\hat{\Lambda}_{\theta}$.

##  Appendix C Fock-basis Representation of Two-mode Squeezer

To derive the representation in (42) of the two-mode squeezing operator
$\hat{S}(\omega_{1})$ in the two-mode Fock (photon number) basis
$\left\\{\left|km\right>:k,m=0,1,\ldots\right\\}$, note that [63, Eq.
(1.233)]:

$\displaystyle S(\omega_{1})$
$\displaystyle=e^{-\omega_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}+\omega_{1}^{*}\hat{a}_{R}\hat{a}_{I}}$
(152)
$\displaystyle=e^{-\tau_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}}\nu_{1}^{-\hat{a}_{R}^{\dagger}\hat{a}_{R}-\hat{a}_{I}^{\dagger}\hat{a}_{I}-\hat{I}}e^{\tau_{1}^{*}\hat{a}_{R}\hat{a}_{I}},$
(153)

where $\hat{I}$ denotes identity operator, and

$\displaystyle\tau_{1}$
$\displaystyle=\frac{\omega_{1}}{|\omega_{1}|}\tanh|\omega_{1}|$ (154)
$\displaystyle\nu_{1}$ $\displaystyle=\cosh|\omega_{1}|.$ (155)

Now,

$\displaystyle\IEEEeqnarraymulticol{3}{l}{S(\omega_{1})\left|km\right>}$ (161)
$\displaystyle=$ $\displaystyle
e^{-\tau_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}}\nu_{1}^{-\hat{a}_{R}^{\dagger}\hat{a}_{R}-\hat{a}_{I}^{\dagger}\hat{a}_{I}-\hat{I}}e^{\tau_{1}^{*}\hat{a}_{R}\hat{a}_{I}}\left|km\right>$
$\displaystyle=$ $\displaystyle
e^{-\tau_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}}\nu_{1}^{-\hat{a}_{R}^{\dagger}\hat{a}_{R}-\hat{a}_{I}^{\dagger}\hat{a}_{I}-\hat{I}}$
$\displaystyle\times\sum_{u=0}^{\infty}\frac{(\tau_{1}^{*})^{u}}{u!}(\hat{a}_{R}\hat{a}_{I})^{u}\left|km\right>$
$\displaystyle=$
$\displaystyle\sum_{u=0}^{\min(k,m)}e^{-\tau_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}}\nu_{1}^{-\hat{a}_{R}^{\dagger}\hat{a}_{R}-\hat{a}_{I}^{\dagger}\hat{a}_{I}-\hat{I}}\frac{(\tau_{1}^{*})^{u}}{u!}$
$\displaystyle\times\sqrt{\frac{k!m!}{(k-u)!(m-u)!}}\left|k-u,m-u\right>$
$\displaystyle=$
$\displaystyle\sum_{u=0}^{\min(k,m)}e^{-\tau_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}}\nu_{1}^{-k-m+2u-1}\frac{(\tau_{1}^{*})^{u}}{u!}$
$\displaystyle\times\sqrt{\frac{k!m!}{(k-u)!(m-u)!}}\left|k-u,m-u\right>$
$\displaystyle=$
$\displaystyle\sum_{u=0}^{\min(m,n)}\sum_{l=0}^{\infty}\frac{(-\tau_{1})^{l}}{l!}(\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger})^{l}\nu_{1}^{-k-m+2u-1}$
$\displaystyle\times\frac{(\tau_{1}^{*})^{u}}{u!}\sqrt{\frac{k!m!}{(k-u)!(m-u)!}}\left|k-u,m-u\right>$
$\displaystyle=$
$\displaystyle\sum_{u=0}^{\min(k,m)}\sum_{l=0}^{\infty}\frac{(-\tau_{1})^{l}}{l!}\nu_{1}^{-k-m+2u-1}\frac{(\tau_{1}^{*})^{u}}{u!}$
$\displaystyle\phantom{=}\times\sqrt{\frac{(k-u+l)!(m-u+l)!}{(k-u)!(m-u)!}}$
$\displaystyle\phantom{=}\times\sqrt{\frac{k!m!}{(k-u)!(m-u)!}}$
$\displaystyle\phantom{=}\times\left|k-u+l,m-u+l\right>,$

where (161) is the power series representation of the operator exponential
$e^{\tau_{1}^{*}\hat{a}_{R}\hat{a}_{I}}$ and (161) is from applying
annihilation operators $\hat{a}_{R}\hat{a}_{I}$ $u$ times on the two-mode Fock
state $\left|km\right>$. The upper limit on the sum in (161) is because
$\hat{a}\left|0\right>=0$. Furthermore, (161) follows from the power series of
exponential and Fock states being eigenstates of the photon number operator
$\hat{a}^{\dagger}\hat{a}$, (161) is the power series representation of the
operator exponential
$e^{-\tau_{1}\hat{a}_{R}^{\dagger}\hat{a}_{I}^{\dagger}}$, and (161) is from
applying creation operators $u$ times. By orthonormality of the Fock states,
$\left<st\right|S(\omega_{1})\left|km\right>$ is not zero only when

$\displaystyle s$ $\displaystyle=k-u+l\text{~{}and~{}}t=m-u+l.$ (162)

We eliminate the summation over $l$ in (161) by solving for $l$ in (162).
Since (162) also implies that $l-u=s-k=t-m$, rearranging the terms yields
(42).

## Acknowledgment

The authors benefited from discussions with Amit Ashok, Animesh Datta, Prajit
Dhara, and Janis Nötzel, as well as comments from Gail Bash and the anonymous
reviewers. This material is based upon High Performance Computing (HPC)
resources supported by the University of Arizona TRIF, UITS, and Research,
Innovation, and Impact (RII) and maintained by the UArizona Research
Technologies department.

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