# Classes of Gaussian States for Squeezing Estimation

Leonardo A. M. Souza<EMAIL_ADDRESS>Federal University of Viçosa –
Campus Florestal, LMG 818 Km6 S/N,Florestal,35690-000, Minas Gerais, Brazil

###### Abstract

This study explores a detailed examination of various classes of single- and
two-mode Gaussian states as key elements for an estimation process,
specifically targeting the evaluation of an unknown squeezing parameter
encoded in one mode. To quantify the efficacy of each probe, we employ the
concept of Average Quantum Fisher Information (AvQFI) as a robust metric to
quantify the optimal performance associated with specific classes of Gaussian
states as input. For single-mode probes, we identify pure squeezed single-mode
states as the optimal choice and we explore the correlation between Coherence
and AvQFI. Also, we show that pure two-mode squeezed states exhibit behavior
resembling their single-mode counterparts for estimating the encoded squeezing
parameter, and we studied the interplay between entanglement and AvQFI. This
paper presents both analytical and numerical results that encompass all the
studied classes, offering valuable insights for quantum estimation processes.

##  I Introduction

Estimation theory encompasses methods and principles used for making
predictions based on limited or incomplete data kubacek2012foundations . This
theory, often called Metrology, is vital for technological advances in various
fields kubacek2012foundations ; kay1993fundamentals . The core idea of an
estimation strategy, be it classical or quantum, is to achieve the the best
accuracy in determining a parameter encoded in the system. This often involves
using multiple copies of the system for encoding and conducting a precise
measurement to approach the true parameter value kubacek2012foundations ;
kay1993fundamentals ; helstrom1976quantum ; holevo2003statistical ;
ballester2005estimation ; paris2009quantum .

In estimation theory, we can utilize classical or quantum systems for an
estimation task and compare their performance, in order to determine if there
is a “quantum advantage” in estimating unknown parameters. The extensive
literature on this subject delves comparisons between classical and quantum
systems, shedding light on their individual strengths and responses within the
domain of estimation tasks helstrom1976quantum ; holevo2003statistical ;
paris2009quantum ; adesso2009optimal ; AdessoIP ; Luca ; luo2022environmental
; predko2020time ; assad2020accessible ; bradshaw2018ultimate ;
morelli2021bayesian ; ge2020operational ; escher2011general ;
escher2011quantum ; maleki2023speed . It is important to emphasize that within
the Estimation Theory, the estimation precision of a parameter is quantified
by the Fisher Information (FI), or by its quantum version, the Quantum Fisher
Information (QFI) helstrom1976quantum ; holevo2003statistical . Typically,
higher FI or QFI values correspond to better parameter precision. Also, one
may indeed obtain a precision incrementation using quantum resources
adesso2009optimal ; escher2011general ; schnabel2010quantum ;
demkowicz2013fundamental ; pezze2018quantum ; giovannetti2011advances ;
giovannetti2006quantum .

Quantum Metrology, or quantum estimation theory, the use of quantum systems
and measurement apparatus to perform an estimation task, plays a pivotal role
in our technological advancements, particularly exemplified by the Laser
Interferometer Gravitational-Wave Observatory (LIGO) abbott2020prospects :
LIGO’s profound success in detecting gravitational waves, as predicted by
Einstein’s theory of relativity, can potentially achieve even greater success
by relying on quantum-based precision measurements yu2020quantum .
Additionally, the utilization of squeezed light in the LIGO experiment can
significantly enhance the sensitivity of the detector aasi2013enhanced ;
dwyer2022squeezing ; grote2013first . Furthermore, it is fundamental in the
design and optimization of quantum-enhanced sensors niezgoda2021many ;
marciniak2022optimal ; degen2017quantum  and metrology techniques, both
theoretical and experimental pezze2018quantum .

Within the scope of estimation theory using quantum systems and measurements,
one can specify the analysis in discrete and continuous variables (the latter
is commonly abbreviated to CV) adesso2009optimal ; pezze2018quantum . Finally,
for continuous variables systems it is possible to subdivide the analysis for
non-Gaussian states (i.e. those that cannot be completely discribed by their
first and second statistical moments) and Gaussian states (the group we will
be concerned in this work, where the first and second statistical moments
completely describe the state, that in turn possess a Gaussian characteristic
function, by definition) adesso2009optimal ; Luca . While in each studied case
a specific quantum system can perform better or worse than other kind of state
(for example, adesso2009optimal ), there is no “rule” dictating that some
‘special’ quantum system is, in fact and in general, the golden state to
perform all tasks better than classical ones.

As already mentioned aasi2013enhanced ; dwyer2022squeezing ; grote2013first ,
the use of squeezed light in technological devices are of great importance
nowadays. With this motivation as our driving force, in this work we focus our
attention in the Quantum estimation problem within a Gaussian scenario:
single- or two-mode states subjected to Gaussian dynamics. We are interested
in the estimation of an unknown squeezing parameter encoded in one of the
modes. We follow the strategy proposed by the authors in Luca , where the mode
to be encoded is sent to an Squeezer, and the other mode is an ancilla. The
precision of estimating the squeezing parameter is quantified using the
Average Quantum Fisher Information (AvQFI), which is computed by averaging
over the phase acquired by the mode during its evolution. In Luca  the authors
studied this problem both in complete generality and also for some classes of
Gaussian states. Here we explicitly studied in details the most important
classes of single- and two-mode Gaussian states as probes for this estimation
proccess. For single-mode states we studied the interplay of the AvQFI and a
_Coherence_ quantifier, as defined in xu2016quantifying . Finally, for two-
mode Gaussian states, we analliticaly and numerically investigate the AvQFI as
a function of Entanglement.

This paper is organized as following: in section II we review in great details
the formalism of Gaussian states, quantum estimation theory and the estimation
strategy used in this work (following closely Luca ); in section III we
present our results, sorting by each class of Gaussian state we have chosen.
Still in section III we studied the AvQFI as a function of Coherence (for
single-mode states), and as a function of the Logarithmic Negativity (for two-
mode states). Finally, we conclude our work in section IV.

##  II Preliminaries

Our work is focused on estimating a parameter, specifically the squeezing
parameter encoded in one mode of a Gaussian State. Therefore, in this section,
we will review some important concepts concerning Gaussian States and Quantum
Estimation Theory.

###  II.1 Gaussian States

In this work we are interested in single- or two-mode Gaussian States
subjected to the so called Gaussian dynamics, i.e. dynamics that preserve the
“Gaussianity” of the state (adesso2007entanglement ). A CV system of two modes
$A$ and $B$ (with annihilation operators $\hat{a}$ and $\hat{b}$
respectively), can be defined by the quadrature vector
$\hat{\boldsymbol{O}}=\\{\hat{q}_{A},\hat{p}_{A},\hat{q}_{B},\hat{p}_{B}\\}$,
where $\hat{q}_{k}=(\hat{a}_{k}+\hat{a}_{k}^{\dagger})$ and
$\hat{p}_{k}=i(\hat{a}_{k}^{\dagger}-\hat{a}_{k})$, where $k=A,B$ (assuming
natural units, $\hbar=2$). We encourage the reader to pay attention to
equations and definitions, since one can find both works with $\hbar=1$ and
$\hbar=2$ in the literature. The quadratures obey the canonical commutation
relations $[\hat{O}_{j},\hat{O}_{k}]=2i\Omega_{jk}$, with the two-mode
symplectic form

$\boldsymbol{\Omega}=\left(\begin{tabular}[]{cc}0&1\\\ -1&0\\\
\end{tabular}\right)^{\bigoplus 2}.$ (1)

The relations for single-mode Gaussian states are, obviouslly, the same,
changing the dimension of the vectors and matrices.

A Gaussian state $\rho_{AB}$ adesso2007 ; Adesso2014ext ; cerf2007 ; paris2005
is represented by a Gaussian characteristic function in phase space, and is
completely characterized by its first and second statistical moments of the
quadrature vector, given respectively by the displacement vector
$\boldsymbol{\varepsilon}_{AB}=(\varepsilon_{j})$ and the covariance matrix
$\boldsymbol{\sigma}_{AB}=(\sigma_{jk})$, where
$\varepsilon_{j}=\langle\hat{O}_{j}\rangle$ and
$\sigma_{jk}=\frac{1}{2}\langle\hat{O}_{j}\hat{O}_{k}+\hat{O}_{k}\hat{O}_{j}\rangle-\langle\hat{O}_{j}\rangle\langle\hat{O}_{k}\rangle$.
A _bona fide_ condition satisfied by all physical Gaussian states is the
Robertson-Schrödinger uncertainty relation, given by

$\boldsymbol{\sigma}_{AB}+i\boldsymbol{\Omega}\geq 0.$ (2)

For our purpose (since we will deal with mode B as an ancilla, as one can see
soon in section II.3), a general covariance matrix for a two-mode Gaussian
state can be written as Luca :

$\boldsymbol{\sigma}_{AB}=\left(\begin{tabular}[]{cccc}$a_{1}$&$g$&$c$&0\\\
$g$&$b_{1}$&0&$d$\\\ $c$&0&$b_{2}$&0\\\ 0&$d$&0&$b_{2}$\end{tabular}\right),$
(3)

where the coeficients are defined such that (3) satisfies the physical
constraint (2). Its worth recalling that, by local symplectic operations
(equivalent to local changes of basis on the state), every two-mode covariance
matrix can be transformed to a standard form, with diagonal $2\times 2$
subblocks that can be written as:

$\boldsymbol{\sigma}_{AB}=\left(\begin{tabular}[]{cc}$\boldsymbol{\alpha}$&$\boldsymbol{\gamma}$\\\
$\boldsymbol{\gamma}^{T}$&$\boldsymbol{\beta}$\\\ \end{tabular}\right),$ (4)

where $\alpha=\textrm{diag}\\{a,a\\}$, $\beta=\textrm{diag}\\{b,b\\}$,
$\gamma=\textrm{diag}\\{c,d\\}$, such that $a,b\geq 1$,
$c\geq\left|d\right|\geq 0.$ For future purposes, we define here the
symplectic invariants:
$A=\det\boldsymbol{\alpha};B=\det\boldsymbol{\beta};C=\det\boldsymbol{\gamma};D=\det\boldsymbol{\sigma}_{AB}.$

The total mean number of excitations (proportional to the total mean energy)
of a two-mode Gaussian state can be defined as:
$E\equiv\bar{n}_{A}+\bar{n}_{B}=2\bar{n},$ where
$\bar{n}_{A}=(\operatorname{tr}[\boldsymbol{\alpha}]-2)/4+(\varepsilon_{x,A}^{2}+\varepsilon_{p,A}^{2})/4$
and
$\bar{n}_{B}=(\operatorname{tr}[\boldsymbol{\beta}]-2)/4+(\varepsilon_{x,B}^{2}+\varepsilon_{p,B}^{2})/4$
are the mean number of excitations in modes $A$ and $B$ respectively
(considering the displacement vector as non-null, since we will deal with such
states soon), and $\bar{n}_{i}$ denotes the mean number of excitations per
mode. Throughout this manuscript, we will consistently uphold the physical
requirement that any initial state must be subject to a _finite_ mean energy
constraint.

The various categories of Gaussian states that we address in this study will
be detailed (including their covariance matrices, displacement vectors, etc.)
in dedicated sections that follow.

To provide a comprehensive background on Gaussian states for the reader,
ensuring a proper understanding of this work, we introduce two quantities that
quantify important characteristics of Gaussian states. First, we will discuss
an entanglement quantifier for Gaussian states, specifically the Logarithmic
Negativity.

The _Logarithmic Negativity_ , $\mathcal{E}_{N}$, is a decreasing function of
the smallest symplectic eigenvalue $\tilde{\nu}$ of the partial transpose of
the covariance matrix:

$\mathcal{E}_{N}=\max\\{0,-\ln\tilde{\nu}\\}.$ (5)

Here $\tilde{\nu}$ are the symplectic eigenvalues of the _partially transposed
covariance matrix_ (for details we suggest the reader to check chapter 3 of
adesso2007 ), and one can show that the $\tilde{\nu}$ can be written in
function of the symplectic invariants AdessoIP :

$2\tilde{\nu}^{2}=H-\sqrt{H^{2}-4D}$ (6)

with $H=A+B-2C$. The Logarithmic Negativity is a measure of Entanglement in
composed Gaussian systems.

Finally, we introduce a coherence measure that we will use to analyze single-
mode states. Coherence is typically associated with the concepts of
interference and superposition states. However, how can we quantify a state’s
ability to formally exhibit _Coherence_? This question was addressed a few
years ago, in the discrete case by baumgratz2014quantifying  and for Gaussian
states by xu2016quantifying . Given a N-mode Gaussian state $\rho$, with
covariance matrix $\boldsymbol{\sigma}$ and displacement vector
$\boldsymbol{\varepsilon}$, the _Coherence_ of the state can be quantified by
xu2016quantifying :

$C(\rho)=-S(\rho)+\sum_{i=1}^{N}[(\bar{n_{i}}+1)\log_{2}(\bar{n_{i}}+1)-\bar{n_{i}}\log_{2}\bar{n_{i}}],$
(7)

where $\bar{n}_{i}$ is the mean photon number of the $i$-esim state, and
$S(\rho)$ is the entropy of the state $\rho$:

$S(\rho)=-\sum_{i=1}^{N}\left[\left(\frac{\nu_{i}-1}{2}\right)\log_{2}\left(\frac{\nu_{i}-1}{2}\right)-\left(\frac{\nu_{i}+1}{2}\right)\log_{2}\left(\frac{\nu_{i}+1}{2}\right)\right].$
(8)

In equation (8), $\\{\nu_{i}\\}_{i=1}^{N}$ are the symplectic eigenvalues of
the Covariance Matrix (please do not confuse $\nu$ with $\tilde{\nu}$, since
the latter represents the symplectic eigenvalues of the partially transposed
covariance matrix). In this framework, maximally coherent states are pure
states. Furthermore, it is intuitive to observe that states exhibiting
squeezing can have Coherence values (as defined in equation 7) greater than
those of the so-called coherent states. As a final remark, we stress that we
will differentiate _Coherence_ (the measure of Coherence as in Equation 7)
from coherent (concerning coherent states) by capitalizing the first letter of
the former.

###  II.2 Quantum Estimation Theory

For completeness, this subsection is dedicated to introducing Quantum
Estimation Theory and the concept of Quantum Fisher Information (QFI). We
encourage readers with a keen interest in exploring this topic further to
consult the references such as helstrom1976quantum ; holevo2003statistical ;
ballester2005estimation ; paris2009quantum .

The fundamental concept behind any estimation strategy, whether classical or
quantum, is to obtain information about a parameter encoded in the system with
the highest possible accuracy. Typically, this information is acquired by
sending N copies of the system to the encoding stage. In the final stage, the
most precise measurement is performed on the system, aiming to obtain a value
for the parameter that approaches the ‘actual’ value. Consequently, an
estimation strategy is grounded in statistical theory.

In this scenario, the idea is to encounter a probability distribution that
depends on a parameter, denoted here as $p(x|\epsilon)$. The objective is to
achieve the most accurate estimate of the parameter $\epsilon$ by repeatedly
sampling the random variable $x$, which follows the distribution
$p(x|\epsilon)$. An important principle in classical estimation theory, known
as the _Cramér-Rao bound_ , asserts that the variance of any unbiased
estimator $\hat{\epsilon}$ for the parameter $\epsilon$ must satisfy the
following inequality:

$\text{VAR}\left(\hat{\epsilon}\right)\geq\frac{1}{NF_{\epsilon}^{M}}.$ (9)

where $N$ is the number of samplings, $\text{VAR}\left(\hat{\epsilon}\right)$
is the variance of the estimator $\hat{\epsilon}$ and $F_{\epsilon}^{M}$ is
the classical Fisher information (where we explicitly showed the dependence of
$F_{\epsilon}^{M}$ on the measurement M), defined by:

$F_{\epsilon}^{M}=\int\text{d}x\,p(x|\epsilon)\left[\partial_{\epsilon}\log
p(x|\epsilon)\right]^{2}.$ (10)

In a quantum version of the previously mentioned situation, the parameter
$\epsilon$ is encoded in a quantum state $\rho_{\epsilon}$, typically by
applying a quantum map $\Phi_{\epsilon}$ to a known input probe $\rho$:
$\rho_{\epsilon}=\Phi_{\epsilon}[\rho].$ To acquire information about the
system and, consequently, about $\epsilon$, a generic Positive Operator-Valued
Measurement (POVM) must be executed on $\rho_{\epsilon}$. The maximum
precision achievable within the bounds of quantum mechanics for unbiased
estimation of $\epsilon$ is attained through optimization across all possible
POVMs.

This approach leads to the derivation of the quantum Cramér-Rao bound, which
stipulates:

$\text{VAR}(\hat{\epsilon})\geq\frac{1}{NH_{\epsilon}(\rho)}.$ (11)

Here, $H_{\epsilon}$ represents the Quantum Fisher Information (QFI) linked to
the encoded state $\rho_{\epsilon}$, derived from $\rho$. The Quantum Fisher
Information is defined as:

$H_{\epsilon}[\rho]=\mathrm{\text{Tr}}\left[\rho_{\epsilon}L_{\epsilon}^{2}\right].$
(12)

$L_{\epsilon}$ is the so called symmetric logarithmic derivative (SLD), an
Hermitian operator that satisfies the relation:

$\rho_{\epsilon}L_{\epsilon}+L_{\epsilon}\rho_{\epsilon}=2\,\partial_{\epsilon}\rho_{\epsilon}.$
(13)

It can be showed ballester2005estimation ; de2009estudo  that the Quantum
Fisher Information (QFI) is linked to the second-order expansion of the Bures
distance, or equivalently, the Uhlmann fidelity
${\mathcal{F}(\rho_{1},\rho_{2})=\left(\mathrm{\text{Tr}}\left[\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}\right]\right)^{2}}$
:

$H_{\epsilon}[\rho]=8\lim_{\text{d}\epsilon\to
0}\frac{1-\sqrt{\mathcal{F}(\rho_{\epsilon},\rho_{\epsilon+\text{d}\epsilon})}}{\text{d}\epsilon^{2}}.$
(14)

It’s worth noticing two interesting aspects: (i) In general, Fisher
Information is less than or equal to Quantum Fisher Information:
$F_{\epsilon}^{M}\leq H_{\epsilon}$. (ii) By harnessing quantum resources such
as entanglement, squeezing, “genuine” quantum states like Fock states, and
others (as exemplified in adesso2009optimal ; giovannetti2011advances ;
giovannetti2006quantum ), one can surpass the Standard Limit of estimation
(equation 9) (often referred to as the shot noise limit or standard quantum
limit), and achieve the Heisenberg limit, i.e., reaching the saturation of
equation 11. Moreover, employing quantum resources can lead to a quadratic
improvement in estimation problems compared to using classical resources.
While this quadratic gain is not a strict rule, it serves as a guiding
principle in quantum system research aimed at parameter estimation.

We can attribute the following interpretation to Fisher information: it
quantifies the sensitivity of a probability distribution (or a quantum state,
in the quantum version) to small changes in the parameter $\epsilon$. If we
make a small alteration $\theta$ to the probability distribution, and it
becomes substantially different from the original distribution, this results
in a higher Fisher information. Consequently, by the Cramér-Rao bound (and the
Bures distance showed above), we can infer that we are moving away from the
original probability distribution (or the original state), and therefore
Fisher information quantifies the error with respect to the estimator: the
higher the Fisher information, more the Fisher information will “capture” any
deviation from the original distribution or state, the lower the variance with
respect to the estimator.

###  II.3 Estimation Strategy

Figure 1: (Color online)Proposed Estimation Strategy. The state is initially
prepared as a single- or two-mode Gaussian state. Mode A is sent to a
Squeezer, where the parameter $\epsilon$ is encoded. Mode B propagates freely.
After the action of the Squeezer, mode A is brought back to be measured with
mode B. Information about the phase $\theta$ acquired by the modes during
their journey from state preparation to the measurement stage is utilized to
establish the strategy. The average Quantum Fisher Information,
$\overline{H_{\epsilon}}(\rho)$, is constructed by averaging over all phases
$\theta$ in order to estimate the squeezing parameter $\epsilon$. More details
are provided in section II.3 of the text.

In this work we follow the estimation strategy as proposed in Luca , depicted
in Figure 1. Initially, the state is prepared as a single- or two-mode
Gaussian state, $\rho_{AB}$, depending on the specific case under
consideration. In this context, we designate mode A as the mode where the
squeezing parameter will be encoded, and mode B as an ancilla. Naturally, mode
B will not be considered when studying a single-mode Gaussian probe. In the
next step, mode A is sent to a Squeezer, while mode B evolves freely. Just
before the action of the Squeezer, the evolved state is given by:
$(\mathcal{R}_{A}\otimes\mathcal{R}_{B})\rho_{AB}(\mathcal{R}_{A}\otimes\mathcal{R}_{B})^{\dagger},$
where $\mathcal{R}_{A(B)}=e^{-i\theta_{A(B)}(t)a^{\dagger}a}$ is a unitary
single-mode rotation operator acting on mode A(B), and acquired during the
‘flight’ between the stages:

$\mathcal{R}_{\theta}^{(A)}[\rho_{A}]=U_{\theta}^{(A)}[\rho_{A}]=e^{-i\theta
a^{\dagger}a}\rho_{A}e^{i\theta
a^{\dagger}a}=\left(\begin{tabular}[]{cc}$\cos\theta$&$\sin\theta$\\\
$-\sin\theta$&$\cos\theta$\end{tabular}\right).$

After the first ‘flight’, the Squeezer
$\mathcal{S}_{\epsilon}^{(A)}=\exp[{\frac{\epsilon}{2}(a^{2}-(a^{\dagger})^{2})}]$
acts on mode A, encoding the parameter $\epsilon$ that we are interested in
estimate. Explicitly:

$\mathcal{S}_{\epsilon}^{(A)}[\rho_{A}]=U_{\epsilon}[\rho_{A}]=e^{-\frac{\epsilon}{2}(a^{\dagger
2}-a^{2})}\rho_{A}e^{+\frac{\epsilon}{2}(a^{\dagger
2}-a^{2})}=\left(\begin{tabular}[]{cc}$e^{\epsilon}$&$0$\\\
$0$&$e^{-\epsilon}$\end{tabular}\right).$

Mode A now returns, in order to be measured, acquiring another phase
$\mathcal{R}_{A}$, and mode B a phase $\mathcal{R}_{B}$. The state can be
written as:

$(\mathcal{R}_{A}\mathcal{S}_{\epsilon}^{(A)}\mathcal{R}_{A}\otimes\mathcal{R}_{B}^{2})\rho_{AB}(\mathcal{R}_{A}\mathcal{S}_{\epsilon}^{(A)}\mathcal{R}_{A}\otimes\mathcal{R}_{B}^{2})^{\dagger}.$

If $\theta_{A,B}(t)$ is known by parts A and B, a unitary operation
$(\mathcal{R}_{A}^{\dagger 2}\otimes\mathcal{R}B^{\dagger 2})$ can be applied
by both parts. It is worth stressing this point of the proposed strategy: the
knowledge of $\theta{A(B)}$ allows parts A and B to apply this unitary
operation $(\mathcal{R}_{A}^{\dagger 2}\otimes\mathcal{R}_{B}^{\dagger 2})$ in
such a way as to ‘remove’ the dynamical phase introduced in mode B (the
ancillary mode), leaving only the dynamics to which mode A is subjected. One
can now write the state as:

$(\mathcal{R}_{A}^{\dagger}\mathcal{S}_{\epsilon}^{(A)}\mathcal{R}_{A}\otimes\mathds{1}_{B})\rho_{AB}(\mathcal{R}_{A}^{\dagger}\mathcal{S}_{\epsilon}^{(A)}\mathcal{R}_{A}\otimes\mathds{1}_{B})^{\dagger}.$

The total dynamical map describing the full evolution of the state is:
$\Phi_{\epsilon,\theta}[\rho_{AB}]=\Phi_{\epsilon,\theta}^{A}\otimes\mathds{1}^{B}[\rho_{AB}]$.

Since we are dealing with estimation theory paris2009quantum , an estimator
$\hat{\epsilon}$ for the applied squeezing can now be obtained (usually the
maximum likelihood estimator is constructed), after a collection of
measurements (if optimal, they can in principle minimize the error). The
estimation accuracy of $\hat{\epsilon}$ is given by the quantum Cramer-Rao
bound (as detailed in section II.2):

$\delta\hat{\epsilon}\geq\frac{1}{\sqrt{MH_{\epsilon}^{(\theta)}(\rho)}}$ (15)

where $H_{\epsilon}^{(\theta)}(\rho)$ is the Quantum Fisher Information (QFI).

Depending on the phase $\theta$ acquired in flight, the estimation of
$\epsilon$ can be different for each time-of-flight, therefore the
_versatility_ of an input state can be checked by looking at the so called
average QFI (AvQFI):

$\overline{H_{\epsilon}}(\rho)\equiv\int_{0}^{2\pi}H_{\epsilon}^{(\theta)}(\rho)\frac{d\theta}{2\pi}.$
(16)

In Luca  the authors showed several general results concerning
$\overline{H_{\epsilon}}(\rho)$, where we enphasize that by studying
$\overline{H_{\epsilon}}(\rho)$ on can estimate the parameter $\epsilon$
within this approach: the average QFI $\overline{H_{\epsilon}}(\rho)$ sets a
lower bound on the average value of $\delta\hat{\epsilon}$:

$\overline{\delta\hat{\epsilon}}\equiv\int_{0}^{2\pi}\frac{d\theta}{2\pi}\delta\hat{\epsilon}(\theta)\geq\frac{1}{\sqrt{M\overline{H_{\epsilon}}(\rho)}}$
(17)

In this paper we study the average QFI, $\overline{H_{\epsilon}}(\rho)$, for
inportant classes of single and two-mode Gaussian States, within the
estimation strategy proposed in this section.

##  III Results

In this section, we exploit our findings to present relevant analytical and
numerical results. Our investigation covers both single-mode and two-mode
states as a probe in our estimation strategy. For clarity, we have organized
this section into subsections dedicated to each type of state: single-mode and
two-mode states. Within each category, we provide a comprehensive overview of
their general characteristics and present our results.

###  III.1 Single-mode States

A general single-mode Gaussian state is characterized by the following
covariance matrix (CM):

$\boldsymbol{\sigma}_{A}=\left(\begin{tabular}[]{cc}a&g\\\
g&b\end{tabular}\right),$ (18)

and by the displacement vector:
$\boldsymbol{\varepsilon}=(\varepsilon_{x},\varepsilon_{y}).$ The parameters
are chosen such that the CM represent a physical state, i.e. equation 2 is
satisfied. If we consider, as probes, general single-mode Gaussian states, we
obtain the results in Figure 2. In this Figure we show the average QFI
$\overline{H}_{\theta}$ in function of the mean photon number $n_{A}$ of the
mode. Each point is a general single-mode Gaussian state, with parameters of
the CM and displacement vector randomly chosen ($10^{5}$ states).

Figure 2: (Color online) Average QFI $\overline{H}_{\theta}$ in function of
the mean photon number $n_{A}$ of the mode. Each point is a general single-
mode Gaussian state, with parameters of the CM and displacement vector
randomly chosen ($10^{5}$ states). The red dashed line is the upper bound for
this estimation strategy, using $\overline{H}_{\theta}$, and is returned by
pure squeezed states (see section III.1). The pink dashed curve is the lower
bound, achieved by thermal states. The black solid curve is the maximum value
of $\overline{H}_{\theta}$ using coherent states as probes (section III.1.1).

One can see that all states lie within the region bounded by the lower and
upper limiting curves. The upper bound is given by:

$\overline{H_{\theta}}=4n_{A}^{2}+4n_{A}+2,$ (19)

while the lower bound is given by:

$\overline{H_{\theta}}=4\frac{(2n_{A}+1)^{2}}{1+(2n_{A}+1)^{2}}.$ (20)

It is interesting to highlight the physical significance of the upper bound:
this curve is produced by single-mode pure squeezed states. Consequently, this
class of states serves as the optimal probe for estimating the squeezing
parameter $\varepsilon$ within the proposed strategy. One possible
interpretation is that a pure squeezed state will be more sensitive to any
parameter introduced into the state. The lower bound is produced by mixed
thermal states.

####  III.1.1 Coherent States

Now we restrict our analysis to single-mode Gaussian coherent states as probe
states, without squeezing, with covariance matrix:

$\boldsymbol{\sigma}_{A}=\left(\begin{tabular}[]{cc}a&0\\\
0&a\end{tabular}\right),$ (21)

and displacement vector
$\boldsymbol{\varepsilon}=(\varepsilon_{x},\varepsilon_{y}).$ Working out the
same procedure with this class of states, we can see that the maximum value of
$\overline{H}_{\theta}$ is given by the black solid curve in Figure 3.
Explicitly, this curve is given by:

$\overline{H_{\theta}}=4n_{A}+2.$ (22)

It is interesting that pure squeezed states approaches the Heisenberg limit
Luca ; escher2011general , since $\overline{H}_{\theta}\sim n_{A}^{2}$, while
using coherent states we can obtain $\overline{H}_{\theta}\sim n$, the so
called Standard Quantum Limit (or Shot Noise Limit). This can be viewed as a
“quantum advantage” in this metrology scheme.

Figure 3: (Color online) Average QFI $\overline{H}_{\theta}$ in function of
the mean photon number $n_{A}$ of the mode. Each point is a single-mode
Gaussian coherent state, with parameters of the CM and displacement vector
randomly chosen ($10^{5}$ states). The red dashed line is the upper bound for
this estimation strategy, using $\overline{H}_{\theta}$, and is returned by
pure squeezed states. The pink dashed curve is the lower bound, achieved by
thermal states. The black solid curve is the maximum value of
$\overline{H}_{\theta}$ using coherent states as probes (section III.1.1).

####  III.1.2 Relation between the Average QFI and Coherence of the state

A Coherence quantifier for Gaussian states was proposed in reference
xu2016quantifying  and we briefly summarized in section II.1. The quantifier
is given by equation 7:

$C(\rho)=-S(\rho)+\sum_{i=1}^{N}[(\bar{n_{i}}+1)\log_{2}(\bar{n_{i}}+1)-\bar{n_{i}}\log_{2}\bar{n_{i}}],$
(23)

with $S(\rho)$ the relative entropy. Here we establish a relation between
$\overline{H}_{\theta}$ and $C(\rho)$ for single-mode states. Our results are
presented in Figure 4.

Figure 4: (Color online) Average Quantum Fisher Information
($\overline{H}_{\theta}$) as a function of Coherence ($C(\rho)$). The blue
dots represent general single-mode states, while the red dots represent
coherent single-mode states (a total of $10^{5}$ states were randomly chosen
for each case). We varied the mean photon number ($n_{A}$) in each plot: (a)
$n_{A}=3$; (b) $n_{A}=5$; (c) $n_{A}=10$; (d) $n_{A}=100$. It is evident that
as $n_{A}$ increases, the value of $\overline{H}_{\theta}$ also increases.
Furthermore, single-mode states with squeezing surpass coherent states in this
estimation problem when they possess the same level of Coherence.

In Figure 4, one can observe that coherent states achieve lower values of
$\overline{H}_{\theta}$ compared to general single-mode states (which
generally may exhibit some degree of squeezing) for the same amount of
Coherence as defined in Equation 7. This corroborates the intuition that
coherent states, being "quasi-classical" states, cannot attain the same level
of sensitivity in this estimation problem when compared to squeezed states,
which possess a “quantum” advantage.

###  III.2 Two-mode States

As mentioned before (equation 3), a general two-mode Gaussian state can be
characterized by the following CM:

$\boldsymbol{\sigma}_{AB}=\left(\begin{tabular}[]{cccc}$a_{1}$&$g$&$c$&0\\\
$g$&$b_{1}$&0&d\\\ $c$&0&$b_{2}$&0\\\ 0&$d$&0&$b_{2}$\end{tabular}\right),$
(24)

with displacement vector given by:
$\boldsymbol{\varepsilon}=(\varepsilon_{x},\varepsilon_{y},0,0).$ Tipically,
general two-mode states has the same behavior of single-mode states. The upper
and lower bounds are given by the same states, pure two-mode squeezed states
and mixed thermal states, and by the exact same equations, Eq. 19 and Eq. 20,
respectively. Figure 5 depicts how $10^{5}$ randomly chosen states are
distributed, when we study $\overline{H}_{\theta}$ as a function of $n_{A}$.
However, using two-mode states one can access more physics as we can see in
the next sections.

Figure 5: (Color online) Average QFI $\overline{H}_{\theta}$ in function of
the mean photon number $n_{A}$ of mode A. Each point is a general two-mode
Gaussian state, with parameters of the CM and displacement vector randomly
chosen ($10^{5}$ states). The red dashed line is the upper bound for this
estimation strategy, using $\overline{H}_{\theta}$, and is returned by pure
two-mode squeezed states (see section III.2.3). The pink dashed curve is the
lower bound, achieved by mixed thermal states. The black solid curve is the
maximum value of $\overline{H}_{\theta}$ using states containing discord tipe
correlations as probes (section III.2.2). Finally, the solid green curve is
the best possible estimation value of $\overline{H}_{\theta}$ using states
without any type of correlations (section III.2.1).

####  III.2.1 Separable States in the standard form

We start with the simplest case: separable states with no correlations between
modes A and B. Separable states in the standard form are characterized, within
our approach, by:

$\boldsymbol{\sigma}_{AB}=\left(\begin{tabular}[]{cccc}$a_{1}$&0&0&0\\\
0&$b_{1}$&0&0\\\ 0&0&$b_{2}$&0\\\ 0&0&0&$b_{2}$\end{tabular}\right),$ (25)

with displacement vector given by:
$\boldsymbol{\varepsilon}=(\varepsilon_{x},\varepsilon_{y},0,0).$ Our results
for this class of states are depicted in Figure 6. Interesting enough, for the
case of separable states in the standard form as Equation 25, while obviously
the lower bound is given by mixed thermal states (equation 20), the upper
bound for separable states is given by (the solid green curve of Figure 6):

$\overline{H_{\theta}}=3-\frac{1}{1+2n_{A}}+2n_{A}.$ (26)

This class of state can be thought as been worst than single-mode coherent
states for this estimation problem. Naturally, if we allow our probe to be in
the form of equation 24, with $c=d=0$, the state will behave as a single-mode
coherent state and achieve the same result of equation 22. We can conjecture
that, for this case of separable states in the standard form, some level of
Coherence in mode A is required to reach greater level of squeezing
estimation.

Figure 6: (Color online) Average QFI $\overline{H}_{\theta}$ in function of
the mean photon number $n_{A}$ of mode A. Each point is an uncorrelated two-
mode Gaussian state with CM given by equation 25, with parameters of the CM
and displacement vector randomly chosen ($10^{5}$ states). The red dashed line
is the upper bound for this estimation strategy, using
$\overline{H}_{\theta}$, and is returned by pure two-mode squeezed states (see
section III.2.3). The pink dashed curve is the lower bound, achieved by mixed
thermal states. The black solid curve is the maximum value of
$\overline{H}_{\theta}$ using states containing discord tipe correlations as
probes (section III.2.2). Finally, the solid green curve is the best possible
estimation value of $\overline{H}_{\theta}$ using states without any type of
correlations (section III.2.1).

####  III.2.2 Two-mode states with Discord type correlation

Quantum Discord represents quantum correlations within composite systems that
do not necessarily involve entanglement zurek2000einselection ;
ollivier2001quantum ; henderson2001classical . In the literature one can
encounter a wide range of options to quantify and study Discord-type
correlations. For example, it has been proven that, in CV systems, the so-
called interferometric power is a measure of discord-type correlations for
general mixed states $\rho_{AB}$, reducing to a measure of entanglement in the
particular case of pure states AdessoIP ; Bera2014 . A Discordant Gaussian
state can be characterized by the following CM:

$\boldsymbol{\sigma}_{AB}=\left(\begin{tabular}[]{cccc}a&0&c&0\\\ 0&a&0&c\\\
c&0&b&0\\\ 0&c&0&b\end{tabular}\right),$ (27)

with displacement vector given by:
$\boldsymbol{\varepsilon}=(\varepsilon_{x},\varepsilon_{y},0,0),$ and $c\neq
0$. For this class of states our results are shown in Figure 7. One can see
clearly that this class of state reach the same maximum value for
$\overline{H}_{\theta}$ as single-mode coherent states, given by equation 22.

Figure 7: (Color online) Average QFI $\overline{H}_{\theta}$ in function of
the mean photon number $n_{A}$ of mode A. Each point is a two-mode Gaussian
state with CM given by equation 27, with parameters of the CM and displacement
vector randomly chosen ($10^{5}$ states). The red dashed line is the upper
bound for this estimation strategy, using $\overline{H}_{\theta}$, and is
returned by pure two-mode squeezed states (see section III.2.3). The pink
dashed curve is the lower bound, achieved by mixed thermal states. The black
solid curve is the maximum value of $\overline{H}_{\theta}$ using states
containing discord tipe correlations as probes (section III.2.2). Finally, the
solid green curve is the best possible estimation value of
$\overline{H}_{\theta}$ using states without any type of correlations (section
III.2.1).

####  III.2.3 Entangled Two-mode States

In this section, we examine the last category of Gaussian states we have
investigated in this study. We dedicate our attention now to the important
class of Entangled states in the standard form, where the CM is given by:

$\boldsymbol{\sigma}_{AB}=\left(\begin{tabular}[]{cccc}a&0&c&0\\\ 0&a&0&-c\\\
c&0&b&0\\\ 0&-c&0&b\end{tabular}\right),$ (28)

with displacement vector given by: $\boldsymbol{\varepsilon}=(0,0,0,0).$ For
this important class, that include the states that saturate our measure
$\overline{H}_{\theta}$ (pure two-mode-squeezed states), our results show the
same upper bound as single-mode Gaussian states (equation 19):

$\overline{H_{\theta}}=4n_{A}^{2}+4n_{A}+2,$ (29)

while the lower bound is, in agreement with previous sections, the upper bound
for separable states in the standard form (equation 26). Figure 8 depicts our
results for $\overline{H}_{\theta}$ in terms of $n_{A}$.

Figure 8: (Color online) Average QFI $\overline{H}_{\theta}$ in function of
the mean photon number $n_{A}$ of mode A. Each point is an Entangled two-mode
Gaussian state with CM given by equation 28, with parameters of the CM and
displacement vector randomly chosen ($10^{5}$ states). The red dashed line is
the upper bound for this estimation strategy, using $\overline{H}_{\theta}$,
and is returned by pure two-mode squeezed states (see section III.2.3). The
pink dashed curve is the lower bound, achieved by mixed thermal states. The
black solid curve is the maximum value of $\overline{H}_{\theta}$ using states
containing discord tipe correlations as probes (section III.2.2). Finally, the
solid green curve is the best possible estimation value of
$\overline{H}_{\theta}$ using states without any type of correlations (section
III.2.1).

####  III.2.4 Relation between the Average QFI and Entanglement

In order to obtain some physical intuition concerning the interplay between
Entanglement and this estimation problem, we studied the relation between the
Average QFI, $\overline{H}_{\theta}$, and the Entanglement quantifier
mentioned in section II.1, explicitly the Logarithmic Negativity
$\mathcal{E}_{N}$ AdessoIP . We first note that our result is Energy
dependent, i.e., we were able to obtain results that depend explicitly on the
energy of mode A $n_{A}$. We focused our study in states with CM as in
equation 28. In Figure 9 we show our results for different values of $n_{A}$.
The overall value of $\overline{H}_{\theta}$ increase with $n_{A}$, an
intuitive result, since if one use a higher value of Energy, more one can
access the paramenter to be estimated.

In Figure 9, it is evident that as the system’s Entanglement, measured by
$\mathcal{E}_{N}$, increases, so does the value of $\overline{H}_{\theta}$. An
important result of our work is that the upper bound for Figures 9 is given by
pure two-mode squeezed states, and we were able to obtain an analytical
expression for this dependence as:

$\overline{H_{\theta}}=2+\frac{8n_{A}(1+n_{A})}{1+(2+4n_{A}-\tilde{\nu})\tilde{\nu}}=2+\frac{8n_{A}(1+n_{A})}{1+(2+4n_{A}-\mathrm{e}^{-\mathcal{E}_{\mathcal{N}}})\mathrm{e}^{-\mathcal{E}_{\mathcal{N}}}},$
(30)

where it is clear the dependence of $\overline{H}_{\theta}$ on the
Entanglement measure $\mathcal{E}_{N}$ and also on the Energy of mode A,
$n_{A}$.

Figure 9: (Color online) Average Quantum Fisher Information
($\overline{H}\theta$) as a function of the Logarithmic Negativity
($\mathcal{E}_{N}$). The dots represent Entangled two-mode states (a total of
$10^{5}$ states were randomly chosen for each case). We varied the mean photon
number ($n_{A}$) in each plot: (a) $n_{A}=3$; (b) $n_{A}=5$; (c) $n_{A}=10$;
(d) $n_{A}=100$. It is evident that as $n_{A}$ increases, the value of
$\overline{H}_{\theta}$ also increases. The black dashed curves are the upper
bound for each case, returned by pure squeezed two-mode states. The dotdashed
black curves are the lower bound for each energy, and were obtained
numerically. Details about the upper and lower bounds in the text.

To investigate the lower bound, numerical methods were necessary, as it is not
straightforward to determine which states yield the lower bounds. For each
dataset comprising $(\mathcal{E}_{N},\overline{H}_{\theta})$, we conducted a
numerical regression analysis to model the lower bound, aiming to obtain a
curve that approximates this bound. After conducting this analysis, we derived
the following expression for the lower bound:

$\overline{H}_{\theta}(\mathcal{E}_{N})=A_{1}\exp\left[B_{1}\mathcal{E}_{N}\right]+A_{2}.$
(31)

The specific values of $A_{1}$, $A_{2}$, and $B_{1}$ depend on each individual
case, corresponding to different energy levels $n_{A}$. In Table 1, we present
the numerical results for each $n_{A}$ displayed in Figure 9. Additionally, we
provide the Mean Square Error (MSE) and the standard deviation of MSE (which
we call in this work $\Delta_{MSE}$). It is noteworthy that the lower bound
for $n_{A}=10$ and $n_{A}=100$ are the same due to the characteristics of our
numerical approach.

Table 1: Numerical analysis of the lower bound of $\overline{H}_{\theta}$ in function of $\mathcal{E}_{N}$ $n_{A}$ | $A_{1}$ | $A_{2}$ | $B_{1}$ | $MSE$ | $\Delta_{MSE}$
---|---|---|---|---|---
$3$ | $0.169$ | $1.778$ | $2.187$ | $0.002$ | $0.486$
$5$ | $0.215$ | $1.695$ | $2.037$ | $0.007$ | $0.586$
$10$ | $0.193$ | $1.964$ | $2.089$ | $0.402$ | $0.963$
$100$ | $0.193$ | $1.964$ | $2.089$ | $0.402$ | $0.963$

It can be observed that the MSE increases significantly as the energy values
rise, while for low energy values, it remains close to zero. This indicates
that our results are quite accurate for lower energy levels when analyzing the
lower bound. Conversely, for higher values of $n_{A}$, both the MSE and
$\Delta_{MSE}$ increase, and we understand that our analysis is enough for the
context of this work. Finally we mention that it would be interest for the
community if the specific class of states that return the lower bound could be
obtained, in the same sense that pure two-mode squeezed states return the
upper bound.

##  IV Conclusion

In this work we exploit important classes of single- and two-mode Gaussian
states, for the specific problem of estimate with the higher precision the
squeezing parameter $\epsilon$ in one of the modes. After quite complete
sections reviewing Gaussian states, Estimation theory we detailed presented
the estimation strategy studied in this work, within the approach as
originally proposed by Luca  and followed closely here.

For single-mode states we showed that pure squeezed states are the best probes
for this estimation problem, approaching the so called Heisenberg limit.
Coherent states, been a “quasi-classical” class of states, achieve the linear
(in relation to the Energy) behavior of the Shot Noise limit (or Quantum
Standard limit). Still for single-mode states, we studied a relation between
the state’s Coherence and the Average QFI, showing that even among states with
similar levels of coherence, those with squeezing capabilities can outperform
coherent states in terms of estimation precision.

For two-mode Gaussian states, our results demonstrate the significance of
entanglement in squeezing estimation. Pure two-mode squeezed states have the
potential to approach the Heisenberg limit ($\overline{H}_{\theta}\sim
n^{2}$), while states exhibiting Discord-type correlations yield
$\overline{H}_{\theta}\sim n$. We have also explored other classes of state,
including separable states in their standard form, and provided analytical
results for upper and lower bounds in each category (in the analysis of
$\overline{H}_{\theta}$ as a function of $n_{A}$). Furthermore, we conducted
an investigation into the relationship between entanglement and the Average
QFI, revealing intriguing findings: (i) As energy levels increase, the Average
QFI also rises, enhancing the precision of the squeezing estimation problem.
(ii) Once again, pure two-mode squeezed states emerge as the optimal probes
for squeezing estimation, constituting the upper bound for states concerning
the interplay between the Average QFI and Logarithmic Negativity. We also
presented an analytical result for this upper bound. (iii) Numerical results
for the lower bound within the study of the Average QFI and Entanglement
(measured by $\mathcal{E}_{N}$) were presented.

We encourage the community to investigate whether there are classes of
Gaussian states that meet this lower bound, similar to how pure two-mode
squeezed states satisfy the upper bound.

## Acknowledgments

L.A.M.S. would like to express gratitude to INCT-IQ (Instituto Nacional de
Ciência e Tecnologia - Informação Quântica) for their financial support during
the VIII Paraty Quantum Information School and Workshop. Special thanks are
extended to Ana Mizher, Rodrigo Dias, and Guilherme Dinnebier for their
invaluable contributions through insightful discussions on data analysis.
Additionally, L.A.M.S. extends appreciation to Carlos Henrique S. Vieira and
Irismar G. da Paz for their helpful insights and discussions regarding
Coherence quantifiers. This work makes use of the QuGIT toolbox QuGIT .

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