# Quantum Fisher Information Perspective on Sensing in Anti-PT Symmetric
Systems

J. Wang  Department of Physics and Astronomy, Texas A&M University, College
Station, Texas 77843, USA  Institute for Quantum Science and Engineering and
Department of Biological and Agricultural Engineering, Texas A&M University,
College Station, Texas 77843, USA  D. Mukhopadhyay  Department of Physics and
Astronomy, Texas A&M University, College Station, Texas 77843, USA  Institute
for Quantum Science and Engineering and Department of Biological and
Agricultural Engineering, Texas A&M University, College Station, Texas 77843,
USA  G. S. Agarwal  Department of Physics and Astronomy, Texas A&M University,
College Station, Texas 77843, USA  Institute for Quantum Science and
Engineering and Department of Biological and Agricultural Engineering, Texas
A&M University, College Station, Texas 77843, USA  Department of Biological
and Agricultural Engineering, Texas A&M University, College Station, Texas
77843, USA

###### Abstract

The efficient sensing of weak environmental perturbations via special
degeneracies called exceptional points in non-Hermitian systems has gained
enormous traction in the last few decades. However, in contrast to the
extensive literature on parity-time (PT) symmetric systems, the exotic
hallmarks of anti-PT symmetric systems are only beginning to be realized now.
Very recently, a characteristic resonance of vanishing linewidth in anti-PT
symmetric systems was shown to exhibit tremendous sensitivity to intrinsic
nonlinearities. Given the primacy of sensing in non-Hermitian systems, in
general, and the immense topicality of anti-PT symmetry, we investigate the
statistical bound to the measurement sensitivity for any arbitrary
perturbation in a dissipatively coupled, anti-PT symmetric system. Using the
framework of quantum Fisher information and the long-time solution to the full
master equation, we analytically compute the Cramér-Rao bound for the system
properties like the detunings and the couplings. As an illustrative example of
this formulation, we inspect and reaffirm the role of a long-lived resonance
in dissipatively interacting systems for sensing applications.

##  I I. INTRODUCTION

The Hamiltonian of a physical system characterizes its energy spectrum and
time evolution, and is thus, of fundamental importance in quantum theory. In
nature, all systems are invariably dissipative as they interface with their
environment and relax into thermal equilibrium. As a consequence, an open-
system formulation in terms of the master equation becomes imperative. Such a
description leads to the system dynamics being described in terms of an
effective non-Hermitian Hamiltonian bender1 ; bender2 . Over the last few
years, certain symmetry properties of these non-Hermitian Hamiltonians have
facilitated enhanced sensing of weak perturbations. For instance, the sensing
potential of exceptional points (EPs) vis-à-vis linear perturbations in both
parity-time (PT) symmetric and anti-PT symmetric systems has been brought to
the fore wiersig1 ; wiersig2 ; hodaei ; demnchrist ; lyang ; miri ; zhang .
For example, in a two-mode system, the square-root singularity at an EP
engenders a heightened sensitivity in the spectral splitting to perturbative
influences. That is to say, the normal mode splitting scales as the square
root of the perturbation, which produces a divergent behavior in the first-
order derivative with respect to the perturbation parameter. Recently, in the
specific context of anti-PT symmetric systems, a protocol to efficiently
detect weak nonlinearities was also proposed nair . The scheme exploited a
real spectral singularity of the linear response function which strongly
inhibits the linewidth of a resonance, thereby drawing out a spectacular
response. It was shown that, near the singular point, the nonlinear response
$\mathscr{R}$ behaved as $\delta\mathscr{R}/\delta U\propto U^{-5/3}$, where
$U$ quantifies the strength of Kerr nonlinearity. Unlike PT symmetry, anti-PT
symmetry does not require any extrinsic gain, and is predisposed to simple
laboratory realization. A wide array of physical systems have been tailored to
exhibit anti-PT symmetry in a multitude of settings including, but not limited
to, rapidly moving atomic vapor peng , dressed atomic lattices wu1 ; wu2 ,
driven cold atoms wang ; chuang , laser-cooled atomic ensembles jiang ,
metamaterials ge , nonlinear optical systems anton , waveguides konotop ; yang
, electrical LRC resonators choi , diffusive systems li , coupled cavity-
magnon systems hu  and many more, revealing exotic physical phenomena.
Recently, Jingwei et. al. wen  demonstrated anti-PT symmetry in a quantum
circuit model with three qubits using nuclear spins.

In the context of sensing, it must be acknowledged that the developments
heretofore on the sensing potential near special singularities of the system,
such as the EP, have been quite remarkable. It is also to be noted that
empirical measurements suffer inherently from statistical imprecision, and
therefore, uncertainties in the measured response impose fundamental
restrictions to the estimation of system parameters. For example, the
indelible impact of quantum-limited intensity noise on EP-based sensing
protocols was acknowledged in a recent comment by Wiersig wiersig3 . In order
to address the inevitability of statistical errors, we bring in a quantum
statistical outlook to the problem of sensing, framed and contextualized in
anti-PT symmetric configurations. Employing the well-known framework of
quantum Fisher information (QFI) fisher , we formulate the sensitivity measure
in terms of the Cramér-Rao bound cramer ; rao , which sets down a theoretical
constraint to parameter estimation. The concept of Fisher information was
conceived as a tool to quantify the amount of information encoded in an
observable quantity about an unknown variable through statistical modeling.
With the statistical properties of a quantum system described by its density
matrix, the notion was subsequently generalized to the quantum formalism
helstrom ; holevo ; caves2 ; davi1 ; davi2 ; davi3 ; braun2 ; jordan ; chen ;
liu , and has been applied for diverse practical ends, such as for quantum
metrology in lossy open systems paris ; braun1 ; souza ; jiaxuan . In this
work, we report the exact precision bound to parameter estimation in anti-PT
symmetric systems and thereby, provide an overarching statistical framework
for the sensing of weak perturbative effects.

The paper is structured as follows. In section II, we briefly overview the
basics of anti-PT symmetry from a classical perspective. In section III, we
employ the quantum apparatus and show that anti-PT symmetric systems
engineered through vacuum-induced dissipative couplings relax into a coherent
state in the long-time limit. In section IV, we work out a general formula for
the QFI of a coherent state which provides a lower bound to the sensitivity of
an arbitrary perturbation variable.

##  II II. SENSING CAPABILITIES OF CLASSICAL ANTI-PT SYMMETRIC SYSTEMS

According to the coupled-mode theory in classical optics, the temporal
evolution of an open system of two harmonic modes can be described via a
$2\crossproduct 2$ non-Hermitian Hamiltonian matrix $\mathscr{H}$. In response
to any electromagnetic excitation, the dynamics of the system, in the rotating
frame of the drive, can be encoded as

$\begin{split}\begin{pmatrix}\dot{\alpha_{0}}\\\
\dot{\beta_{0}}\end{pmatrix}=-i\mathscr{H}\begin{pmatrix}\alpha_{0}\\\
\beta_{0}\end{pmatrix}+\mathcal{E}\begin{pmatrix}1\\\
0\end{pmatrix},\end{split}$ (1)

where $(\alpha_{0},\beta_{0})$ signifies the classical amplitudes of the two
modes, $\mathcal{E}$ is some generalized Rabi frequency, and $\omega_{d}$ is
the laser frequency. The generic $2\crossproduct 2$ matrix $\mathscr{H}$ lends
itself to two very interesting symmetries, namely PT-symmetry, which is
defined by the condition $[PT,H]=0$, and anti-PT symmetry, which subscribes to
$\\{PT,H\\}=0$. Throughout this manuscript, we tailor our analysis around the
constraint of anti-PT symmetry as we set forth some information theoretic
restrictions to sensing in systems exhibiting anti-PT symmetry. Such symmetry
can be reconciled with the parameter description
$\mathscr{H}_{11}=\Delta-i\gamma$, $\mathscr{H}_{22}=-\Delta-i\gamma$,
$\mathscr{H}_{12}=\mathscr{H}_{21}=-i\Gamma$, where $(\Delta,\gamma,\Gamma)$
are all real. The coherence-inducing off-diagonal elements of $\mathscr{H}$
characterize a purely dissipative form of interaction between the two modes.
The diagonal terms indicate equal damping rates but opposite detunings in the
two modes. The eigenvalues for this Hamiltonian are obtained to be
$-i\gamma\pm\sqrt{\Delta^{2}-\Gamma^{2}}$ for $\absolutevalue{\Delta}>\Gamma$
and $-i\gamma\pm i\sqrt{\Gamma^{2}-\Delta^{2}}$ (broken anti-PT) for
$\absolutevalue{\Delta}<\Gamma$. The points of transition $\Delta=\pm\Gamma$,
where the eigenvalues coalesce, are known as exceptional points, which have
been found useful in the context of sensing. In addition to that, such systems
possess a real singularity at $\Delta\rightarrow 0$,
$\gamma\rightarrow\Gamma$, which has immense sensing potential, allowing
efficient detection of both linear and nonlinear perturbations in the system.
The key measurement parameter, in this case, is the residual response of the
system characterized by the amplitudes $\alpha_{0}(\varepsilon)$ and
$\beta_{0}(\varepsilon)$, assumed to be functions of a perturbation
$\varepsilon$. Subject to the fulfillment of the stability criterion, viz.
$\gamma>\Gamma$, the system reduces to a steady state in the long-time limit,
yielding the amplitudes

$\displaystyle\alpha_{0}$
$\displaystyle=-i\frac{{\mathscr{H}_{22}}}{\det\mathscr{H}}\mathcal{E}=-\frac{\gamma-i\Delta}{\Gamma^{2}-\Delta^{2}-\gamma^{2}}\mathcal{E},$
$\displaystyle\beta_{0}$
$\displaystyle=i\frac{\mathscr{H}_{21}}{\det\mathscr{H}}\mathcal{E}=\frac{\Gamma}{\Gamma^{2}-\Delta^{2}-\gamma^{2}}\mathcal{E}.$
(2)

In realistic scenarios, however, there would be some discord with the
watertight conditions of anti-PT symmetry, no matter how small the error can
be made. Since the ensuing mode amplitudes are proportional to $\mathcal{E}$
and sensitive to the system parameters $(\Delta,\gamma,\Gamma)$, the
application of a probe field yields nonzero values of the derivatives
$\frac{\partial(\alpha_{0},\beta_{0})}{\partial(\Delta,\gamma,\Gamma)}$. This
gives us a way of sensing any small perturbations to these parameters. Let us
consider, for instance, the particular case when one of the frequency
detunings is zero but there is only a small mismatch in the magnitudes of the
two detunings, i.e., $\mathscr{H}_{11}=-i\gamma$ and
$\mathscr{H}_{22}=-s-i\gamma$, where $s$ is a small parameter. One could get
an estimate of this mismatch by measuring the complex amplitudes

$\displaystyle\alpha_{0}(s)$ $\displaystyle=-\frac{\gamma-
is}{\Gamma^{2}+is\gamma-\gamma^{2}}\mathcal{E},$ $\displaystyle\beta_{0}(s)$
$\displaystyle=\frac{\Gamma}{\Gamma^{2}+is\gamma-\gamma^{2}}\mathcal{E}.$ (3)

The pertinent sensitivities could, then, be obtained in terms of

$\displaystyle\frac{\partial\alpha_{0}(s)}{\partial s}$
$\displaystyle=\frac{i\Gamma^{2}}{(\Gamma^{2}+is\gamma-\gamma^{2})^{2}}\mathcal{E},$
$\displaystyle\frac{\partial\beta_{0}(s)}{\partial s}$
$\displaystyle=\frac{-i\Gamma\gamma}{(\Gamma^{2}+is\gamma-\gamma^{2})^{2}}\mathcal{E}.$
(4)

Clearly, the coherence-induced singularity in the limit
$\gamma\rightarrow\Gamma$ gets translated into an infinite theoretical
potential for sensing any arbitrary perturbation around this singularity.
Supposing that $\gamma=(1+\xi)\Gamma$ for $\xi\ll 1$, both the derivatives
would scale as $1/s^{2}$, if $\xi\ll s/\Gamma$. This would imply an augmented
sensing capability around this point. Alternatively, there could be some weak
dispersive coupling between $a$ and $b$, which modifies the off-diagonal terms
in $\mathscr{H}$ into $\mathscr{H}_{12}=\mathscr{H}_{21}=g-i\Gamma$, where
$g\ll\Gamma$. An example of this case from integrated photonics would be a
string of two dipolar emitters embedded onto a waveguide and separated by a
distance that is scale-wise similar to the resonant wavelength. Following the
same approach as above, we can compute the corresponding sensitivities as

$\displaystyle\frac{\partial\alpha_{0}(g)}{\partial g}$
$\displaystyle=2i\frac{(\Gamma+ig)(\gamma-i\Delta)}{[(\Gamma+ig)^{2}-\Delta^{2}-\gamma^{2}]^{2}}\mathcal{E},$
$\displaystyle\frac{\partial\beta_{0}(g)}{\partial g}$
$\displaystyle=-i\frac{(\Gamma+ig)^{2}+\Delta^{2}+\gamma^{2}}{[(\Gamma+ig)^{2}-\Delta^{2}-\gamma^{2}]^{2}}\mathcal{E}.$
(5)

Once again, for $\xi\ll g/\Gamma$, the derivatives scale as $1/g^{2}$ in the
limit $\Delta\rightarrow 0$. Both these observations are, however, not only
classical but also free from the nuance of statistical considerations. In the
subsequent sections, we delve into the quantum statistical framework for
treating anti-PT symmetric systems and examine the sensing scheme from the
perspective of the QFI.

##  III III. THE QUANTUM MODEL: STEADY-STATE RESPONSE

The sensitivity bounds, termed as the Cramér-Rao bounds, pertaining to
precision measurements are furnished in terms of the QFI of the system,
thereby necessitating a quantum formulation. In the quantum mechanical
picture, the dynamics of a non-interacting two-mode system is underpinned by
the Hamiltonian operator

$\begin{split}\mathcal{H}/\hbar=\omega_{a}a^{\dagger}a+\omega_{b}b^{\dagger}b,\end{split}$
(6)

where $\omega_{a}$ and $\omega_{b}$ denote the bare resonances of the
individual modes $a$ and $b$. If mode $a$ is now irradiated by a laser drive
of Rabi frequency $\mathcal{E}$, the interaction Hamiltonian can be reduced to
the form

$\displaystyle\mathcal{H}_{int}/\hbar=i(\mathcal{E}e^{-i\omega_{d}t}a^{\dagger}-\mathcal{E}^{*}e^{i\omega_{d}t}a).$
(7)

To stamp out the harmonic time variation in the interaction, it is convenient
to rotate variables as $(a,b)\rightarrow(a,b)e^{-i\omega_{d}t}$, leading to a
new Hamiltonian

$\displaystyle\mathcal{H}_{eff}/\hbar=\Delta_{a}a^{\dagger}a+\Delta_{b}b^{\dagger}b+i(\mathcal{E}a^{\dagger}-\mathcal{E}^{*}a).$
(8)

Now, the choice $\Delta_{a}=-\Delta_{b}=\Delta$ would entail a partial
fulfillment of the criteria for anti-PT symmetry. In addition to this, both
the modes could be made to interface with dissipative reservoirs, which can be
appropriately addressed in the master equation formalism. If both the modes
decay at identical rates, anti-PT symmetry would be consummated. We assume,
for simplicity, thermalized reservoirs at zero temperature instill vacuum
noise into the hybrid system. While the noise terms average out to zero, an
effective dissipative coupling is introduced between $a$ and $b$. For a system
with purely dissipative coupling, the master equation can be expressed as

$\frac{\differential\rho}{\differential
t}=-\frac{i}{\hbar}[\mathcal{H}_{eff},\rho]+\kappa\mathcal{L}(a)\rho+\kappa\mathcal{L}(b)\rho+2\Gamma\mathcal{L}(c)\rho,$
(9)

where $\kappa$ stands for the intrinsic damping rate of either of the modes
into its local heat bath, $\mathcal{L}$ is the Liouvillian operator defined by
$\mathcal{L}(\alpha)\rho=2\alpha\rho\alpha^{\dagger}-\alpha^{\dagger}\alpha\rho-\rho\alpha^{\dagger}\alpha$
for any annihilation operator $\alpha$, and $c=(1/{\sqrt{2}})(a+b)$ is the
jump operator corresponding to symmetrical couplings of the modes to a shared
reservoir. This equation holds under the approximation that the phase delay
due to light propagation from one mode to another is an integral multiple of
$2\pi$. We first bring out the correspondence between the quantum mechanical
mean values and the classical amplitudes in the steady state. Since
$\dot{\expectationvalue{a}}=\Tr(\dot{\rho}a)$, the preceding equation
naturally dictates the dynamics of the mean values, which, upon
simplification, reduces to a form commensurate with Eq. (1):

$\begin{split}\begin{pmatrix}\dot{\expectationvalue{a}}\\\
\dot{\expectationvalue{b}}\end{pmatrix}=-i\mathscr{H}\begin{pmatrix}\expectationvalue{a}\\\
\expectationvalue{b}\end{pmatrix}+\mathcal{E}\begin{pmatrix}1\\\
0\end{pmatrix},\end{split}$ (10)

where $\mathscr{H}=\begin{pmatrix}\Delta-i(\kappa+\Gamma)&-i\Gamma\\\
-i\Gamma&-\Delta-i(\kappa+\Gamma)\end{pmatrix}$. The identification
$\gamma=\kappa+\Gamma$ establishes perfect parity with the classical result.

Next, we derive the steady state which would accord not only the solutions to
the mean values but also the relevant quantum fluctuations. This would be
achieved by solving for the full density matrix of the system. Assuming that a
steady state exists, this can be worked out by letting $\dot{\rho}=0$. To that
end, we first argue that the excitation term $\mathcal{H}_{int}$ gets
reflected as a two-mode displacement operation, underscored by the
transformation $(a,b)\rightarrow
D(\alpha_{0},\beta_{0})(a,b)D^{\dagger}(\alpha_{0},\beta_{0})=(\mathcal{A},\mathcal{B)}=(a-\alpha_{0},b-\beta_{0})$,
where
$(\alpha_{0},\beta_{0})=(\expectationvalue{a}_{ss},\expectationvalue{b}_{ss})$
and
$D(\alpha,\beta)=\exp(\alpha_{0}a^{\dagger}-\alpha_{0}^{*}a)\exp(\beta_{0}b^{\dagger}-\beta_{0}^{*}b)$,
and $(\alpha_{0},\beta_{0})$ are given by

$\displaystyle\begin{pmatrix}\alpha_{0}\\\ \beta_{0}\\\
\end{pmatrix}=\begin{pmatrix}\expectationvalue{a}_{ss}\\\
\expectationvalue{b}_{ss}\\\
\end{pmatrix}=-i\mathscr{H}^{-1}\begin{pmatrix}\mathcal{E}\\\ 0\\\
\end{pmatrix}.$ (11)

This is because, such a transformation dispels the driving term in the master
equation, and casts Eq. (9) into the form

$\displaystyle\frac{\differential\rho}{\differential t}=$
$\displaystyle-\frac{i\Delta}{\hbar}[a^{\dagger}a-b^{\dagger}b,\rho]+\kappa\mathcal{L}(\mathcal{A})\rho+\kappa\mathcal{L}(\mathcal{B})\rho+2\Gamma\mathcal{L}(\mathcal{C})\rho$
$\displaystyle=$
$\displaystyle-(\gamma+i\Delta)[\mathcal{A}^{\dagger},\mathcal{A}\rho]-(\gamma-i\Delta)[\mathcal{B}^{\dagger},\mathcal{B}\rho]$
$\displaystyle-\Gamma[\mathcal{A}^{\dagger},\mathcal{B}\rho]-\Gamma[\mathcal{B}^{\dagger},\mathcal{A}\rho]+\textit{h.c.},$
(12)

with h.c. denoting the Hermitian conjugate, and
$\mathcal{C}=(1/{\sqrt{2}})(\mathcal{A}+\mathcal{B})$. Now, in view of the
uniqueness of the steady state, we obtain a solution embodied by the
conditions

$\displaystyle\mathcal{A}\rho=\rho\mathcal{A}^{\dagger}=0,$
$\displaystyle\mathcal{B}\rho=\rho\mathcal{B}^{\dagger}=0,$ (13)

which are conformable only with the bimodal vacuum
$\rho=\ket{0_{\mathcal{A}},0_{\mathcal{B}}}\bra{0_{\mathcal{A}},0_{\mathcal{B}}}$
in the displaced representation. Reverting to the original representation, we
obtain a coherent state

$\displaystyle\rho=\ket{\alpha_{0},\beta_{0}}\bra{\alpha_{0},\beta_{0}}$ (14)

as our solution to the anti-PT symmetric master equation. In view of Eq. (13),
it follows that a calculation of the mean values $\expectationvalue{a}$ and
$\expectationvalue{b}$ with respect to this coherent state coincides with the
classical result laid out in Eq. (2).

The above state can also be inferred from the reformulated master equation
expressed in terms of the corresponding phase-space distribution
$P(\alpha,\beta)$ for a two-mode system. Following the procedure spelled out
in gsabook , we deduce an equivalent master equation in the $P$-representation
as

$\displaystyle\frac{\partial P}{\partial t}$
$\displaystyle=\frac{\partial}{\partial\alpha}[u(\alpha,\beta)P]+\frac{\partial}{\partial\beta}[v(\alpha,\beta)P]+c.c.\,,$
(15)

where $u(\alpha,\beta)=(\gamma+i\Delta)\alpha+\Gamma\beta-\mathcal{E}$,
$v(\alpha,\beta)=\Gamma\alpha+(\gamma-i\Delta)\beta$, and $P(\alpha,\beta)$ is
defined as

$\rho=\int\int P(\alpha,\beta)\ket{\alpha,\beta}\bra{\alpha,\beta}d^{2}\alpha
d^{2}\beta\,,$ (16)

and $c.c.$ stands for the complex conjugate. In the steady state, as
$t\rightarrow\infty$, we require $\frac{\partial P}{\partial t}=0$. This
provides the quantum steady-state solution for the $P$ function to be

$P(\alpha,\beta)=\mathscr{N}\delta^{(2)}(u(\alpha,\beta))\delta^{(2)}(v(\alpha,\beta))\,,$
(17)

where $\delta(.)$ represents the Dirac delta function and $\mathscr{N}$ is
some normalization constant, since $P$ must satisfy $\int\int
P(\alpha,\beta)d^{2}\alpha d^{2}\beta=1$. Considering the linearity of the
functions $u$ and $v$, this is equivalent to

$P(\alpha,\beta)=\delta^{(2)}(\alpha-\alpha_{0})\delta^{(2)}(\beta-\beta_{0})\,,$
(18)

where
$\alpha_{0}=-\frac{(\gamma-i\Delta)}{\Gamma^{2}-\Delta^{2}-\gamma^{2}}\mathcal{E}$
and $\beta_{0}=\frac{\Gamma}{\Gamma^{2}-\Delta^{2}-\gamma^{2}}\mathcal{E}$ is
the solution to $u(\alpha_{0},\beta_{0})=v(\alpha_{0},\beta_{0})=0$.
Substituting this into Eq. (15), we find that the quantum steady state of this
driven anti-PT symmetric system must be a coherent state
$\ket{\alpha_{0},\beta_{0}}\bra{\alpha_{0},\beta_{0}}$. The coherent nature of
this state would remain valid even for small perturbations to any of the
system parameters, as long as the system remains purely dissipative. However,
the amplitudes $(\alpha_{0},\beta_{0})$ would get modified as per the
constraint of Eq. (12). In particular, if the two modes address each other
through a weak dispersive interaction $\hbar g(a^{\dagger}b+ab^{\dagger})$,
the solutions to the response can be found by assigning an imaginary offset to
$\Gamma$, i.e., by letting $\Gamma\rightarrow\Gamma+ig$.

##  IV IV. THE QUANTUM CRAMÉR-RAO BOUND FOR SENSITIVITY

As proved in section III, the steady state for an anti-PT symmetric quantum
system is a two-mode coherent state that can be written as
$\ket{\alpha_{0},\beta_{0}}\bra{\alpha_{0},\beta_{0}}$, where
$\alpha_{0}=-\frac{(\gamma-i\Delta)}{\Gamma^{2}-\Delta^{2}-\gamma^{2}}\mathcal{E}$
and $\beta_{0}=\frac{\Gamma}{\Gamma^{2}-\Delta^{2}-\gamma^{2}}\mathcal{E}$ are
the long-time responses in the two modes induced by the external field. Thus,
information about any arbitrary perturbations to the system parameters would
be encoded in the complex amplitudes $\alpha$ and $\beta$, and as such,
perturbative effects can be sensed via measurements of the response functions.
To put this into information-theoretic perspective, we inspect the Cramér-Rao
bound for the measurement uncertainty in the estimation of a perturbation
parameter. The problem can, in fact, be solved in a general way, and applied
to any unknown parameter $\varepsilon$, as long as the response functions
$\alpha_{0}(\varepsilon)$ and $\beta_{0}(\varepsilon)$ are exactly known.

To start off, we briefly recapitulate the basic concepts of QFI. In the
estimation of an unknown parameter $\varepsilon$, the Cramér-Rao bound reads
$\delta\varepsilon\geq\sqrt{F_{Q}^{-1}(\varepsilon)}$, where
$\delta\varepsilon$ symbolizes the measurement uncertainty or sensitivity of
$\varepsilon$, and $F_{Q}(\varepsilon)=\Tr[\rho L^{2}]$ is the QFI for
$\varepsilon$, pertaining to a system with density matrix $\rho$. The
symmetric logarithmic derivative (SLD) matrix $L$ is calculated from the
equation $\frac{\partial\rho}{\partial\varepsilon}=\frac{1}{2}(L\rho+\rho L)$.
Since we have proved that the anti-PT symmetric system goes over into a
coherent state in the long-time limit, we work out the algebra for the
appropriate steady state. The corresponding density matrix can be expanded out
in the basis of Fock states as

$\rho=e^{-|\alpha_{0}|^{2}}e^{-|\beta_{0}|^{2}}\sum_{mnpq}\frac{\alpha_{0}^{m}\beta_{0}^{n}\alpha_{0}^{*p}\beta_{0}^{*q}}{\sqrt{m!n!p!q!}}\ket{m,n}\bra{p,q}\,.$
(19)

Since each of the parameters $(\alpha_{0}$, $\beta_{0})$ is a function of
$\varepsilon$, we can solve directly for the derivative
$\frac{\partial\rho_{{mn,pq}}}{\partial\varepsilon}$ in terms of the
$\frac{\partial\alpha_{0}}{\partial\varepsilon}$ and
$\frac{\partial\beta_{0}}{\partial\varepsilon}$. Note that $\alpha_{0}$,
$\beta_{0}$ are complex numbers. Therefore, expressing them as
$\alpha_{0}=\absolutevalue{\alpha_{0}}e^{i\theta_{1}}$,
$\beta_{0}=\absolutevalue{\beta_{0}}e^{i\theta_{2}}$, we find the relation

$\displaystyle\frac{\partial\rho_{{mn,pq}}}{\partial\varepsilon}$
$\displaystyle=\bigg{[}m\bigg{(}\frac{\partial
ln|\alpha_{0}|}{\partial\varepsilon}+i\frac{\partial\theta_{1}}{\partial\varepsilon}\bigg{)}+n\bigg{(}\frac{\partial
ln|\beta_{0}|}{\partial\varepsilon}+i\frac{\partial\theta_{2}}{\partial\varepsilon}\bigg{)}$
$\displaystyle+p\bigg{(}\frac{\partial
ln|\alpha_{0}|}{\partial\varepsilon}-i\frac{\partial\theta_{1}}{\partial\varepsilon}\bigg{)}+q\bigg{(}\frac{\partial
ln|\beta_{0}|}{\partial\varepsilon}-i\frac{\partial\theta_{2}}{\partial\varepsilon}\bigg{)}$
$\displaystyle-2|\alpha_{0}|\frac{\partial|\alpha_{0}|}{\partial\varepsilon}-2|\beta_{0}|\frac{\partial|\beta_{0}|}{\partial\varepsilon}\bigg{]}\rho_{mn,pq}\,,$
(20)

which can be analyzed by resolving them into three parts, defined below:

$\displaystyle-2\bigg{(}|\alpha_{0}|\frac{\partial|\alpha_{0}|}{\partial\varepsilon}+|\beta_{0}|\frac{\partial|\beta_{0}|}{\partial\varepsilon}\bigg{)}\rho_{mn,pq}=\frac{1}{2}(L_{1}\rho+\rho
L_{1})_{mn,pq},$ $\displaystyle\bigg{[}(m+p)\frac{\partial
ln|\alpha_{0}|}{\partial\varepsilon}+(n+q)\frac{\partial
ln|\beta_{0}|}{\partial\varepsilon}\bigg{]}\rho_{mn,pq}=\frac{1}{2}(L_{2}\rho+\rho
L_{2})_{mn,pq},$ $\displaystyle
i\bigg{[}(m-p)\frac{\partial\theta_{1}}{\partial\varepsilon}+(n-q)\frac{\partial\theta_{2}}{\partial\varepsilon}\bigg{]}\rho_{mn,pq}=\frac{1}{2}(L_{3}\rho+\rho
L_{3})_{mn,pq}.$ (21)

Following the method discussed in jxn2 , we solve the equations for the
individual components separately. In other words, the corresponding SLD gets
split as

$L=L_{1}+L_{2}+L_{3}\,,$ (22)

where

$\displaystyle L_{1}$
$\displaystyle=-2|\alpha_{0}|\frac{\partial|\alpha_{0}|}{\partial\varepsilon}-2|\beta_{0}|\frac{\partial|\beta_{0}|}{\partial\varepsilon},$
$\displaystyle L_{2}$ $\displaystyle=2\frac{\partial
ln|\alpha_{0}|}{\partial\varepsilon}(a^{+}a)+2\frac{\partial
ln|\beta_{0}|}{\partial\varepsilon}(b^{+}b),$ $\displaystyle L_{3}$
$\displaystyle=2i\frac{\partial\theta_{1}}{\partial\varepsilon}[a^{+}a,\rho]+2i\frac{\partial\theta_{2}}{\partial\varepsilon}[b^{+}b,\rho].$
(23)

This precise form of $L$ pins down the QFI and the corresponding sensitivity
limit. Invoking the definition $F_{Q}(\varepsilon)=\Tr[\rho L^{2}]$ we
conclude that

$F_{Q}(\varepsilon)=4\absolutevalue{\frac{\partial\alpha_{0}}{\partial\varepsilon}}^{2}+4\absolutevalue{\frac{\partial\beta_{0}}{\partial\varepsilon}}^{2}\,.$
(24)

The above expression is absolutely general, regardless of the specifics of the
response function $(\alpha_{0}(\varepsilon),\beta_{0}(\varepsilon))$. Thus,
the QFI, as well as the sensitivity for the system parameters, both depend on
the derivatives
$\absolutevalue{\frac{\partial\alpha_{0}}{\partial\varepsilon}}$ and
$\absolutevalue{\frac{\partial\beta_{0}}{\partial\varepsilon}}$. This result
is specific to an anti-PT symmetric mode hybridization, when the interposing
reservoir has relaxed into the vacuum state at zero temperature. Apropos of
our problem, where the coherent state
$\ket{\alpha_{0},\beta_{0}}\bra{\alpha_{0},\beta_{0}}$ encapsulates the
statistical properties of the measured quantities, we can model the detuning
mismatch $s$ or the weak coupling $g$ as a perturbation. Figs 1 (a) and (b)
depict the sensitivity bounds for these parameters for a set of varying decay
rates. For progressively smaller values of the intrinsic damping rate, the
perturbations can be sensed with incrementally larger efficacy near the
singular point where both the detunings approach zero.

Figure 1: The Cramér-Rao bounds for measurement uncertainties as
$\gamma\rightarrow\Gamma$, scaled in units of $\Gamma$: (a) $\delta s$ for the
sensing of detuning mismatch $s$ for different values of $\xi$; (b) $\delta g$
for the sensing of the dispersive coupling strength $g$, at
$\Delta=0.1\Gamma$. The quantity $\xi$, defined as
$\xi=(\gamma-\Gamma)/\Gamma$, is a dimensionless measure of the rate of
extraneous damping.

##  V VIII. CONCLUSIONS

In summary, we have offered an information-theoretic insight into the subject
of sensing in anti-PT symmetric systems where the coupling between the
participating modes is produced by a common vacuum. We invoked the Fisher
information theory as a statistical tool to circumscribe the theoretical
precision of a single-parameter measurement in terms of the Cramér-Rao bound,
which serves as a well-rounded metric for the sensitivity. We argued that
anti-PT symmetric systems decay into a coherent state in the long-time limit,
which renders the QFI calculation tractable. Then, by treating the steady-
state response functions as sensing signals, we derived an exact statistical
expression for the sensitivity of the perturbation variable. To exemplify the
role of the QFI, we applied this approach to the sensing of a detuning
mismatch as well as that of weak coherent coupling between the two modes
around a real singularity of the hybrid system. The vanishingly small
sensitivity bounds enabled by the emergence of a long-lived resonance
reinforced the impact of a strong vacuum induced coherence on precision
measurements.

##  VI ACKNOWLEDGMENTS

We thank the support of Air Force Office of Scientific Research (Award No
FA-9550-20-1-0366) and the Robert A Welch Foundation (A-1943-20210327).

## References

  * (1) C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry”, Phys. Rev. Lett. 80, 5243 (1998).
  * (2) C. M. Bender, “Making sense of non-Hermitian Hamiltonians”, Rep. Prog. Phys. 70, 947 (2007).
  * (3) J. Wiersig, “Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection”, Phys. Rev. Lett. 112, 203901 (2014).
  * (4) W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity”, Nature 548, 192 (2017).
  * (5) H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points”, Nature 548, 187 (2017).
  * (6) R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S.Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nature Physics, 14(1), 11-19 (2018).
  * (7) Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nature materials 18(8), 783-798 (2019).
  * (8) M.-A. Miri and A. Alu, “Exceptional points in optics and photonics”, Science 363 (2019).
  * (9) H. Zhang, R. Huang, S.-D. Zhang, Y. Li, C.-W. Qiu, F. Nori, and H. Jing, “Breaking anti-PT symmetry by Spinning a Resonator”, Nano Letters 20, 7594 (2020).
  * (10) J. M. P. Nair, D. Mukhopadhyay, and G. S. Agarwal, “Enhanced Sensing of Weak Anharmonicities through Coherences in Dissipatively Coupled Anti-PT symmetric Systems”, Phys. Rev. Lett 126, 180401 (2021).
  * (11) P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms”, Nature Physics, 12, 1139-1145 (2016).
  * (12) J. H. Wu, M. Artoni, and G.C. La Rocca, “Non-Hermitian Degeneracies and Unidirectional Reflectionless Atomic Lattices”, Phys. Rev. Lett. 113, 123004 (20014).
  * (13) J. H. Wu, M. Artoni, and G.C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain”, Phys. Rev. A 91, 033811 (2015).
  * (14) X. Wang and J. -H. Wu, “Optical PT-symmetry and PT-antisymmetry in coherently driven atomic lattices”, Opt. Express, 24, 4289-4298 (2016).
  * (15) Y. -L. Chuang, Ziauddin, and R. K. Lee, “Realization of simultaneously parity-time-symmetric and parity-time-antisymmetric susceptibilities along the longitudinal direction in atomic systems with all optical controls”, Opt. Express 26, 21969-21978 (2018).
  * (16) Y. Jiang, Y. Mei, Y. Zhuo, Y. Zhai, J. Li, J. Wen, and S. Du, “Anti-Parity-Time Symmetric Optical Four-Wave Mixing in Cold Atoms”, Phys. Rev. Lett. 123, 193604 (2019).
  * (17) L. Ge, and H. E. Türeci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials”, Phys. Rev. A 88, 053810 (2013).
  * (18) D. A. Antonosyan, A. S. Solntsev, and A. A. Sukhorukov, “Parity-time anti-symmetric parametric amplifier” Opt. Lett. 40, 4575-4578 (2015).
  * (19) V. V. Konotop, and D. A. Zezyulin, “Odd-Time Reversal PT Symmetry Induced by an Anti-PT-Symmetric Medium”, Phys. Rev. Lett. 120, 123902 (2018).
  * (20) F. Yang,, Y. -C. Liu, and L. You, “Anti-$\mathcal{PT}$ symmetry in dissipatively coupled optical systems”, Phys. Rev. A 96, 053845 (2017).
  * (21) Y. Choi, C. Hahn, J. W. Yoon, “Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators”, Nat. Commun 9, 2182 (2018).
  * (22) Y. Li,Y. -G. Peng, L. Han, M.-A. Miri, W. Li, M. Xiao, X. -F. Zhu, J. Zhao A. Alu, S. Fan and C.-W. Qiu, “Anti-parity-time symmetry in diffusive systems”, Science 364 6436, 1261-1265, (2021).
  * (23) Y. Yang, Y. -P. Wang, J. W. Rao, Y. S. Gui, B. M. Yao, W. Lu, and C. -M. Hu, “Unconventional Singularity in Anti-Parity-Time Symmetric Cavity Magnonics”, Phys. Rev. Lett. 125 147202 (2020).
  * (24) J. W. Wen, G. Qin, and C. Zheng, S. Wei, X. Kong, T. Xin, and G. Long, “Observation of information flow in the anti-PT-symmetric system with nuclear spins”, Npj Quantum Information 6, 28 (2020).
  * (25) J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing”, Nat. Commun 11, 2454 (2020).
  * (26) R. A. Fisher, “On the Dominance Ratio,” Proc. R. Soc. Edinburgh 42, 321-341 (1922).
  * (27) H. Cramér, “Mathematical Methods of Statistics,” (Princeton University Press, 1946, pp. 500).
  * (28) C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters”, Bulletin of the Calcutta Mathematical Society, 37, 81-89, (1945)
  * (29) C. W. Helstrom, “Quantum Detection and Estimation Theory,” (Academic Press, 1976, Chap. VIII.4).
  * (30) A. S. Holevo, “Probabilistic and Statistical Aspects of Quantum Theory,” (North-Holland, 1982).
  * (31) S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439-3443 (1994).
  * (32) B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406-411 (2011).
  * (33) C. L. Latune, B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “Quantum limit for the measurement of a classical force coupled to a noisy quantum-mechanical oscillator,” Phys. Rev. A 88, 042112 (2013).
  * (34) B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum Metrological Limits via a Variational Approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
  * (35) L. J. Fiderer, J. M. E. Fraisse, and D. Braun, “Maximal Quantum Fisher Information for Mixed States,” Phys. Rev. Lett. 123, 250502 (2019).
  * (36) A. N. Jordan, J. M.-Rincon and J. C. Howell, “Technical Advantages for Weak-Value Amplification: When Less Is More,” Phys. Rev. X 4, 011031 (2014).
  * (37) Y. Chen, H. Yuan, “Maximal quantum Fisher information matrix,” New J. Phys. 19(6), 063023 (2017).
  * (38) J. Liu, H. Yuan, X. M. Lu, and X. Wang, “Quantum Fisher information matrix and multiparameter estimation,” J. Phys. A: Math. Theor. 53(2), 023001 (2019).
  * (39) A. Monras and M. G. Paris, “Optimal Quantum Estimation of Loss in Bosonic Channels,” Phys. Rev. Lett. 98, 160401 (2007).
  * (40) P. Binder, and D. Braun, “Quantum parameter estimation of the frequency and damping of a harmonic oscillator,” Phys. Rev. A 102(1), 012223 (2020).
  * (41) G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, and L. A. M. Souza, “Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states,” Phys. Rev. A 79, 040305 (2009).
  * (42) J. Wang, L. Davidovich, and G. S. Agarwal, “Quantum sensing of open systems: Estimation of damping constants and temperature,” Phys. Rev. Research 2(3), 033389 (2020).
  * (43) G. S. Agarwal, “Quantum Optics” (Cambridge University Press, 2013, Chap. 10.2).
  * (44) J. Wang, and G. S. Agarwal, “Quantum Fisher Information Bounds on Precision Limits of Circular Dichroism,” arXiv:2108.13970 (2021).