# Optimal detection of losses by thermal probes

Carmen Invernizzi<EMAIL_ADDRESS>Dipartimento di Fisica
dell’Università degli Studi di Milano, I-20133 Milano, Italy  Matteo G. A.
Paris<EMAIL_ADDRESS>Dipartimento di Fisica dell’Università
degli Studi di Milano, I-20133 Milano, Italy  CNISM, Udr Milano, I-20133
Milano, Italy  Stefano Pirandola<EMAIL_ADDRESS>Department of Computer
Science, University of York, York YO10 5GH, United Kingdom

###### Abstract

We consider the discrimination of lossy bosonic channels and focus to the case
when one of the values for the loss parameter is zero, i.e., we address the
detection of a possible loss against the alternative hypothesis of an ideal
lossless channel. This discrimination is performed by inputting one-mode or
two-mode squeezed thermal states with fixed total energy. By optimizing over
this class of states, we find that the optimal inputs are pure, thus
corresponding to single- and two-mode squeezed vacuum states. In particular,
we show that for any value of the damping rate smaller than a critical value
there is a threshold on the energy that makes the two-mode squeezed vacuum
state more convenient than the corresponding single-mode state, whereas for
damping larger than this critical value two-mode squeezed vacua are always
better. We then consider the discrimination in realistic conditions, where it
is unlikely to have pure squeezing. Thus by fixing both input energy and
squeezing, we show that two-mode squeezed thermal states are always better
than their single-mode counterpart when all the thermal photons are directed
into the dissipative channel. Besides, this result also holds approximately
for unbalanced distribution of the thermal photons. Finally, we also
investigate the role of correlations in the improvement of detection. For
fixed input squeezing (single-mode or two-mode), we find that the reduction of
the quantum Chernoff bound is a monotone function of the two-mode entanglement
as well as the quantum mutual information and the quantum discord. We thus
verify that employing squeezing in the form of correlations (quantum or
classical) is always a resource for loss detection whenever squeezed thermal
states are taken as input.

###### pacs:

03.67.-a, 42.50.Dv, 42.50.Ex

##  I Introduction

One of the main obstacles to the development of quantum technologies is the
decoherence associated to losses and absorption processes occurring during the
propagation of a quantum signal. The description of the dynamics of systems
subject to noisy environments Breuer , as well the detection, quantification
and estimation of losses and, more generally, the characterization of lossy
channels at the quantum level, received much attention in the recent years
Breuer ; DecoRev ; Dau06 ; Zurek . An efficient characterization of
decoherence is relevant for quantum repeaters Brask , quantum memories Jensen
, cavity QED systems Haroche08 , superconducting quantum circuits Wang ,
quantum teleportation Qtele , quantum cryptography QKD1 ; QKD2  and secret-key
capacities Devetak ; PirSKcapacity .

In this paper we address the discrimination of lossy channels, i.e., we
consider a situation where the loss (damping) rate of a channel may assume
only two possible values and we want to discriminate between them by probing
the channel with a given class of states. In particular, we address the
discrimination of lossy channels for bosonic systems using squeezed thermal
states as probing states, and focus attention to the case when one of the
values for the loss parameter is zero, i.e., we address the detection of a
possible loss against the alternative hypothesis of an ideal lossless channel.
Such kind of discrimination is crucial since a recent analysis Mon11  has
revealed the importance of assessing the deviation from ideal conditions,
i.e., the identity channel, in implementing large-scale quantum communication.

Despite the discrimination of lossy channels has been already considered in
the literature, the approach of this paper is novel. Previous works have in
fact considered this kind of discrimination by constraining the energy
irradiated over the unknown lossy channel but not the total energy employed in
preparing the input states. For instance, in the quantum reading of digital
memories QreadingPRL , the discrimination of lossy channels has been analyzed
by fixing the mean total number of photons irradiated over the channel,
independently on the number of probing modes (this approach has been also
followed by the recent Ref. NairLAST ). Previously, in the quantum
illumination of targets Qillumination ; QilluminationOTHERS , the channel
discrimination was performed by fixing the mean number of photons in each of
the modes probing the channel. In both these models there was no restriction
for the energy involved in the use of ancillary modes. Our approach considers
the discrimination of lossy channels by constraining the total energy of the
input state, thus including both the probing mode (irradiated over the unknown
lossy channel) and a possible ancillary mode (bypassing the lossy channel and
detected by the output measurement). Thus, while previous models were more
focussed on restricting the energy irradiated over the channel, we address the
problem from the point of view of the input source, i.e., considering the
global effort in preparing this source. By fixing the input energy, we then
optimize over an important class of Gaussian states, i.e., single- and two-
mode squeezed thermal states. The choice of these states relies in their
experimental accessibility, being routinely generated in today’s quantum
optics laboratories where they can be reliably controlled PRLtwo-mode .
Furthermore, because of the squeezing, they represent important examples of
non-classical states, i.e., states with non-positive P representation Glauber
. This is another feature which diversifies our study from previous works
QreadingPRL ; Qillumination , where the main goal was the comparison between
non-classical states and classical states (i.e., with positive P
representation). Note also that our problem, involving a discrete channel
discrimination, is completely different from problems of parameter estimation,
where one has to infer a parameter taking continuous values. The estimation of
the damping constant of a bosonic channel has been recently addressed by Refs.
Freyberger ; Mon07 ; AdessoR ; Mon10 .

It is clear that, for a given input state, any problem of channel
discrimination collapses into a problem of state discrimination Helstrom ;
Chefles ; QSE ; QSE2 ; Kargin , where we have to compute the minimum error
probability in identifying one of two possible output states. By assuming $M$
identical copies of the input state and a memoryless quantum channel, we have
$M$ output states which are exact replicas of two equiprobable states. In this
case, the minimum error probability is well-approximated by the quantum
Chernoff bound (QCB) PRAQCB ; PRLQCB ; Nuss ; Aud ; Pirandola . Now the
crucial step is to vary the input state, trying to optimize the value of the
output QCB. In the case of two lossy bosonic channels, this kind of
optimization must be constrained, meaning that we have to fix some crucial
parameters of the input states, in particular, their energy. As we have
already mentioned before, in our investigation we optimize the QCB on the
class of single- and two-mode squeezed thermal states by fixing their total
energy. Single-mode thermal states are sent through the lossy channel, while
two-mode thermal states probe the channel with the one of the modes (probing
mode) while bypassing the channel and assisting the output measurement with
the other mode (reference mode). In this scenario, we find that the pure
version of these states, i.e., single- and two-mode squeezed vacuum states,
are optimal for detecting losses, i.e., discriminating a lossy from a lossless
channel. Furthermore, we are able to show that, for any value of the damping
rate smaller than a certain critical value, there is a threshold on the energy
that makes the two-mode squeezed vacuum state more convenient than the
corresponding single-mode state. More interestingly, for damping rates larger
than the critical value, the two-mode squeezed vacuum state performs always
better than the single-mode squeezed vacuum state with exactly the same
energy.

In order to stay close to schemes which are feasible with current technology,
we also analyze the effect of the mixedness in the probe states. In this case,
we study the channel discrimination by fixing not only the total energy of the
input state but also its total amount of squeezing. Then, we are able to show
that two-mode squeezed thermal states are always better than their single-mode
counterpart when all the thermal photons are directed into the dissipative
channel. We have numerically verified that this result also holds,
approximately, for unbalanced distribution of the thermal photons. Finally, we
also investigate the role of correlations in the improvement of loss
detection. In order to quantify correlations, besides entanglement and mutual
information, we exploit the recent results on the quantum discord Disc1 ;
Disc2 ; Ferraro10 ; GQDSCa ; GQDSCb ; Vedral , which has been defined with the
aim of capturing quantum correlations in mixed separable states that are not
quantified by entanglement. Thus, at fixed input squeezing, we study the
reduction of the QCB as a function of various correlation quantifiers, i.e.,
quantum mutual information, entanglement and quantum discord. This analysis
allows us to conclude that employing squeezing in the form of correlations,
either quantum or classical, is beneficial for the task of loss detection
whenever squeezed thermal states are considered as input probes.

The paper is structured as follows. In Sec. II we review the discrimination of
quantum states together with the general definition of the QCB. In Sec. III we
give the basic notation for Gaussian states and in Sec. IV we review the
formula of the QCB for Gaussian states. Then, in Sec. V we discuss the
discrimination of lossy channels by considering single-mode and two-mode
squeezed thermal states. In particular, we provide the details on how to
compute the QCB for distinguishing an ideal lossless channel from a lossy
channel. Section VI reports the main results regarding the single- and two-
mode states as a function of the total energy and squeezing. Finally, in
Section VII we analyze the role of correlations in enhancing the
discrimination. Section VIII closes the paper with some concluding remarks.

##  II Quantum State Discrimination and Quantum Chernoff Bound

As we have mentioned before, the problem of quantum channel discrimination
collapses to the problem of quantum state discrimination when we fix the input
state. As a result, mathematical tools such as the quantum fidelity and the
QCB are fundamental in our analysis. Despite it has been introduced only very
recently, the QCB has been already a crucial tool in several areas of quantum
information: it has been exploited as a distinguishability measure between
qubits and single-mode Gaussian states PRAQCB ; PRLQCB , to evaluate the
degree of nonclassicality for single-mode Gaussian states Boca  or the
polarization of a two-mode state Ghiu . It has also been applied in the theory
of quantum phase transitions to distinguish between different phases of the
$XY$ model at finite temperature AbastoQCM , and to the discrimination of two
ground states or two thermal states in the quantum Ising model JMOQCB . For
continuous variables systems the quantum discrimination of Gaussian states is
a central point in view of their experimental accessibility and their
relatively simple mathematical description GSinQI ; QITCV . In fact, in the
case of Gaussian states, the QCB can be computed from their first and the
second statistical moments. A first formula, valid for single-mode Gaussian
states, was derived in PRAQCB . Later, Ref. Pirandola  provided a general
closed formula for multimode Gaussian states, relating the QCB bound to their
symplectic spectra. Furthermore, from these spectra one can derive larger
upper bounds which are easier to compute than the QCB Pirandola .

In this section, we start by establishing notation and reviewing the problem
of quantum state discrimination, together with the general definition of QCB.
Then, from the next section, we will specialize our attention to the case of
Gaussian states and we will review the formula for Gaussian states in Sec. IV.

In its simplest formulation, the problem of quantum state discrimination
consists in distinguishing between two possible states, $\rho_{A}$ and
$\rho_{B}$, which are equiprobable for a quantum system. We suppose that $M$
identical copies of the quantum system are available. Then, we have the
following equiprobable hypotheses, $H_{A}$ and $H_{B}$, about the global state

$\displaystyle H_{A}$
$\displaystyle:\rho_{A}^{M}=\underbrace{\rho_{A}\otimes\ldots\otimes\rho_{A}}_{M}$
$\displaystyle H_{B}$
$\displaystyle:\rho_{B}^{M}=\underbrace{\rho_{B}\otimes\ldots\otimes\rho_{B}}_{M}.$

In order to discriminate between these two hypotheses, one can measure the
global system by using a two-outcome positive operator valued measure (POVM)
$\\{E_{A},E_{B}\\}$, with $E_{A}+E_{B}={\mathbb{I}}$ and $E_{A},E_{B}\geq 0$.
After observing the outcome $j=A$ or $B$, the observer infers that the state
of the system was $\rho_{j}^{M}$. The error probability of inferring the state
$\rho_{j}^{M}$ when the actual state is $\rho_{k}^{M}$ is thus given by the
Born rule $P_{jk}=\hbox{Tr}\left[\rho_{k}^{M}E_{j}\right]$. As a result, the
optimal POVM for this discrimination problem is the one minimizing the overall
probability of misidentification, i.e., $P_{e}=\frac{1}{2}(P_{BA}+P_{AB})$.
Since $E_{A}={\mathbb{I}}-E_{B}$, we have

$\displaystyle P_{e}$
$\displaystyle=\frac{1}{2}\hbox{Tr}[\rho_{A}^{M}E_{B}]+\frac{1}{2}\hbox{Tr}[\rho_{B}^{M}E_{A}]$
$\displaystyle=\frac{1}{2}\left(1-\hbox{Tr}\left[E_{B}\Lambda\right]\right)~{},$
(1)

where

$\Lambda=\rho_{B}^{M}-\rho_{A}^{M}\,,$

is known as the Helstrom matrix Helstrom . Now, the error probability $P_{e}$
has to be minimized over $E_{B}$. Since $\hbox{Tr}[\Lambda]=0$, the Helstrom
matrix has both positive and negative eigenvalues and the minimum $P_{e}$ is
attained if $E_{B}$ is chosen as the projector over $\Lambda_{+}$, i.e., the
positive subspace of $\Lambda$. Assuming this optimal operator we have
$\hbox{Tr}[E_{B}\Lambda]=\mathop{\text{Tr}}\nolimits[\Lambda_{+}]=\frac{1}{2}\mathop{\text{Tr}}\nolimits|\Lambda|$
with $|\Lambda|=\sqrt{\Lambda^{\dagger}\Lambda}$. Thus the minimal error
probability is given by

$P_{e}=\frac{1}{2}\left[1-T(\rho_{A}^{M},\rho_{B}^{M})\right]~{},$

where

$T(\rho,\sigma)=\frac{1}{2}\mathop{\text{Tr}}\nolimits|\rho-\sigma|$

is the so-called trace distance. The computation of the trace distance may be
rather difficult. For this reason, one can resort to the QCB that gives an
upper bound to the probability of error $P_{e}$ PRAQCB ; PRLQCB ; Nuss ; Aud ;
Pirandola

$P_{e}\leq\frac{Q^{M}}{2}~{},$ (2)

where

$Q=\inf_{0\leq s\leq
1}\mathop{\text{Tr}}\nolimits\left[\rho_{A}^{s}\rho_{B}^{1-s}\right]~{}.$ (3)

The bound of Eq. (2) is attainable asymptotically in the limit
$M\rightarrow\infty$ as follows from the results in PRLQCB ; Nuss . One may
think that the trace distance has a more natural operational meaning than the
QCB. In spite of this, it does not adapt to the case of many copies; indeed,
one can find states $\rho,\sigma,\rho^{\prime},\sigma^{\prime}$ such that

$T(\rho,\sigma)<T(\rho^{\prime},\sigma^{\prime})\quad\hbox{but}\quad
T(\rho^{\prime}{}^{M},\sigma^{\prime}{}^{M})<T(\rho^{M},\sigma^{M})\,.$

By contrast, the QCB does resolve this problem since

$Q(\rho,\sigma)<Q(\rho^{\prime},\sigma^{\prime})\;\Longrightarrow\;Q(\rho^{M},\sigma^{M})<Q(\rho^{\prime
M},\sigma^{\prime M})\,.$

Because of this property, the minimization of the QCB over single-copy states
($\rho$ and $\sigma$) implies the minimization over multi-copy states
($\rho^{M}$ and $\sigma^{M}$). This is true as long as the minimization is
unconstrained or if the constraints regard single-copy observables (e.g., the
mean energy per copy).

Finally, note that there is a close relation between the QCB and the Uhlmann
fidelity

$F(\rho_{A},\rho_{B})=\mathop{\text{Tr}}\nolimits\left(\sqrt{\sqrt{\rho_{A}}\rho_{B}\sqrt{\rho_{A}}}\right)^{2}$

which is one of the most popular measures of distinguishability for quantum
states. In fact, for the single-copy state discrimination ($M=1$) we have
PRAQCB ; Fuchs ; Nielsen ; Pirandola

$\frac{1-\sqrt{1-F(\rho_{A},\rho_{B})}}{2}\leq
P_{e}\leq\frac{Q}{2}\leq\frac{\sqrt{F(\rho_{A},\rho_{B})}}{2}.$ (4)

More generally, by exploiting the multiplicativity of the fidelity under
tensor products of density operators, i.e.,

$F(\rho_{A}\otimes\sigma_{A},\rho_{B}\otimes\sigma_{B})=F(\rho_{A},\rho_{B})F(\sigma_{A},\sigma_{B})~{},$

we can write

$F(\rho_{A}^{M},\rho_{B}^{M})=F(\rho_{A},\rho_{B})^{M}=F^{M}~{}.$

This leads to the general multi-copy version of Eq. (4) which is given by
QreadingSUPP

$\frac{1-\sqrt{1-F^{M}}}{2}\leq
P_{e}\leq\frac{Q^{M}}{2}\leq\frac{F^{M/2}}{2}~{}.$ (5)

From the previous inequalities, it is clear that the QCB gives a tighter bound
than the quantum fidelity. However, if one of the two states is pure, then the
QCB just equals the fidelity, i.e., we have

$Q(\rho_{A},\rho_{B})=F(\rho_{A},\rho_{B})=\mathop{\text{Tr}}\nolimits[\rho_{A}\,\rho_{B}]~{}.$

##  III Gaussian states

In this section we give the basic notions on bosonic systems and Gaussian
states, ending with the definition of squeezed thermal states.

An $n$-mode bosonic system is described by a tensor-product Hilbert space
$\mathcal{H}^{\otimes n}$ and a vector of canonical operators
$\boldsymbol{R}=(q_{1},p_{1},\ldots,q_{n},p_{n})^{T}$ satisfying the
commutation relations

$[R_{l},R_{m}]=i\Omega_{lm}~{},$

where $l,m=1,\cdots,2n$ and $\Omega_{lm}$ are the elements of the symplectic
form

$\boldsymbol{\Omega}=\bigoplus_{k=1}^{n}\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)~{}.$ (6)

Alternatively, we can use the mode operators $a_{k}$ which are given by the
cartesian decomposition of the canonical operators, i.e.,

$q_{k}=\frac{1}{\sqrt{2}}(a_{k}+a_{k}^{\dagger})~{},~{}p_{k}=\frac{1}{i\sqrt{2}}(a_{k}-a_{k}^{\dagger})~{}.$

These operators satisfy the commutation relations
$[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}$ with
$k,k^{\prime}=1,\cdots,n$.

An arbitrary quantum state $\rho$ of the system is equivalently described by
the characteristic function

$\chi[\rho](\boldsymbol{\lambda})=\mathop{\text{Tr}}\nolimits[\rho
D(\boldsymbol{\lambda})]$

where $D(\boldsymbol{\lambda})=\otimes_{k=1}^{n}D_{k}(\lambda_{k})$ is the
$n$-mode displacement operator, with
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{n})^{T}$,
$\lambda_{k}\in\mathbb{C}$, and
$D_{k}(\lambda_{k})=\exp\\{\lambda_{k}a_{k}^{\dagger}-\lambda_{k}^{\ast}a_{k}\\}$
is the single mode displacement operator. A state $\rho$ is called Gaussian if
the corresponding characteristic function is Gaussian

$\chi[\rho](\boldsymbol{\Lambda})=\exp\left\\{-\frac{1}{2}\boldsymbol{\Lambda^{T}\sigma\Lambda}+\boldsymbol{X^{T}\Omega\Lambda}\right\\}$
(7)

where $\boldsymbol{\Lambda}$ is the real vector

$\boldsymbol{\Lambda}=(\text{{Re}}\lambda_{1},\text{{{Im}}}\lambda_{1}\mathrm{,\ldots,{Re}}\lambda_{n}\mathrm{,{Im}}\lambda_{n})^{T}~{}.$

In this case, the state is described by its first two statistical moments,
i.e., the vector of mean values $\boldsymbol{X}$ and the covariance matrix
(CM) $\boldsymbol{\sigma}$, whose elements are defined as

$\displaystyle X_{l}=$ $\displaystyle\langle R_{l}\rangle$
$\displaystyle\sigma_{lm}=$
$\displaystyle\frac{1}{2}\langle\\{R_{l},R_{m}\\}\rangle-\langle
R_{l}\rangle\langle R_{m}\rangle$ (8)

where $\\{A,B\\}=AB+BA$ denotes the anti-commutator, and $\langle
O\rangle=\mathop{\text{Tr}}\nolimits[\rho O]$ is the mean value of the
operator $O$.

In the remainder of this section, we consider only zero-mean Gaussian states,
i.e., Gaussian states with $\boldsymbol{X}=0$, which are therefore fully
specified by their CM. The properties of these states may be expressed in very
simple terms by introducing the symplectic transformations. A matrix
$\boldsymbol{S}$ is called symplectic when preserves the symplectic form of
Eq. (6), i.e.,

$\boldsymbol{S\Omega S^{T}}=\boldsymbol{\Omega}.$

Then, according to the Williamson’s theorem, for every CM
$\boldsymbol{\sigma}$, there exists a symplectic matrix $\boldsymbol{S}$ such
that

$\boldsymbol{\sigma}=\boldsymbol{SWS^{T}}$ (9)

where

$\boldsymbol{W}=\bigoplus_{k=1}^{n}d_{k}\left(\begin{array}[c]{cc}1&0\\\
0&1\end{array}\right)~{},$

and the $d_{k}$’s are called the symplectic eigenvalues of
$\boldsymbol{\sigma}$. The physical statement implied by the decomposition of
Eq. (9) is that every zero-mean Gaussian state $\rho$ can be obtained from a
thermal state by performing the unitary transformation $U_{\boldsymbol{S}}$
associated with the symplectic matrix $\boldsymbol{S}$, i.e., we have

$\rho=U_{\boldsymbol{S}}~{}\boldsymbol{\nu~{}}U_{\boldsymbol{S}}^{{\dagger}}$

where $\boldsymbol{\nu}=\nu_{1}\otimes\ldots\otimes\nu_{n}$ is a tensor
product of single-mode thermal states

$\nu_{k}=\frac{1}{\bar{n}_{k}+1}\sum_{m}\left(\frac{\bar{n}_{k}}{\bar{n}_{k}+1}\right)^{m}|m\rangle_{k}\langle
m|$

with average number of photons given by $\bar{n}_{k}=d_{k}-1/2$. For a single-
mode system the most general zero-mean Gaussian state may be written as

$\rho=S(\zeta)\nu S^{\dagger}(\zeta)$

where $S(\zeta)=\exp\\{\frac{1}{2}({\zeta
a^{\dagger}}^{2}-\zeta^{\ast}{a}^{2})\\}$ is the single-mode squeezing
operator and $\zeta=re^{i\phi}\in\mathbb{C}$. The corresponding covariance
matrix is given by

$\boldsymbol{\sigma}=\left(\begin{array}[c]{cc}a&c\\\ c&b\end{array}\right)$
(10)

where

$\displaystyle a$
$\displaystyle=(\bar{n}+\frac{1}{2})\left[\cosh(2r)-\sinh(2r)\cos\phi\right]$
$\displaystyle b$
$\displaystyle=(\bar{n}+\frac{1}{2})\left[\cosh(2r)+\sinh(2r)\cos\phi\right]$
$\displaystyle c$ $\displaystyle=(\bar{n}+\frac{1}{2})\sinh(2r)\sin\phi~{}.$
(11)

In particular, we can consider the case of a real squeezing parameter, e.g.,
by fixing $\zeta=-r$ Segno . In this case, the previous expressions of Eq.
(11) simplify into the following

$\displaystyle a$ $\displaystyle=\frac{1}{2}(2\bar{n}+1)\exp(2r)$
$\displaystyle b$ $\displaystyle=\frac{1}{2}(2\bar{n}+1)\exp(-2r)$
$\displaystyle c$ $\displaystyle=0~{}.$ (12)

This state defines the single-mode squeezed thermal state. It depends on two
real parameters only, i.e., we have

$\rho=S(r)\nu S^{\dagger}(r)=\rho(r,\bar{n})~{}.$

In particular, for $\bar{n}=0$ the state is pure and corresponds to a single-
mode squeezed vacuum state $\rho(r,0)=S(r)\left|0\right\rangle\left\langle
0\right|S^{\dagger}(r)$.

Now let us consider two-mode (zero-mean) Gaussian states. They are completely
characterized by their $4\times 4$ CM

$\boldsymbol{\sigma}=\left(\begin{array}[c]{cc}\mathbf{A}&\mathbf{C}\\\
\mathbf{C}^{T}&\mathbf{B}\end{array}\right)$ (13)

where $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are $2\times 2$ blocks. It
is useful to introduce the symplectic invariants

$\displaystyle I_{1}$
$\displaystyle=\det\mathbf{A}~{},~{}I_{2}=\det\mathbf{B}\,,$ $\displaystyle
I_{3}$ $\displaystyle=\det\mathbf{C}~{},~{}I_{4}=\det\boldsymbol{\sigma}~{}.$
(14)

By means of these invariants, we can simply write the two symplectic
eigenvalues as

$d_{\pm}=\sqrt{\frac{\Delta\pm\sqrt{\Delta^{2}-4I_{4}}}{2}}~{},$

where $\Delta=I_{1}+I_{2}+2I_{3}$ DiagSympl ; PirCMs . By means of local
symplectic operations, the CM of Eq. (13) can be recast in the standard form,
where the three blocks $\mathbf{A}$ and $\mathbf{B}$ are proportional to the
identity and $\mathbf{C}$ is diagonal DiagSympl . In the particular case of a
two-mode squeezed thermal state, we can write

$\boldsymbol{\sigma}=\frac{1}{2}\left(\begin{array}[c]{cc}A{\mathbb{I}}_{2}&C\sigma_{z}\\\
C\sigma_{z}&B{\mathbb{I}}_{2}\end{array}\right)$ (15)

where

$\displaystyle A$
$\displaystyle=\cosh(2r)+2\bar{n}_{1}\cosh^{2}r+2\bar{n}_{2}\sinh^{2}r$
$\displaystyle B$
$\displaystyle=\cosh(2r)+2\bar{n}_{1}\sinh^{2}r+2\bar{n}_{2}\cosh^{2}r$
$\displaystyle C$ $\displaystyle=(1+\bar{n}_{1}+\bar{n}_{2})\sinh 2r,$ (16)

with ${\mathbb{I}}_{2}$ the $2\times 2$ identity matrix and
$\sigma_{z}=\text{diag}(1,-1)$ the z-Pauli matrix. This corresponds to
considering a density operator of the form

$\rho=S_{2}(r)\left(\nu_{1}\otimes\nu_{2}\right)S_{2}(r)^{\dagger}\,,$

where $S_{2}(r)=\exp\\{r(a^{\dagger}b^{\dagger}-ab)\\}$ is the two-mode
squeezing operator. This state depends on three real parameters: the squeezing
parameter and the two thermal numbers, i.e., we have

$\rho=\rho(r,\bar{n}_{1},\bar{n}_{2})~{}.$

In particular, for $\bar{n}_{1}=\bar{n}_{2}=0$ the state is pure and
corresponds to a two-mode squeezed vacuum state
$\rho(r,0,0)=S_{2}(r)(\left|0\right\rangle_{1}\left\langle
0\right|\otimes\left|0\right\rangle_{2}\left\langle
0\right|)S_{2}^{\dagger}(r)$.

##  IV Quantum Chernoff bound for Gaussian states

Here we review the formula of the QCB for multimode Gaussian states Pirandola
. In particular, we adapt this formula to our notation and physical units
(here the vacuum noise is $1/2$, while in Ref. Pirandola  it was equal to
$1$). Let us consider two Gaussian states $\rho$ (with statistical moments
$\boldsymbol{X}$ and $\boldsymbol{\sigma}$) and $\rho^{\prime}$ (with
statistical moments $\boldsymbol{X}^{\prime}$ and
$\boldsymbol{\sigma}^{\prime}$). The CMs of these two states can be decomposed
as

$\displaystyle\boldsymbol{\sigma}$
$\displaystyle=\boldsymbol{S~{}\boldsymbol{W}}(\bar{n}_{1},\cdots,\bar{n}_{n})~{}\boldsymbol{S^{T}}$
(17) $\displaystyle\boldsymbol{\sigma}^{\prime}$
$\displaystyle=\boldsymbol{S}^{\prime}~{}\boldsymbol{\boldsymbol{W}}(\bar{n}_{1}^{\prime},\cdots,\bar{n}_{n}^{\prime})~{}\boldsymbol{S}^{\prime}\boldsymbol{{}^{T}},$
(18)

where $\\{\bar{n}_{k}\\}$ and $\\{\bar{n}_{k}^{\prime}\\}$ are their thermal
numbers, and

$\boldsymbol{W}(x_{1},\cdots,x_{n})=\bigoplus_{k=1}^{n}(2x_{k}+1){\mathbb{I}}_{2}~{}.$

Now let us define the functions

$G_{s}(x)=\frac{1}{(x+1)^{s}-x^{s}}~{},$

and

$\Lambda_{s}(x)=\frac{x^{s}}{(x+1)^{s}-x^{s}}~{}.$

Then, the QCB is given by

$Q=\inf_{0\leq s\leq 1}Q_{s}~{},$

where

$Q_{s}=\frac{\Pi_{s}}{\sqrt{\det\boldsymbol{\Sigma}_{s}}}\exp\left(-\frac{1}{2}\mathbf{d}^{T}\boldsymbol{\Sigma}_{s}^{-1}\mathbf{d}\right)~{}.$
(19)

In the formula of Eq. (19), we have
$\mathbf{d}=\boldsymbol{X}-\boldsymbol{X}^{\prime}$,

$\Pi_{s}=\prod_{k=1}^{n}G_{s}(\bar{n}_{k})G_{1-s}(\bar{n}_{k}^{\prime})~{},$

and

$\displaystyle\boldsymbol{\Sigma}_{s}=$
$\displaystyle\boldsymbol{S~{}W}[\Lambda_{s}(\bar{n}_{1}),\cdots,\Lambda_{s}(\bar{n}_{n})]~{}\boldsymbol{S^{T}}$
$\displaystyle+\boldsymbol{S}^{\prime}~{}\boldsymbol{W}[\Lambda_{1-s}(\bar{n}_{1}^{\prime}),\cdots,\Lambda_{1-s}(\bar{n}_{n}^{\prime})]~{}\boldsymbol{S}^{\prime
T}~{}.$

###  IV.1 Discrimination of squeezed thermal states

For the discrimination of squeezed thermal states, the previous formula
simplifies a lot. First of all, since they are zero-mean Gaussian states, we
have $\mathbf{d}=0$ and, therefore, the exponential factor in Eq. (19)
disappears. Then, the symplectic decompositions in Eqs. (17) and (18) are
achieved using symplectic matrices $\boldsymbol{S}$ and
$\boldsymbol{S}^{\prime}$ which are just one-parameter squeezing matrices,
i.e., $\boldsymbol{S}=\boldsymbol{S}(r)$ and
$\boldsymbol{S}^{\prime}=\boldsymbol{S}^{\prime}(r^{\prime})$.

Thus, let us consider the discrimination of single-mode squeezed thermal
states $\rho=\rho(r,\bar{n})$ and
$\rho^{\prime}=\rho^{\prime}(r^{\prime},\bar{n}^{\prime})$. In this case, the
QCB can be computed using

$Q_{s}=\frac{\Pi_{s}(\bar{n},\bar{n}^{\prime})}{\sqrt{\det\boldsymbol{\Sigma}_{s}(r,\bar{n},r^{\prime},\bar{n}^{\prime})}}~{},$
(20)

where

$\Pi_{s}(\bar{n},\bar{n}^{\prime})=G_{s}(\bar{n})G_{1-s}(\bar{n}^{\prime})~{},$

and

$\displaystyle\boldsymbol{\Sigma}_{s}(\bar{n},\bar{n}^{\prime},r,r^{\prime})$
$\displaystyle=\boldsymbol{S}(r)\boldsymbol{W}[\Lambda_{s}(\bar{n})]\boldsymbol{S}(r)\boldsymbol{{}^{T}}$
$\displaystyle+\boldsymbol{S}(r^{\prime})\boldsymbol{W}[\Lambda_{1-s}(\bar{n}^{\prime})]\boldsymbol{S}(r^{\prime})\boldsymbol{{}^{T}}.$

For the discrimination of two-mode squeezed thermal states
$\rho=\rho(r,\bar{n}_{1},\bar{n}_{2})$ and
$\rho^{\prime}=\rho^{\prime}(r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})$,
we can use

$Q_{s}=\frac{\Pi_{s}(\bar{n}_{1},\bar{n}_{2},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})}{\sqrt{\det\boldsymbol{\Sigma}_{s}(r,\bar{n}_{1},\bar{n}_{2},r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})}}~{},$
(21)

where

$\Pi_{s}(\bar{n}_{1},\bar{n}_{2},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})=G_{s}(\bar{n}_{1})G_{s}(\bar{n}_{2})G_{1-s}(\bar{n}_{1}^{\prime})G_{1-s}(\bar{n}_{2}^{\prime})~{},$

and

$\displaystyle\boldsymbol{\Sigma}_{s}(r,\bar{n}_{1},\bar{n}_{2},r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})=\boldsymbol{S}(r)\boldsymbol{W}[\Lambda_{s}(\bar{n}_{1}),\Lambda_{s}(\bar{n}_{2})]\boldsymbol{S}(r)\boldsymbol{{}^{T}}$
$\displaystyle+\boldsymbol{S}(r^{\prime})\boldsymbol{W}[\Lambda_{1-s}(\bar{n}_{1}^{\prime}),\Lambda_{1-s}(\bar{n}_{2}^{\prime})]\boldsymbol{S}(r^{\prime})\boldsymbol{{}^{T}}.$

##  V Detection of losses by thermal probes

In what follows, we study the evolution of a Gaussian state in a dissipative
channel $\mathcal{E}_{\Gamma}$ characterized by a damping rate $\Gamma$, which
may result from the interaction of the system with an external environment, as
for example a bath of oscillators, or from an absorption process. We consider
the problem of detecting whether or not the dissipation dynamics occurred.
Given an input state $\rho$, this corresponds to discriminating between an
output state identical to the input $\rho$, and another output state storing
the presence of loss $\mathcal{E}_{\Gamma}(\rho)$. Lossy channels are Gaussian
channels, meaning that they tansform Gaussian states into Gaussian states.
Furthermore, if the input is a squeezed thermal state, then the output state
is still squeezed thermal (this is discussed in detail afterwards).

In general, we consider the schematic diagram depicted in Fig.1. In order to
detect loss, we consider either a single-mode squeezed thermal state evolving
in the lossy channel with parameter $\Gamma$ followed by a measurement at the
output, or a two-mode squeezed thermal state with the damping process
occurring in only one of the two modes (the probing mode), followed by a
measurement on both of the modes. Our aim is to minimize the error probability
in discriminating between the ideal case $\Gamma=0$ and the lossy case
$\Gamma>0$. In the next section, this will be done by fixing some important
parameters of the input state, such as total energy and squeezing ContQCB .

Figure 1: Single- and two-mode schemes for the detection of losses. Top: a
single-mode squeezed thermal state $\rho$ enters the lossy channel with
damping rate $\Gamma$. A measurement apparatus detects the output state
$\rho^{\prime}$. Bottom: the lossy channel acts on the probing mode (1) of a
two-mode squeezed thermal state $\rho$, while the reference mode (2) bypasses
the channel. The output state $\rho^{\prime}$ of both the modes is then
measured.

The propagation of a mode of radiation in a lossy channel corresponds to the
coupling of the mode $a$ with a zero temperature reservoir made of large
number of external modes. By assuming a Markovian reservoir and weak coupling
between the system and the reservoir the dynamics of the system is described
by the Lindblad Master equation Walls

$\dot{\rho}=\frac{\Gamma}{2}\mathcal{L}[a]\rho$ (22)

where $\mathcal{L}[a]\rho=2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a$. The general solution may be expressed by using the operator-sum
representation of the associated completely-positive map, i.e., upon writing
$\eta=e^{-\Gamma t}$, we have

$\varrho(\eta)=\sum_{m}V_{m}\varrho V_{m}^{{\dagger}}$

where

$V_{m}=\sqrt{\frac{(1-\eta)^{m}}{m!}}a^{m}\eta^{\frac{1}{2}(a^{{\dagger}}a-m)}\,,$

and $\varrho$ is the initial state.

###  V.1 Single-mode case

Let us now start with single-mode states. Eq. (22) can be recast into a
Fokker-Planck equation for the Wigner function in terms of the quadrature
variables $q$ and $p$,

$\dot{W}=\frac{\Gamma}{2}\left[\partial_{\boldsymbol{X}}^{T}\boldsymbol{X}+\partial_{\boldsymbol{X}}^{T}\boldsymbol{\sigma}_{\infty}\partial_{\boldsymbol{X}}^{T}\right]W$

where $\boldsymbol{X}=(q,p)^{T}$,
$\partial_{\boldsymbol{X}}=(\partial_{q},\partial_{p})^{T}$ and we introduced
the diffusion matrix $\boldsymbol{\sigma}_{\infty}=\text{diag}(1/2,1/2)$.
Solving the equation for the Wigner function of a single-mode Gaussian state
one can obtain the evolution equation for the CM $\boldsymbol{\sigma}$. This
is given by AS1

$\dot{\boldsymbol{\sigma}}=-\Gamma(\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\infty})~{},$

which yields to

$\boldsymbol{\sigma}(t)=e^{-\Gamma t}\boldsymbol{\sigma}_{0}+(1-e^{-\Gamma
t}){\boldsymbol{\sigma}}_{\infty}~{}.$

The latter equation describes the evolution of an initial Gaussian state with
CM $\boldsymbol{\sigma}_{0}$ towards the stationary state given by the
Gaussian state of the environment with CM $\boldsymbol{\sigma}_{\infty}$. For
simplicity, from now on we omit the index of $\boldsymbol{\sigma}_{0}$, we
replace $\boldsymbol{\sigma}(t)\rightarrow\boldsymbol{\sigma}^{\prime}$, and
we insert the time $t$ into the damping parameter $\Gamma$. Thus, the evolved
CM of the single mode case simply reads

$\boldsymbol{\sigma}^{\prime}=e^{-\Gamma}\boldsymbol{\sigma}+(1-e^{-\Gamma})\,\boldsymbol{\sigma}_{\infty}~{}.$

Now let us consider the specific case of an input squeezed thermal state
$\rho=\rho(r,n_{T})$ with squeezing $r$ and thermal number $n_{T}$. According
to Eqs. (10) and (12), its CM is given by

$\boldsymbol{\sigma}=\left(\begin{array}[c]{cc}a&0\\\
0&b\end{array}\right)~{},$

with

$a=\frac{1}{2}\left(1+2n_{\scriptstyle
T}\right)e^{2r}~{},~{}b=\frac{1}{2}\left(1+2n_{\scriptstyle
T}\right)e^{-2r}\,~{}.$ (23)

At the output of the channel, the state $\rho^{\prime}$ has CM

$\boldsymbol{\sigma}^{\prime}=\left(\begin{array}[c]{cc}a_{\Gamma}&0\\\
0&b_{\Gamma}\end{array}\right)~{},$

where

$a_{\Gamma}=\frac{1}{2}\left(1+2n_{\Gamma}\right)e^{2r_{\Gamma}}~{},~{}b_{\Gamma}=\frac{1}{2}\left(1+2n_{\Gamma}\right)e^{-2r_{\Gamma}}~{},$
(24)

and

$\displaystyle n_{\Gamma}$
$\displaystyle=\sqrt{\det[\boldsymbol{\sigma}^{\prime}]}-1/2$ (25)
$\displaystyle r_{\Gamma}$
$\displaystyle=\frac{1}{4}\log\left[\frac{e^{-\Gamma}a+(1-e^{-\Gamma})/2}{e^{-\Gamma}b+(1-e^{-\Gamma})/2}\right].$
(26)

Thus, we still have a squeezed thermal state
$\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\Gamma})$ with squeezing
$r_{\Gamma}$ and thermal number $n_{\Gamma}$. Now, the discrimination between
a lossless ($\Gamma=0$) and a lossy channel ($\Gamma>0$) corresponds to the
discrimination between the input state $\rho=\rho(r,n_{T})$ and the output one
$\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\Gamma})$. In order to estimate the
error probability affecting this discrimination, we can compute the quantum
Chernoff bound. This is achieved by replacing

$(r,\bar{n})\rightarrow(r,n_{T})\;\;\hbox{and}\;\;(r^{\prime},\bar{n}^{\prime})\rightarrow(r_{\Gamma},n_{\Gamma})\,$

in Eq. (20).

###  V.2 Two-mode case

According to the scheme of Fig. 1, the map describing the evolution of a two-
mode state is $\mathcal{E}_{\Gamma}\otimes\mathcal{I}$, where the lossy
channel $\mathcal{E}_{\Gamma}$ acts on the probing mode while the identity
channel $\mathcal{I}$ acts on the reference mode. At the level of the CM it
corresponds to the following transformation

$\displaystyle\boldsymbol{\sigma}^{\prime}=$
$\displaystyle\left(e^{-\Gamma/2}{\mathbb{I}}_{2}\oplus{\mathbb{I}}_{2}\right)\boldsymbol{\sigma}\left(e^{-\Gamma/2}{\mathbb{I}}_{2}\oplus{\mathbb{I}}_{2}\right)$
$\displaystyle+({\mathbb{I}}_{4}-e^{-\Gamma}{\mathbb{I}}_{2}\oplus{\mathbb{I}}_{2})\boldsymbol{\sigma}_{\infty}~{}.$
(27)

As input state, let us consider a two-mode squeezed thermal state
$\rho=\rho(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$. Its CM is
provided in Eq. (15) with the elements given in Eq. (16) by replacing
$(r,\bar{n}_{1},\bar{n}_{2})\rightarrow(r,n_{\scriptstyle
T_{1}},n_{\scriptstyle T_{2}})$. The CM of the output state can be derived
using the Eq. (27). This CM can be put in the normal form of Eq. (15) with
elements given by Eq. (16) by replacing
$(r,\bar{n}_{1},\bar{n}_{2})\rightarrow(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$.
Here the squeezing parameter $r_{\Gamma}$ and the thermal numbers
$n_{\scriptstyle\Gamma_{1}}$ and $n_{\scriptstyle\Gamma_{2}}$ are function of
the input parameters $r$, $n_{\scriptstyle T_{1}}$ and $n_{\scriptstyle
T_{2}}$ (the explicit expression is too long to be shown here). Thus, the
output state is still a two-mode squeezed thermal state
$\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$.

As before, the discrimination between a lossless ($\Gamma=0$) and a lossy
channel ($\Gamma>0$) corresponds to the discrimination between the input state
$\rho=\rho(r,n_{\scriptstyle T_{1}},n_{\scriptstyle T_{2}})$ and the output
one
$\rho^{\prime}=\rho^{\prime}(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$.
The error probability affecting this discrimination is estimated by the QCB
which is computed by replacing
$(r,\bar{n}_{1},\bar{n}_{2})\rightarrow(r,n_{\scriptstyle
T_{1}},n_{\scriptstyle T_{2}})$ and
$(r^{\prime},\bar{n}_{1}^{\prime},\bar{n}_{2}^{\prime})\rightarrow(r_{\Gamma},n_{\scriptstyle\Gamma_{1}},n_{\scriptstyle\Gamma_{2}})$
in Eq. (21).

##  VI Optimization of the thermal probes

In this section, we optimize the discrimination of a lossless ($\Gamma=0$)
from a lossy channel ($\Gamma>0$) by maximizing over thermal probes, i.e.,
single- and two-mode squeezed thermal states. For this sake, we evaluate the
QCB as a function of the most important parameters of the input state, i.e.,
its total energy and squeezing. In our first analysis, we show that for fixed
total energy, single- and two-mode squeezed vacuum states are optimal. In
particular, we show the conditions where the two-mode state outperforms the
single-mode counterpart. Then, by fixing both the total energy and squeezing,
we will find the optimal squeezed thermal state. According to Sec. II the
minimization of the QCB over single-copy states implies the minimization over
multi-copy states (when the minimization is unconstrained or subject to
single-copy constraints). This implies that finding the optimal input state
$\rho$ at fixed energy automatically assures that
$\rho\otimes\rho\otimes\cdots$ is the optimal multi-copy state at fixed energy
per copy when we consider a multiple access to the unknown (memoryless)
channel.

In order to perform our investigation we introduce a suitable parametrization
of the input energy. Given a single-mode squeezed thermal state
$\rho=\rho(r,n_{T})$, its energy (mean total number of photons) can be written
as

$N_{1}=n_{\scriptstyle T}+n_{\scriptstyle S}+2n_{\scriptstyle
S}n_{\scriptstyle T}~{},$ (28)

where $n_{\scriptstyle T}$ accounts for the mean number of thermal photons,
$n_{\scriptstyle S}=\sinh^{2}r$ quantifies the squeezing, and $n_{\scriptstyle
S}n_{\scriptstyle T}$ is a cross term. Alternatively, we can introduce a
squeezing factor $\beta_{1}\in[0,1]$ such that

$\displaystyle n_{\scriptstyle S}$ $\displaystyle=\beta_{1}N_{1}$ (29)
$\displaystyle n_{\scriptstyle T}$
$\displaystyle=\frac{(1-\beta_{1})N_{1}}{1+2\beta_{1}N_{1}}~{}.$ (30)

Thus the single-mode squeezed thermal state can be parametrized as
$\rho=\rho(N_{1},\beta_{1})$, i.e., in terms of its total energy $N_{1}$ and
the squeezing factor $\beta_{1}$. Note that for $\beta_{1}=0$ the state is
completely thermal with energy $N_{1}=n_{\scriptstyle T}$, while for
$\beta_{1}=1$ the state is a squeezed vacuum with energy
$N_{1}=n_{\scriptstyle S}$. In our problem of loss detection ($\Gamma=0$
versus $\Gamma>0$), we denote by $Q_{1}(N_{1},\beta_{1})$ the output QCB which
is computed by using the input state $\rho=\rho(N_{1},\beta_{1})$.

Now, given a two-mode squeezed thermal state $\rho=\rho(r,n_{\scriptstyle
T_{1}},n_{\scriptstyle T_{2}})$, its total energy can be written as

$N_{2}=n_{\scriptstyle T_{1}}+n_{\scriptstyle T_{2}}+2n_{\scriptstyle
S}+2n_{\scriptstyle S}(n_{\scriptstyle T_{1}}+n_{\scriptstyle T_{2}})~{},$
(31)

where $n_{\scriptstyle T_{1}}$ ($n_{\scriptstyle T_{2}}$) quantifies the
thermal photons in the probing (reference) mode, $n_{\scriptstyle
S}=\sinh^{2}r$ quantifies the two-mode squeezing energy, and the last
energetic term is a cross term. In this case, besides the squeezing factor
$\beta_{2}$, we can also introduce an asymmetry parameter $\gamma\in[0,1]$
which quantifies the fraction of thermal energy used for the probing mode. In
other words, we can write

$\displaystyle n_{\scriptstyle S}$ $\displaystyle=\frac{1}{2}\beta_{2}N_{2}$
(32) $\displaystyle n_{\scriptstyle T_{1}}$
$\displaystyle=\gamma\,\frac{(1-\beta_{2})N_{2}}{1+\beta_{2}N_{2}}$ (33)
$\displaystyle n_{\scriptstyle T_{2}}$
$\displaystyle=(1-\gamma)\frac{(1-\beta_{2})N_{2}}{1+\beta_{2}N_{2}}.$ (34)

Thus the two-mode squeezed thermal state can be parametrized as
$\rho=\rho(N_{2},\beta_{2},\gamma)$, i.e., in terms of the total energy
$N_{2}$, the squeezing factor $\beta_{2}$, and the asymmetry parameter
$\gamma$. Note that for $\beta_{2}=0$ we have two thermal states, one
describing the probing mode with thermal energy $n_{\scriptstyle
T_{1}}=\gamma\,N_{2}$, and the other one describing the reference mode with
thermal energy $n_{\scriptstyle T_{2}}=(1-\gamma)N_{2}$. For $\beta_{2}=1$ we
have instead a two-mode squeezed vacuum state with total energy
$N_{2}=2n_{\scriptstyle S}$. In this case, the thermal energy is zero and
$\gamma$ can be therefore arbitrary. In our problem of loss detection
($\Gamma=0$ versus $\Gamma>0$), we denote by $Q_{2}(N_{2},\beta_{2},\gamma)$
the output QCB which is computed by using the input state
$\rho=\rho(N_{2},\beta_{2},\gamma)$.

###  VI.1 Optimal input at fixed total energy

In our first investigation we fix the mean total number of photons of the
input state. In other words we fix

$N_{1}=N_{2}=N~{}.$ (35)

Then we minimize the output QCB among single-mode and two-mode squeezed
thermal states. As a first step we compute the optimal quantities

$\displaystyle Q_{1}(N)$ $\displaystyle:=\inf_{\beta_{1}}Q_{1}(N,\beta_{1})$
(36) $\displaystyle Q_{2}(N)$
$\displaystyle:=\inf_{\beta_{2},\gamma}Q_{2}(N,\beta_{2},\gamma)~{}.$ (37)

Then, we compare $Q_{1}(N)$ with $Q_{2}(N)$.

Figure 2: Output QCB$\ Q_{1}(N,\beta_{1})$ optimized over input single-mode
squeezed thermal states $\rho=\rho(N,\beta_{1})$. From left to right we
consider different values of the transmissivity: $\eta=0.1$ (left panel),
$\eta=0.5$ (middle panel) and $\eta=0.9$ (right panel). In each panel, we plot
$Q_{1}(N,\beta_{1})$ as function of the energy $N$ for different values of
$\beta_{1}$. From top to bottom: $\beta_{1}=0.1$ (dashed line),
$\beta_{1}=0.5$ (dotted line) and $\beta_{1}=1$ (solid line). The minimum
curve is always achieved for $\beta_{1}=1$, i.e., for an input single-mode
squeezed vacuum state. Figure 3: Output QCB$\ Q_{2}(N,\beta_{2},\gamma)$
optimized over input two-mode squeezed thermal states
$\rho=\rho(N,\beta_{2},\gamma)$. From left to right we consider different
values of the transmissivity: $\eta=0.1,$ $0.5$ and $0.9$. From top to bottom,
we consider different values of the asymmetry parameter $\gamma=0,$ $0.5$ and
$1$. In each panel, we then plot $Q_{2}$ as function of the energy $N$ for
different values of $\beta_{2}$. From top to bottom: $\beta_{2}=0.1$ (dashed
line), $\beta_{2}=0.5$ (dotted line) and $\beta_{2}=1$ (solid line). The
minimum curve is always achieved for $\beta_{2}=1$ corresponding to an input
two-mode squeezed vacuum state.

According to our findings, in the Eqs. (36) and (37) the infima are achieved
for $\beta_{1}=\beta_{2}=1$. This is numerically shown in Fig. 2 for the
single-mode case and in Fig. 3 for the two-mode case. Thus, we have found
that, at fixed input energy $N$, the optimal thermal probes are given by
single- and two-mode squeezed vacuum states. In this case, the input state is
pure and the QCB corresponds to the fidelity (which is the case when the
s-overlap in Eq. (3) is minimized for $s$ approching the border). Let us adopt
the transmissivity $\eta=e^{-\Gamma}$ to quantify the damping of the channel,
so that $\Gamma=0$ (ideal channel) corresponds to $\eta=1$, and $\Gamma>0$
(lossy channel) corresponds to $0\leq\eta<1$. Then, for single-mode we can
write

$Q_{1}(N)=\frac{1}{\sqrt{1+N(1-\eta^{2})}}~{},$ (38)

and for two-modes we derive

$Q_{2}(N)=\frac{4}{\left[2+N(1-\sqrt{\eta})\right]^{2}}~{}.$ (39)

In Fig. 4, we show the behaviors of the single-mode QCB $Q_{1}(N)$ and two-
mode QCB $Q_{2}(N)$ as function of the input energy $N$ for several values of
transmissivity $\eta$ (or, equivalently, the damping rate $\Gamma$). As
expected the discrimination improves by increasing the input energy $N$ and
decreasing the transmissivity $\eta$.

Figure 4: (Color online). Single-mode QCB $Q_{1}(N)$ (solid lines) and two-
mode QCB $Q_{2}(N)$ (dashed lines) as a function of the input energy $N$ for
different damping rates. From top to bottom $\Gamma=0.1,0.3,1$ (red, green and
blue, respectively) corresponding to $\eta\simeq 0.9,0.74,0.37$. By comparing
curves with the same color (fixed damping $\Gamma$), we can see that
$Q_{2}(N)$ outperforms $Q_{1}(N)$ only after a certain value of the input
energy $N$. Figure 5: Threshold energy as a function of the transmissivity
$N_{th}=N_{th}(\eta)$ (solid curve dividing the dark and the white areas). The
dark area indicates the values of the energy $N$ for which the two-mode
squeezed vacuum state is optimal. The white region indicates where the single-
mode squeezed vacuum state is optimal. The dashed line denotes the behavior of
the threshold energy $N_{th}$ close to the critical transmissivity
$\eta_{c}\simeq 0.296$.

As we can see from Fig. 4, for a given value of the transmissivity $\eta$, the
two-mode QCB $Q_{2}(N)$ outperforms the single-mode QCB $Q_{1}(N)$ only after
a threshold energy. In fact, for any value of the transmissivity $\eta$ larger
than a critical value $\eta_{c}$ there is a threshold energy
$N_{th}=N_{th}(\eta)$ that makes the two-mode squeezed vacuum state more
convenient than the single-mode counterpart. This threshold energy decreases
for decreasing values of $\eta$. In particular, for transmissivities less than
the critical value $\eta_{c}$, the threshold energy becomes zero, i.e., the
two-mode state is always better than single-mode state. We have numerically
evaluated the critical value $\eta_{c}\simeq 0.296$ (corresponding to
$\Gamma_{c}\simeq 1.22$). This phenomenon is fully illustrated in Fig. 5,
where we have plotted the threshold energy as function of the transmissivity
$N_{th}=N_{th}(\eta)$. For $N>N_{th}$ (dark area), the optimal state is the
two-mode squeezed vacuum state, while for $N<N_{th}$ (white area) it is the
single-mode squeezed vacuum state. In particular, note that $N_{th}=0$ at
$\eta=\eta_{c}$. Close to the critical transmissivity we have OtherCurve

$N_{th}\simeq 4(\eta-\eta_{c})+5.5(\eta-\eta_{c})^{2}~{}.$ (40)

###  VI.2 Optimal input at fixed energy and squeezing

It should be said that, in realistic conditions, it is unlikely to have pure
squeezing. For this reason, it is important to investigate the performances of
the squeezed thermal states by fixing this physical parameter together with
the total energy. Thus, in this section, we fix both the input energy and
squeezing, i.e., we set

$\displaystyle N_{1}$ $\displaystyle=N_{2}=N,$ $\displaystyle\beta_{1}$
$\displaystyle=\beta_{2}=\beta~{}~{}(0\leq\beta\leq 1)~{}.$ (41)

Then, we compare the single-mode squeezed thermal state $\rho=\rho(N,\beta)$
with the two-mode squeezed thermal states $\rho=\rho(N,\beta,\gamma)$ for
various values of $\gamma$. In other words, we compare $Q_{1}(N,\beta)$ and
$Q_{2}(N,\beta,\gamma)$.

For fixed $N$ and $\beta$, we find that the minimum of $Q_{2}(N,\beta,\gamma)$
is achieved for $\gamma=1$ (easy to check numerically). This means that two-
mode discrimination is easier when all the thermal photons are sent through
the lossy channel. In this case we find numerically that

$Q_{2}(N,\beta,1)<Q_{1}(N,\beta)\,,$

for every values of the input parameters $N$ and $\beta$, and every value of
damping rate $\Gamma$ in the channel. In other words, at fixed energy and
squeezing, there is a two-mode squeezed thermal state (the asymmetric one with
$\gamma=1$) able to outperform the single-mode squeezed thermal state in the
detection of any loss. In order to quantify the improvement we introduce the
QCB reduction

$\Delta Q=Q_{1}(N,\beta)-Q_{2}(N,\beta,1).$

The more positive this quantity is, the more convenient is the use of the two-
mode state instead of the single-mode one. In Fig. 6 we show the behavior of
$\Delta Q$ as function of the input energy and squeezing for two different
values of the damping. As one can see from the plot, the QCB reduction is
always positive. Its value increases with the energy while reaching a maximum
for intermediate values of the squeezing. By comparing the two panels of Fig.
6, we can also note that the QCB reduction increases for increasing damping
$\Gamma$ (i.e., decreasing transmissivity).

Figure 6: (Color online) Density plot of the QCB reduction $\Delta Q$ as
function of the input energy $N$ and the squeezing $\beta$. The left plot is
for $\Gamma=0.1$ and the right one for $\Gamma=0.9$.

Thus, we have just shown that, for fixed values of $N$ and $\beta$, the
asymmetric two-mode squeezed thermal state ($\gamma=1$) is the optimal thermal
probe in the detection of any loss $\Gamma$. Here we also show that this is
approximately true for $\gamma\lesssim 1$. In other words, we show that the
inequality $Q_{2}(N,\beta,\gamma)<Q_{1}(N,\beta)$ is robust against
fluctuations of $\gamma$ below the optimal value $\gamma=1$. This property is
clearly important for practical implementations. To study this situation, let
us consider the $\gamma$-dependent QCB reduction

$\Delta Q_{\gamma}=Q_{1}(N,\beta)-Q_{2}(N,\beta,\gamma)~{}.$ (42)

In Fig. 7 we have specified this quantity for different values of the
asymmetry parameter $\gamma$ (each panel refers to a different value of
$\gamma$). Then, for every chosen $\gamma$, we have computed $\Delta
Q_{\gamma}$ over a sample of $10^{3}$ random values of $N$, $\beta$, and
$\Gamma$ (in each panel). As one can see from the figure, the quantity $\Delta
Q_{\gamma}$ is approximately positive also when $\gamma$ is quite different
from the unity.

Figure 7: (Color online) QCB reduction $\Delta Q_{\gamma}$ for different
values of $\gamma$ (top left $\gamma=0.99$, top right $\gamma=0.9$, bottom
left $\gamma=0.8$, bottom right $\gamma=0.7$). In each panel, $\Delta
Q_{\gamma}$ is computed over a sample of $10^{3}$ random values of $N$,
$\beta$, and $\Gamma$.

##  VII Analysis of the correlations

Since two-mode squeezed thermal states are able to outperform the single-mode
counterpart under several physical conditions, it is natural to investigate
this improvement directly in terms of the correlations of the input state. The
quantification of the correlations is realized by using the entanglement, the
quantum discord and the quantum mutual information. In order to quantify the
degree of entanglement of a two-mode Gaussian state, we can use the
logarithmic negativity. Let us consider a bipartite Gaussian state with CM
given in Eq. (13). It is easy to derive the symplectic eigenvalues of the
partially transposed state. These are given by

$\tilde{d}_{\pm}=\sqrt{\frac{\tilde{\Delta}-\sqrt{\tilde{\Delta}^{2}-4I_{4}}}{2}}~{},$

where $\tilde{\Delta}=I_{1}+I_{2}-2I_{3}$, and the symplectic invariants
$I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ are defined in Eq. (14). From the
smallest of these symplectic eigenvalues, we can compute the logarithmic
negativity, which is equal to

$E=\max\\{0,-\log 2\tilde{d}_{-}\\}~{}.$

A bipartite Gaussian state is entangled iff $\tilde{d}_{-}<1/2$, so that the
logarithmic negativity gives positive values for all the entangled states and
$0$ otherwise.

The quantum discord is defined as the mismatch of two different quantum
analogues of classically equivalent expressions of the mutual information and
may be used to quantify quantum correlations in mixed separable states. For a
two-mode squeezed thermal state with CM as in Eq. (15), the quantum discord
may be written as GQDSCa

$\displaystyle D=$ $\displaystyle h(\sqrt{I_{2}})-h(d_{-})-h(d_{+})$
$\displaystyle+h\left(\frac{\sqrt{I_{1}}+2\sqrt{I_{1}I_{2}}+2I_{3}}{1+2\sqrt{I_{2}}}\right)$
(43)

where

$h(x)=\left(x+\frac{1}{2}\right)\log\left(x+\frac{1}{2}\right)-\left(x-\frac{1}{2}\right)\log\left(x-\frac{1}{2}\right)$

is the binary Shannon entropy. We have that, for $0\leq D\leq 1$, the state
may be either entangled or separable, whereas all the states with $D>1$ are
entangled GQDSCa ; GQDSCb .

Finally, the quantum mutual information, which quantifies the amount of total,
classical plus quantum, correlations, is given by
$I=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})$, where
$S(\rho)=-\mathop{\text{Tr}}\nolimits[\rho\log\rho]$ is the von Neumann
entropy of the state $\rho$ and
$\rho_{A(B)}=\mathop{\text{Tr}}\nolimits_{B(A)}[\rho_{AB}]$ are the partial
traces over the two subsystems. For a two-mode squeezed thermal state with CM
as in Eq. (15) the quantum mutual information can be computed using the
formula

$I=\frac{1}{2}\left[h(\sqrt{I_{1}})+h(\sqrt{I_{2}})-h(d_{+})-h(d_{-})\right]\,.$

Figure 8: (Color online) Upper panels. QCB reduction $\Delta Q_{\bar{\gamma}}$
(with $\bar{\gamma}=0.999$) as a function of the three correlation quantifiers
X=I,D,E where I is the quantum mutual information (dotted red), D is the
quantum discord (dashed blue) and E is the entanglement (solid black). The
plots are for fixed squeezing: $\beta=0.1$ for the left panel and $\beta=0.9$
for the right one. For each quantifier we plot three different curves
corresponding to different values of the damping (from top to bottom
$\Gamma=0.9,0.5$ and $0.1$). Each curve is generated by varying the input
energy $N$ between $0$ and $5$ photons. Middle panels. Density plots of the
QCB reduction $\Delta Q_{\bar{\gamma}}$ as a function of the input discord and
entanglement. The plots are for fixed damping: $\Gamma=0.2$ in the left panel
and $\Gamma=0.8$ in the right one. In each panel, the density plot is
generated by varying the squeezing $0\leq\beta\leq 1$ and the energy $0\leq
N\leq 5$. Lower panels. Entanglement (left) and discord (right) as a function
of the quantum mutual information. Plots are generated by taking a random
sample of $10^{4}$ two-mode squeezed thermal states, i.e., random values of
$N$ and $\beta$ with $\gamma=\bar{\gamma}$.

For pure states the previous three measures are equivalent, whereas for mixed
states, as in the case under investigation in this section, they generally
quantify different kind of correlations. Here we consider the QCB reduction
$\Delta Q_{\bar{\gamma}}=Q_{1}(N,\beta)-Q_{2}(N,\beta,\bar{\gamma})$ between a
single-mode squeezed thermal state $\rho=\rho(N,\beta)$ and a two-mode
squeezed thermal state $\rho=\rho(N,\beta,\bar{\gamma})$ with
$\bar{\gamma}=0.999$. By fixing the input squeezing $\beta$ and varying the
input energy $N$, we study the behaviour of $\Delta Q_{\bar{\gamma}}$ as
function of the three correlation quantifiers, i.e., quantum mutual
information, quantum discord and entanglement (computed over the input two-
mode state).

As shown in the upper panels of Fig. 8, the QCB reduction $\Delta
Q_{\bar{\gamma}}$ is an increasing function of all the three correlation
quantifiers for fixed input squeezing ($\beta=0.1$ for the left panel and
$\beta=0.9$ for the right one). Note that, in each panel and for each
quantifier, we plot three different curves corresponding to different values
of the damping $\Gamma=0.9,$ $0.5$ and $0.1$. The monotonicity of the QCB
reduction in all the correlation quantifiers suggests that the presence of
correlations should definitely be considered as a resource for loss detection,
whether these correlations are classical or genuinely quantum, i.e., those
quantified by entanglement. In other words, employing the input squeezing in
the form of correlations is always beneficial for loss detection when we
consider squeezed thermal states as input sources. The importance of
correlations is confirmed by the plots in the middle panels. Here we consider
again the QCB reduction $\Delta
Q_{\bar{\gamma}}=Q_{1}(N,\beta)-Q_{2}(N,\beta,\bar{\gamma})$ for
$\bar{\gamma}=0.999$. Then, by varying input squeezing $\beta$ and energy $N$,
we study $\Delta Q_{\bar{\gamma}}$ as function of both discord and
entanglement (damping is $\Gamma=0.2$ in the left panel, and $\Gamma=0.8$ in
the right one). These plots show how the QCB reduction is approximately an
increasing function of both discord and entanglement. Finally, in the lower
panels of Fig. 8, we also show how entanglement (left) and discord (right) are
increasing functions of the quantum mutual information with good approximation
(these plots are generated by choosing a random sample of $10^{4}$ two-mode
squeezed thermal states).

##  VIII Conclusions

In this paper we have considered the quantum discrimination of lossy channels.
In particular, we have focused to the case when one of the two channels is the
identity, i.e., the problem of discriminating the presence of a damping
process from its absence (loss detection). For this kind of discriminination
we have considered thermal probes as input, i.e., single- and two-mode
squeezed thermal states. The performance of the channel discrimination has
been quantified using the QCB, computed over the two possible states at the
output of the unknown channel for a given input state. Finding the optimal
input state $\rho$ which minimizes this bound gives automatically the optimal
multi-copy state $\rho\otimes\rho\otimes\cdots$ when we consider many accesses
to the unknown channel (under the assumption of single-copy constraints). In
this scenario, we have fixed the total energy of the input state and optimized
the discrimination (detection of loss) over the class of single- and two-mode
squeezed thermal states. We have found numerically that the optimal states are
pure, thus corresponding to single- and two-mode squeezed vacuum states.
Furthermore, we have determined the conditions where the two-mode state
outperforms the single-mode counterpart. This happens when the energy exceeds
a certain threshold, which becomes zero for suitably low values of the
transmissivity (i.e., high values of damping).

It is worth noticing that our approach (where we fix the total energy of
probing and reference modes) also gives a sufficient condition for the problem
where only the probing energy is fixed. In fact, if a two-mode state
outperforms a single-mode state above a certain threshold value $N_{th}$ of
the total energy, this also happens when just the energy of the probing mode
is above that value $N_{th}$. This is a trivial consequence of the fact that
the total energy is bigger than the probing energy for two-mode states
($N_{2}>N_{2}^{\mathrm{probe}}$) while the two quantities are the same for
single-mode states ($N_{1}=N_{1}^{\mathrm{probe}}$). Thus,
$N_{2}^{\mathrm{probe}}=N_{1}^{\mathrm{probe}}>N_{th}$ can be written as
$N_{2}>N_{1}>N_{th}$ which is a stronger condition than $N_{2}=N_{1}>N_{th}$,
since the QCB is decreasing in the total energy, as one can see from Eqs. (38)
and (39).

In our investigation we have then considered the problem of loss detection in
more realistic conditions, where it is unlikely to have pure squeezing. In
this case, we have studied the optimal state for fixed total energy and
squeezing, i.e., by fixing all the relevant resources needed to create the
input state. Under these constraints, we have shown that a two-mode squeezed
thermal state which conveys all the thermal photons in the dissipative channel
is the optimal thermal probe. In addition, this result is robust against
fluctuations, i.e., it holds approximately also when the thermal photons are
distributed in a more balanced way between the probe mode (sent through the
dissipative channel) and the reference mode (bypassing the channel).

Finally we have closely investigated the role of correlations in our problem
of loss detection. We have found that, for fixed input squeezing, the
reduction of the QCB is an increasing function of several correlation
quantifiers, such as the quantum entanglement, the quantum discord and the
quantum mutual information. We then verify that employing the input squeezing
in the form of correlations (quantum or classical) is always beneficial for
the detection of loss by means of thermal probes.

The results of our paper provides new elements in the field of quantum channel
discrimination and can be applied to a wide range applications, including the
characterization of absorbing materials. In particular, they are relevant in
all the situations where the physical constraints regard the creation of the
input resources rather than the channel to be discriminated.

## Acknowledgments

The authors thank Marco Genoni, Stefano Olivares and Samuel L. Braunstein for
useful discussions.

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