# Cramér-Rao bound and quantum parameter estimation with non-Hermitian systems

Jianning Li  Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China  Haodi Liu  Center for Quantum
Sciences and School of Physics, Northeast Normal University, Changchun 130024,
China  Zhihai Wang  Center for Quantum Sciences and School of Physics,
Northeast Normal University, Changchun 130024, China  Xuexi Yi
<EMAIL_ADDRESS>Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China  Center for Advanced Optoelectronic
Functional Materials Research, and Key Laboratory for UV-Emitting Materials
and Technology of Ministry of Education, Northeast Normal University,
Changchun 130024, China

###### Abstract

The quantum Fisher information constrains the achievable precision in
parameter estimation via the quantum Cramér-Rao bound, which has attracted
much attention in Hermitian systems since the 60s of the last century.
However, less attention has been paid to non-Hermitian systems. In this
Letter, working with different logarithmic operators, we derive two previously
unknown expressions for quantum Fisher information, and two Cramér-Rao bounds
lower than the well-known one are found for non-Hermitian systems. These lower
bounds are due to the merit of non-Hermitian observable and it can be
understood as a result of extended regimes of optimization. Two experimentally
feasible examples are presented to illustrate the theory, saturation of these
bounds and estimation precisions beyond the Heisenberg limit are predicted and
discussed. A setup to measure non-Hermitian observable is also proposed.

Introduction.—Quantum parameter estimation helstrom67 ; helstrom76 ; holevo82
; degen17 ; pezze18  aims at measuring the value of a continuous parameter
$\theta$ encoded in the state $\rho(\theta)$ of a quantum system. It gives
better precision than the same measurement performed in a classical framework
and plays a crucial role in the advancement of physics. The estimation process
generally consists of two steps. In the first step, the state $\rho({\theta})$
is prepared and measured. A simple way to prepare the state $\rho(\theta)$ is
to evolve a reference initial state $\rho_{0}$ under a signal Hamiltonian
$H_{\theta}$ that encodes the parameter $\theta$,
$\rho_{t}(\theta)=e^{-iH_{\theta}t}\rho_{0}e^{iH_{\theta}t}.$ In the second
step, the estimation of $\theta$ is obtained by data processing the
measurement outcomes, aiming at the smallest uncertainty $\Delta\theta$ given
finite resources such as time and number of particles. The uncertainty is
bounded by the Cramér-Rao bound (CRB), which expresses a lower limit on the
variance of unbiased estimations, stating that the variance of any such
estimation is at least as high as the inverse of the Fisher information
fisher25 ; matson06 .

The quantum Cramér-Rao bound works in general with Hermitian quantum systems
frieden90 ; braunstein94 ; liu20  in order to meet the requirement of quantum
mechanics. However, recent studies found that non-Hermitian systems with
unbroken PT symmetries also possess a real spectrum bender98 . In fact, non-
Hermiticity is ubiquitous in the quantum world bender07 ; guo09 , including
systems with gain and/or loss konotop16m ; feng17 ; ganainy19 ; miri19 ;
ozdemir19 , many-body localization and dynamical stability of non-Hermitian
systems hamazaki19 , as well as sensors designed with non-Hermitian systems
berry04 ; heiss12 .

These facts give rise to a question that in non-Hermitian quantum systems
moiseyev11  what is the Cramér-Rao bound? This question was answered about 5
decades ago by Yuen and Lax in Ref. yuen73 , where the authors proposed an
idea to estimate a complex parameter by measuring a non-Hermitian observable.
Unfortunately, this theory was connected the complex parameter with the
estimator by assuming that the average of the estimator is exactly the
estimate of the parameter. This assumption would lead to a flaw that the error
propagation function is unity, resulting in a less regime of non-Hermitian
observables for optimization and experiments. Consider that the study of non-
Hermitian system and their unique properties have attracted fast growing
interest in the last two decades, revisiting the quantum Cramér bound and its
consequent estimation theory with current technologies is highly desired for
non-Hermitian systems. We should address that there are estimation protocols
(or sensors) based on non-Hermitian system recently wiersig14 ; liu16 ; chen17
; hodaei17 ; djorwe19 ; mao20 , but all analyses so far are based on either
the Fisher information resulting from the Hermitian quantum Cramér-Rao bound,
or the properties of the exceptional points.

In this Letter, we first develop an uncertainty relation for non-Hermitian
operators, then we derive two previously unknown expressions for quantum
Fisher information with different logarithmic derivatives. Two quantum Cramér-
Rao bounds for non-Hermitian systems are defined. We found that the Fisher
information is significantly increased due to the use of non-Hermitian
operators, in particular for systems in mixed states. This is in contrary to
the results of Fisher information with Hermitian symmetric logarithmic
derivatives. Saturation of the two bounds is analyzed and the optimal
measurement to attain the bounds is derived. We elucidate the feature of non-
Hermitian quantum Fisher information with GHZ states of trapped ions pezze18 ;
leibried05  and the phase estimation setup with Mach-Zehnder interferometer.
Comparison with the situation of non-Hermitian signal Hamiltonian is also
presented and discussed.

Non-Hermitian uncertainty relation and Quantum Cramér bound.— Higher
estimation precision demands more resources. The trade-off between the
precision and the resources required is determined by the uncertainty
principle, which constrains to what extent complementary variables maintain
their averaged values and leads to the Heisenberg limit caves81 ; chin12 ;
jarzyna15 ; giovannetti11 ; luis17 ; bai19  in quantum parameter estimation.
One might wonder whether this is the case and how the uncertainty relation
changes in non-Hermitian quantum mechanics moiseyev11 .

To explore the possible change of the uncertainty relation and introduce the
variance for non-Hermitian systems, we introduce two operators $A$ and $B$,
which are linear but not necessarily Hermitian. Defining $\Delta A=A-\langle
A\rangle$, $\Delta B=B-\langle B\rangle$ and $O=\chi\Delta A+i\Delta B$ with
$\chi$ a real parameter, $i^{2}=-1$ and $\langle Z\rangle$ being the
expectation value of operator $Z$, we have $\langle O^{\dagger}O\rangle\geq
0.$ Simple algebra yields supp , $\langle\Delta A^{\dagger}\Delta
A\rangle\langle\Delta B^{\dagger}\Delta B\rangle\geq\frac{1}{4}|\langle
C\rangle|^{2},$ where $C$ was defined by $C=i(A^{\dagger}B-B^{\dagger}A)\equiv
i[A,B]_{a}.$ Operator $C$ is a Hermitian operator regardless of $A$ and $B$
being Hermitian or not. $[X,Y]_{a}$ defines an abnormal commutation for
operator $X$ and $Y$, which can be applied to determine whether a time-
independent operator is a constant of motion for systems governed by non-
Hermitian Hamiltonian rivero20 .

We define $\sigma_{A}^{2}\equiv\langle\Delta A^{\dagger}\Delta A\rangle$ as
the variance for non-Hermitian operator $A$, which reduces to the traditional
variance when $A$ is Hermitian. This uncertainty relation as well as that we
will present in the following also hold for unitary operators $A$ and $B$
bong18 ; yu19 . In non-Hermitian quantum systems, the expectation value of
dynamical variable $A$ might take complex values, $\langle A\rangle=|\langle
A\rangle|e^{i\alpha}$. The absolute value $|\langle A\rangle|$ and its phase
$\alpha$ are both measurable quantities. For example, in scattering
experiments the peaks in the cross sections are obtained when the projectiles
have energy which is equal to the absolute value of the energy, not the real
part of the energy moiseyev11 . This complex expectation of non-Hermitian
operator is equal to the weak value of the positive-semidefinite part of the
operator multiplied by a known complex number pati15 ; nirala19 ; sahoo20 .

A stronger uncertainty relations for non-Hermitian operators $A$ and $B$
follows from the Schwarz inequality, $\langle F|F\rangle\langle
G|G\rangle\geq|\langle F|G\rangle|^{2}$ with $|F\rangle=\Delta A|\Psi\rangle$
and $|G\rangle=\Delta B|\Psi\rangle$ and $|\Psi\rangle$ being an arbitrary
state of system,

$\displaystyle\langle\Delta A^{\dagger}\Delta A\rangle\langle\Delta
B^{\dagger}\Delta B\rangle\geq|\langle A^{\dagger}B\rangle-\langle
A^{\dagger}\rangle\langle B\rangle|^{2}.$ (1)

This is a slightly stronger inequality for the variance of operators $A$ and
$B$, which together with error propagation supp

$(\Delta\theta)^{2}=\frac{\langle\Delta A^{\dagger}\Delta
A\rangle}{\partial\langle
A^{\dagger}\rangle/\partial\theta\cdot\partial\langle
A\rangle/\partial\theta}$ (2)

leads to a quantum Cramér-Rao bound for non-Hermitian system,

$(\Delta\theta)^{2}\geq\frac{1}{F_{nH}^{(1)}},$ (3)

where $F_{nH}^{(1)}=\text{Tr}(\rho L^{(1)\dagger}L^{(1)})$ is defined as non-
Hermitian quantum Fisher information with helstrom73ijtp ; supp

$\frac{\partial\rho}{\partial\theta}=\frac{1}{2}(L^{(1)}\rho+\rho
L^{(1)\dagger}),$ (4)

where $\theta$ is the real parameter to be estimated based on density matrix
$\rho(\theta)$ and Hermitian operator $A$. The slightly stronger uncertainty
relation Eq.(1) reduces to the Robertson-Schrödinger uncertainty relation when
both $A$ and $B$ are Hermitian and returns to the unitary uncertainty relation
bong18 ; yu19  with both $A$ and $B$ being unitary. Eq.(S15) was derived with
an assumption that $A^{\dagger}=A$, but there is no requirement for $L$. This
indicates that Eq.(3) holds for both Hermitian and non-Hermitian $L$. In fact,
when $L$ is Hermitian, the result reduces to the widely used quantum Fisher
information (denoted by $F_{H}$ hereafter) in the literature, while non-
Hermitian $L$ would lead to an enhanced precision for parameter estimations as
shown below.

For non-Hermitian operator $A$, we obtain the other quantum Cramér-Rao bound
leading to non-Hermitian quantum information $F_{nH}^{(2)}=\text{Tr}(\rho
L^{(2)\dagger}L^{(2)}),$ which has the same expression as in Eq.(3) but with
$L^{(2)}$ instead of $L^{(1)}$ helstrom73ijtp ; supp ,

$\frac{\partial\rho}{\partial\theta}=L^{(2)}\rho=\rho L^{(2)\dagger}.$ (5)

There are two fundamental differences between Hermitian and non-Hermitian
system in the quantum Cramér-Rao bounds defined by $F_{nH}^{(1)}$ and
$F_{nH}^{(2)}$, which deserve to be outlined. Firstly, in Fisher information
$F_{nH}^{(1)}$, the observable $A$ is Hermitian but the so-call symmetric
logarithmic derivative $L$ might be not. While both $A$ and $L$ in
$F_{nH}^{(2)}$ are not Hermitian. The state $\rho(\theta)$ that encodes the
parameter is Hermitian, however the signal Hamiltonian $H_{\theta}$ might not
be Hermitian, i.e., $H_{\theta}\neq H_{\theta}^{\dagger}$. Secondly, in non-
Hermitian system, the Fisher information is given by $F_{nH}=\text{Tr}(\rho
L^{(y)\dagger}L^{(y)}),\,y=1,2$ with
$\frac{\partial\rho}{\partial\theta}=L^{(2)}\rho=\rho L^{(2)\dagger}$ and
$\frac{\partial\rho}{\partial\theta}=\frac{1}{2}(L^{(1)}\rho+\rho
L^{(1)\dagger})$ instead of
$\frac{\partial\rho}{\partial\theta}=\frac{1}{2}(L\rho+\rho L)$ in Hermitian
system. As we will show later, $F_{nH}^{(1)}$ recovers the well-known
expression of quantum Fisher information $F_{H}$ for Hermitian system while
$F_{nH}^{(2)}$ can not. Whereas the Cramér-Rao bound defined by $F_{nH}^{(2)}$
can be saturated but that by $F_{nH}^{(1)}$ could not supp .

Fisher information of non-Hermitian system.— One of the main quests in quantum
parameter estimation is to find out the highest achievable precision with
given resources and design schemes that attain that precision. In general,
looking for the optimal resources and schemes is difficult since one needs to
optimize over the input state that encodes the parameter, the measurement that
is performed at the output, and the estimator that assigns a parameter value
to a given measurement outcome. One of the popular ways to obtain useful
bounds in quantum parameter estimations, without the need for cumbersome
optimization, is to use the quantum Fisher information. In the following, we
will give an explicit expression for the non-Hermitian quantum Fisher
information $F_{nH}^{(1)}$ and $F_{nH}^{(2)}$ in terms of the eigenstates and
eigenvalues of the encoding density matrix $\rho(\theta)$.

Consider a $N$-dimensional quantum system. The state of the system
$\rho(\theta)$ is parameterized by the parameter $\theta$ under estimation.
Without loss of generality, we assume the density matrix may not be of full
rank and the spectral decomposition of the density operator $\rho(\theta)$ is,

$\rho(\theta)=\sum_{i}^{M}p_{i}(\theta)|\phi_{i}(\theta)\rangle\langle\phi_{i}(\theta)|,\,M\leq
N.$ (6)

Here $p_{i}(\theta),i=1,2,...,M$ are required to be positive for any value of
$\theta$ and $\sum_{i}^{M}p_{i}(\theta)=1$, since $p_{i}(\theta)$ is the
probability of the system in state $|\phi_{i}(\theta)\rangle$. The following
normalization conditions
$\langle\phi_{i}(\theta)|\phi_{j}(\theta)\rangle=\delta_{ij}$ for all $i$ and
$j$ are imposed. From the definition of Fisher information, we can find that,

$F_{nH}^{(y)}=\sum_{i=1}^{M}\sum_{j=1}^{N}p_{i}(L^{(y)\dagger})_{ij}L^{(y)}_{ji},\,y=1,2.$

Noticing the derivation for the two Fisher information $F_{nH}^{(1)}$ and
$F_{nH}^{(2)}$ is different, we discuss it in the following separately.

We start our derivation of $F_{nH}^{(1)}$ by considering a special case that
$L^{(1)\dagger}=e^{i\beta}L^{(1)}$ with $\beta$ a real parameter. This
assumption together with Eq.(S15) leads to,

$L^{(1)}_{ij}=\langle\phi_{i}|L^{(1)}|\phi_{i}\rangle=\frac{2(\partial_{\theta}\rho)_{ij}}{p_{j}+p_{i}e^{i\beta}}$
(7)

and $L^{(1)\dagger}_{ij}=e^{i\beta}L^{(1)}_{ij}.$ A notation
$(\partial_{\theta}\rho)_{ij}=\left(\frac{\partial\rho(\theta)}{\partial\theta}\right)_{ij}$
was used. Clearly, $L^{(1)}\neq L^{(1)\dagger}$ as we expected. It is worth
noticing that $L^{(1)}_{ij}$ is in principle supported by the full Hilbert
space, not only that spanned by the eigenstates of the density matrix. Thus
the value of $L^{(1)}_{ij}$ for $i,j>N$ can not be established by the above
equations. However, these terms play an important role in the quantum Fisher
information. This problem can be solved by the use of completeness relation,
$\sum_{i=1}^{N}|\phi_{i}\rangle\langle\phi_{i}|=1$, i.e.,
$\sum_{i=M+1}^{N}|\phi_{i}\rangle\langle\phi_{i}|=1-\sum_{i=1}^{M}|\phi_{i}\rangle\langle\phi_{i}|$.
We finally arrive at supp ,

$\displaystyle
F_{nH}^{(1)}=\sum_{i=1}^{M}\frac{2(\partial_{\theta}p_{i})^{2}}{p_{i}(1+\cos\beta)}+\sum_{i=1}^{M}4p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle+\sum_{i=1}^{M}\sum_{j=1}^{M}\left(\frac{4p_{i}(p_{i}-p_{j})^{2}}{p_{i}^{2}+p_{j}^{2}+2p_{i}p_{j}\cos\beta}-4p_{i}\right)\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle,$
(8)

where
$|\partial_{\theta}\phi\rangle\equiv\frac{\partial|\phi\rangle}{\partial\theta}.$
This is one of the main results of this Letter. For pure state, $F_{nH}^{(1)}$
reduce to
$F_{nH}^{(1)}=F_{H}=4\langle\partial_{\theta}\phi|\partial_{\theta}\phi\rangle-4\langle\partial_{\theta}\phi|\phi\rangle\langle\phi|\partial_{\theta}\phi\rangle.$
This is the widely used quantum fisher information in literature. For mixed
states with $\beta\rightarrow 0$, $F_{nH}^{(1)}$ reduces to the Hermitian
Fisher information, whereas for mixed states with $\beta\rightarrow\pi$,

$F_{nH}^{(1)}\simeq\sum_{i=1}^{M}4p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle+F_{c},$
(9)

where
$F_{c}=\sum_{i=1}^{M}\frac{2(\partial_{\theta}p_{i})^{2}}{p_{i}(1+\cos\beta)}|_{\beta\rightarrow\pi}.$
$F_{c}$ approaches to infinity in the limit of $\beta\rightarrow\pi$. This is
not physical, since $L_{ij}$ in Eq. (7) can not be established when
$\beta\rightarrow\pi$ and $i=j$. In the following discussion, we will focus on
the case that $p_{i}$ are $\theta$-independent, such that $F_{c}=0.$

The quantum information $F_{nH}^{(2)}$ can be calculated by the same
technique. Different from the case of $F_{nH}^{(1)}$, here
$(L^{(2)\dagger})_{ij}=\frac{(\partial_{\theta}\rho)_{ij}}{p_{i}}$,
$L^{(2)}_{ji}=\frac{(\partial_{\theta}\rho)_{ji}}{p_{i}}$ for $p_{i}\neq 0.$
Although both $A$ and $L$ are not Hermitian, the Fisher information is real
and given by supp ,

$\displaystyle F_{nH}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\frac{(\partial_{\theta}p_{i})^{2}}{p_{i}}+\sum_{i=1}^{M}p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{M}\frac{p_{j}(p_{j}-2p_{i})}{p_{i}}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle.$

This is the other main result of this Letter. The quantum Fisher information
$F_{nH}^{(2)}$ for pure state $|\phi\rangle$ reduces to

$F_{nH}^{(2)}=\langle\partial_{\theta}\phi|\partial_{\theta}\phi\rangle-\langle\partial_{\theta}\phi|\phi\rangle\langle\phi|\partial_{\theta}\phi\rangle,$
(11)

which is $1/4$ times smaller than $F_{H}$ in Hermitian system for pure stats.
However, this is not the case for mixed states as shown in the examples below.
The saturation of these bounds and the experimental measurement of non-
Hermitian operator will be discussed briefly at the end of this Letter.

Examples.— Considering the Hermitian quantum Fisher information $F_{H}$ and
the non-Hermitian quantum information $F_{nH}^{(1)}$ and $F_{nH}^{(2)}$ are
almost same for pure stats, i.e., $F_{H}=F_{nH}^{(1)}$ and
$F_{H}=4F_{nH}^{(2)}$ for pure states, we focus on the case of mixed states in
the undergoing examples. To simplify the expression, we consider a type of
simplest mixed states—it is constructed by only two orthogonal pure states
with weight $p_{1}$ and $p_{2},$ and $p_{1}+p_{2}=1.$

We first consider the maximally entangled states (or Schrödinger cat state)
$|\phi_{1,2}\rangle=\frac{1}{\sqrt{2}}(|a\rangle^{\otimes
N}\pm|b\rangle^{\otimes N})$ as the input state, which has been created with
up to $N=6$ ${}^{9}Be^{+}$ ions leibried05  and $N=14$ ${}^{40}Ca^{+}$ ions
monz11  in a linear Paul trap. The encoding operator is a spin rotation
defined by $U(\theta)=e^{-i\theta J_{z}}$ with Hermitian signal Hamiltonian
$J_{z}=\sum_{j=1}^{N}s_{z}^{(j)}$. The case of non-Hermitian signal
Hamiltonian will be discussed at the end of examples. The states that encode
the parameter $\theta$ to construct the mixed states are,

$|\phi_{1,2}(\theta)\rangle=\frac{1}{\sqrt{2}}(e^{\frac{i}{2}N\theta}|a\rangle^{\otimes
N}\pm e^{-\frac{i}{2}N\theta}|b\rangle^{\otimes N}).$

The mixed state is
$\rho=\sum_{i=1}^{2}p_{i}|\phi_{i}(\theta)\rangle\langle\phi_{i}(\theta)|.$
Notice that $p_{i},\,i=1,2$ are $\theta$-independent. The rotation
$U(\theta)=e^{-i\theta J_{z}}$ generates a relative phase $N\theta$ between
states $|a\rangle^{\otimes N}$ and $|b\rangle^{\otimes N}$. Straightforward
calculation gives the Hermitian Fisher information $F_{H}$ and the non-
Hermitian Fisher information $F_{nH}^{(1)}$ and $F_{nH}^{(2)}$ (setting
$p_{1}=p$ and $\beta\rightarrow\pi$) supp ,

$\displaystyle F_{H}$ $\displaystyle=$
$\displaystyle\left(1-4p(1-p)\right)N^{2},$ $\displaystyle F_{nH}^{(1)}$
$\displaystyle=$ $\displaystyle N^{2},\quad
F_{nH}^{(2)}=\frac{(2p-1)^{2}}{4p(1-p)}N^{2}.$ (12)

The results are shown in Fig.1. We find that the non-Hermitian Fisher
information are always larger than the Hermitian Fisher information, except
the points $p=0,0.5,1.$

Figure 1: Quantum Fisher information $F_{H}$, $F_{nH}^{(1)}$ and
$F_{nH}^{(2)}$. The Fisher information was plotted in units of $N^{2}$. $p$ is
the eigenvalue of the density matrix.

It is worth addressing that at points $p=0,1$, the state is pure, so the
Fisher information should be calculated by the formula of pure states, i.e.,
$F_{H}=F_{nH}^{(1)}=N^{2}$, and $F_{nH}^{(2)}=0.25N^{2}.$ As $p$ approach to
$1$ and $0$, $F_{nH}^{(2)}$ tends to infinity, manifesting itself as a witness
of transition from mixed states to pure states. The bound defined by
$F_{nH}^{(1)}$ and $F_{nH}^{(2)}$ can be saturated by carefully chosen
measurements. For details, see Supplemental Material supp .

The second example we will show is the phase estimation interferometric
schemes caves81 ; jarzyna12 , which works with coherent and squeezed vacuum
states interfered at the beam-splitter of the Mach-Zehnder interferometer. Let
$a$ and $b$ be the anihilation operators of the two modes.
$|\alpha\rangle=\exp(\alpha b^{\dagger}-\alpha^{*}b)|0\rangle_{b}$ is a
coherent state of mode $b$ and
$|r\rangle=\exp[\frac{1}{2}r^{*}a^{2}-\frac{1}{2}r(a^{\dagger})^{2}]|0\rangle_{a}$
is a squeezed vacuum state of mode $a$ with squeezing parameter $r$. After the
input state supp  has evolved through a beam splitter defined by
$B_{\pi}=\exp[-\frac{i}{2}\pi(ab^{\dagger}+a^{\dagger}b)]$, the estimated
parameter $\theta$ is encoded into the state by relative phase shift operator
$U(\theta)=\exp[-i\theta H],$ where $H$ is the signal Hamiltonian of mode $a$,
$H=a^{\dagger}a$. For the other beam splitter, see Supplemental Materials supp
. We consider the following states that encode the parameter $\theta$,
$\rho=\sum_{i=1}^{2}p_{i}|\phi_{i}(\theta)\rangle\langle\phi_{i}(\theta)|$
with

$\displaystyle|\phi_{1,2}(\theta)\rangle$ $\displaystyle=$ $\displaystyle
U(\theta)BS(r)D(\alpha)|0\rangle_{a}\otimes|X_{1,2}\rangle_{b},$ (13)

where $|X_{1}\rangle_{b}=|0\rangle_{b}$,
$|X_{2}\rangle_{b}=b^{\dagger}|0\rangle_{b},$ and $p_{1}+p_{2}=1.$ Tedious but
straightforward calculations show that (set $p_{1}=p$) supp ,

$\displaystyle F_{H}$ $\displaystyle=$ $\displaystyle
4|\alpha|^{2}(4p^{2}-6p+3),$ $\displaystyle F_{nH}^{(1)}$ $\displaystyle=$
$\displaystyle 4|\alpha|^{4}+(20-16p)|\alpha|^{2}+4(1-p),$ $\displaystyle
F_{nH}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{2p^{3}-2p^{2}+1}{p(1-p)}|\alpha|^{2}.$ (14)

Figure 2: Quantum information as a function of photon number $|\alpha|^{2}$
and the eigenvalue of density matrix $p$.

We have performed numerical calculations for the Fisher information. The
results are shown in Fig. 2. We find that the two non-Hermitian quantum Fisher
information favor $p\sim 0$. This can be understood as that squeezed Fock
states benefit the parameter estimation more than the squeezed vacuum state.
$F_{nH}^{(1)}$ with term $|\alpha|^{4}$ bounds the estimation precision beyond
the Heisenberg limit of $|\alpha|^{2}$.

One might wonder what is the difference between the non-Hermitian Fisher
information and the Hermitian Fisher information with non-Hermitian signal
Hamiltonian. To answer this question, we replace the Hermitian operator
$J_{z}$ by $J_{z}(1-i\gamma)$ supp  in the first example to calculate the
Hermitian Fisher information $F_{H}$, and the result will be denoted by
$F_{H}^{nhs}$. We compare $F_{H}^{nhs}$ with the Hermitian Fisher information
$F_{H}$ with Hermitian signal Hamiltonian $J_{z}$. In other words, the
comparison is conducted between the Hermitian Fisher information with
Hermitian signal and non-Hermitian signal Hamiltonian for pure states. This
comparison together with what we had in the two examples would show the
difference between the Hermitian and non-Hermitian Fisher information.
Calculation supp  finds,

$\displaystyle F_{H}$ $\displaystyle=$ $\displaystyle N^{2},$ $\displaystyle
F_{H}^{nhs}$ $\displaystyle=$
$\displaystyle{|\bar{N}|^{2}}\left[\frac{}{}1-\tanh^{2}(\gamma
N\theta)\right],$ (15)

where $\bar{N}=N(1-i\gamma).$ Numerical results are given in Fig.3. We find
$F_{H}^{nhs}$ is smaller than $F_{H}$ for almost all $\gamma$ and $\theta$,
except $\gamma$ and $\theta$ very close to zero. The difference between
$F_{H}^{nhs}$ and $F_{H}$ can be interpreted by a residual dependence of the
state on the parameter under estimation apart from the statistical manifold of
states. This happens, for example, when the eigenstates of the density matrix
have a parameter-dependent normalization seveso17 .

Figure 3: Hermitian quantum Fisher information $F_{H}$ (yellow-mesh) and the
non-Hermitian Fisher information $F_{H}^{nhs}$(blue-surf) as a function of
estimation parameter $\theta$ and the loss/gain rate $\gamma$. $\theta$ is
chosen in units of $\pi$, and $N=4.$

In the second example, we take $\bar{H}=H(1-i\gamma)$ instead of $H$ as the
signal Hamiltonian to account the effect of decoherence. $F_{H}^{nhs}$ is
calculated with $\bar{H}$, while the Hermitian Fisher information $F_{H}$ is
calculated with $H$,

$\displaystyle F_{H}$ $\displaystyle=$ $\displaystyle 4|\alpha|^{2},$
$\displaystyle F_{H}^{nhs}$ $\displaystyle=$ $\displaystyle
4|\alpha|^{2}(1+\gamma^{2})e^{-2\gamma\theta}.$ (16)

The scaling of Fisher information with the particle number remains unchanged,
indicating there is no significant enhancement to parameter estimation in this
example.

Saturation of the bounds and feasible experiments to measure an non-Hermitian
variable.— To saturate the bounds, the optimal measurement $A^{opt}$ has to
meet $A^{opt}=\gamma L$ and $(A^{opt})^{\dagger}L=L^{\dagger}A^{opt}$
simultaneously supp . These requirements can not be satisfied for bound
$1/F_{nH}^{(1)}$, as $A$ is Hermitian but $L$ is not as we stated earlier.
Even if we lift the constrain on the Hermiticity of $A$, the bound
$1/F_{nH}^{(1)}$ can not be saturated too due to the requirement of
$\beta=\pi$. This claim is confirmed by our numerical simulation, see
Supplemental Material supp . The situation is different for bound
$1/F_{nH}^{(2)}$. Simple analyses show that the measurement variable

$\displaystyle
A^{opt}_{ij}\equiv\langle\phi_{i}|A^{opt}|\phi_{j}\rangle=\gamma\frac{(\partial_{\theta}\rho)_{ij}}{p_{j}}$
(17)

with real $\gamma$ can saturate the bound supp . For the example with trapped
ions, $A^{opt}$ can be measure in a single-shot experiment in an
interferometer setup supp .

Conclusions.—The framework of quantum mechanics in which observable are
associated with Hermitian operators is too narrow to discuss parameter
estimation. Considering in the past two decades the non-Hermitian physics has
attracted fast growing interest in various field of research, we first derived
an uncertainty relation for non-Hermitian operators, then we deduce a
previously unknown expression for non-Hermitian Fisher information. Two
Cramér-Rao bounds that in some cases one of them, and sometimes both of them,
are superior to the previous result are found. The saturation of these bounds
is analysed and the optimal measurement to attain the bounds are given. The
theory was illustrated with two experimentally feasible systems. The setup to
measure non-Hermitian observables is also proposed.

We thank Xiaoming Lu for helpful discussions. This work was supported by the
National Natural Science Foundation of China (NSFC) under Grants No. 11775048,
and No. 12047566.

## References

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Supplementary material for:
Cramér-Rao bound and quantum parameter estimation with non-Hermitian systems

This supplemental material provides detailed derivations and calculations for
the results in the main text. We give numbers to the equations and figures
here with ”S” in contrast with that in the main text, for example, Eq.(S1),
Fig. S1. Numbers without ”S” refer to the items in the main text, e.g., Eq.
(1), Fig. 1. This materials are organized as follows.

  * S1.

We derive two Cramé-Rao bounds from uncertainty relations of non-Hermitian
operators.

  * S2.

Non-Hermitian Fisher information is defined and previously unknown expressions
to calculate the Fisher information is given.

  * S3.

Two examples to illustrate the non-Hermitian Fisher information are presented.

  * S4.

We present discussion and comparison between the non-Hermitian Fisher
information and the Hermitian Fisher information with non-Hermitian signal
Hamiltonian.

  * S5.

We discuss the second example with the other beam splitter.

  * S6.

This section devotes to discussion of saturation of Cramé-Rao bounds defined
by the non-Hermitian Fisher information.

  * S7.

We focus on the measurement of non-Hermitian operators. Detailed experimental
proposal is given.

##  S1 S1. Uncertainly relation and Cramé-Rao bound for non-Hermitian system

The precision of parameter estimation is limited by the quantum Cramér-Rao
bound (CRB), which was an extension of Cramér-Rao bound in classical
statistics to quantum metrology. As an extension of classical method, CRB has
a wide range of applications, meanwhile it lacks a direct physical picture. In
this section, we would derive the quantum CRB from the uncertainty relations
that root deeply in the quantum theory from the beginning of the last century.

Consider two linear operators $A$, $B$ and a real parameter $\chi$, for any
state $|\Psi\rangle$, the following relation holds,

$\langle\Psi|(\chi A^{\dagger}-iB^{\dagger})(\chi A+iB)|\Psi\rangle\geq 0.$
(S1)

Simple algebra yields,

$\chi^{2}\langle\Psi|A^{\dagger}A|\Psi\rangle+\chi\langle\Psi|i(A^{\dagger}B-B^{\dagger}A)|\Psi\rangle+\langle\Psi|B^{\dagger}B|\Psi\rangle\geq
0.$ (S2)

Noticing $\langle\Psi|A^{\dagger}A|\Psi\rangle\geq 0$ regardless of $A$ being
Hermitian or not, and $\langle\Psi|i(A^{\dagger}B-B^{\dagger}A)|\Psi\rangle$
is real as $C\equiv i(A^{\dagger}B-B^{\dagger}A)$ is Hermitian, we obtain

$\langle\Psi|A^{\dagger}A|\Psi\rangle\langle\Psi|B^{\dagger}B|\Psi\rangle\geq\frac{1}{4}|\langle\Psi|C|\Psi\rangle|^{2}.$
(S3)

Define $\Delta A=A-\langle\Psi|A|\Psi\rangle$ and the same for $\Delta B$, and
note the commutation relation, $[A,B]=[\Delta A,\Delta B],$ we have

$\langle\Psi|\Delta A^{\dagger}\Delta A|\Psi\rangle\langle\Psi|\Delta
B^{\dagger}\Delta
B|\Psi\rangle\geq\frac{1}{4}|\langle\Psi|C|\Psi\rangle|^{2}.$ (S4)

This is the equation (not numbered) in the main text.

Although the proof is conducted for pure states, the uncertainty relation
Eq.(S4) holds for mixed states. We prove this by introducing an ancilla $a$,
such that a mixed state $\rho=\sum_{j}q_{j}|\psi_{j}\rangle\langle\psi_{j}|$
can be purified to be

$|\Psi^{\prime}\rangle=\sum_{j}\sqrt{q_{j}}|\psi_{j}\rangle\otimes|\phi_{j}\rangle_{a},$
(S5)

and the state of the system is obtained by tracing $|\Psi^{\prime}\rangle$
over the ancilla,
$\rho=\text{Tr}_{a}|\Psi^{\prime}\rangle\langle\Psi^{\prime}|.$ With this
consideration, Eq. (S4) can be straightforwardly extended to the composite
system consisting of the system and the ancilla,

$\langle\Psi^{\prime}|\Delta A^{\dagger}\otimes I_{a}\Delta A\otimes
I_{a}|\Psi^{\prime}\rangle\langle\Psi^{\prime}|\Delta B^{\dagger}\otimes
I_{a}\Delta B\otimes
I_{a}|\Psi^{\prime}\rangle\geq\frac{1}{4}|\langle\Psi^{\prime}|C\otimes
I_{a}|\Psi^{\prime}\rangle|^{2}.$ (S6)

Here, $I_{a}$ is the identity operator of ancilla $a$. Noticing
$\langle\Psi^{\prime}|\Delta A^{\dagger}\otimes I_{a}\Delta A\otimes
I_{a}|\Psi^{\prime}\rangle=\text{Tr}(\rho\Delta A^{\dagger}\Delta A)$ and
denoting $\text{Tr}(\rho\Delta A^{\dagger}\Delta A)=\langle\Delta
A^{\dagger}\Delta A\rangle$ with Tr representing the trace over the system, we
finish the proof of the weaker uncertainly relation in the main text. The
slightly stronger uncertainty relation in Eq.(1) with mixed states can be
proved in a similar way.

In order to develop a Cramér-Rao bound for non-Hermitian systems, we first
have to extend error propagation from Hermitian to non-Hermitian systems, see
Eq.(2). Assume a quantum state $\rho=\rho(\theta)$ depends on parameter
$\theta$, the expectation value of operator $A^{\dagger}$ and its conjugate
$A$ would then depend on the parameter. Let us discretize the parameter
$\theta$ by $\theta_{1},\theta_{2},\theta_{3},....$ The fluctuation is then,

$\displaystyle\langle\Delta A^{\dagger}\Delta
A\rangle=\sum_{j}p_{j}\left[\langle A^{\dagger}\rangle(\theta_{j})-\langle
A^{\dagger}\rangle(\bar{\theta})\right]\left[\langle
A\rangle(\theta_{j})-\langle A\rangle(\bar{\theta})\right],$ (S7)

where $\bar{\theta}=\sum_{j}p_{j}\theta_{j}$, and $p_{j}$ ($j=1,2,3,...$)
stand for probabilities. Expanding $\langle A\rangle(\theta_{j})$ around
$\bar{\theta}$ as

$\langle A\rangle(\theta_{j})=\langle
A\rangle(\bar{\theta})+\frac{\partial\langle
A\rangle}{\partial\theta}|_{\bar{\theta}}(\theta_{i}-\bar{\theta})+...$

and approximating $\langle\Delta A^{\dagger}\Delta A\rangle$ up to the second
order in ($\theta_{j}-\bar{\theta}$), we arrive at

$\langle\Delta A^{\dagger}\Delta A\rangle=\frac{\partial\langle
A^{\dagger}\rangle}{\partial\theta}|_{\bar{\theta}}\frac{\partial\langle
A\rangle}{\partial\theta}|_{\bar{\theta}}(\Delta\theta)^{2}.$ (S8)

Here, $(\Delta\theta)^{2}=\sum_{j}p_{j}(\theta_{j}-\bar{\theta})^{2}$. This is
the error propagation in Eq.(2) of the main text.

With this error propagation, we now derive the non-Hermitian Cramér-Rao bound
in Eq.(3). Introducing an operator $L$, which is not required to be Hermitian,
we have

$\displaystyle(\Delta\theta)^{2}\frac{\partial\langle
A^{\dagger}\rangle}{\partial\theta}\frac{\partial\langle
A\rangle}{\partial\theta}\langle\Delta L^{\dagger}\Delta L\rangle$
$\displaystyle=$ $\displaystyle\langle\Delta A^{\dagger}\Delta
A\rangle\langle\Delta L^{\dagger}\Delta L\rangle$ (S9) $\displaystyle\geq$
$\displaystyle|\langle A^{\dagger}L\rangle-\langle A^{\dagger}\rangle\langle
L\rangle|^{2}.$

There are few differences between the present study and the earlier
publications in $L$, which deserve to be outlined. First, in most publications
so far, $L$ is the so-called symmetric logarithmic derivative, which is
required to be Hermitian. This requirement is due to the fact that the
uncertainty relation used in the publications till now is for Hermitian
operators. Whereas in our case $L$ is not required to be Hermitian. In fact,
as we show in the main text an anti-Hermitian $L=-L^{\dagger}$ can maximize
the Fisher information. Second, the uncertainty relation used to derive the
Cramér bound is different, and our derivation can recover the results in the
literature.

Until now, the operator $A$ is not required to be Hermitian. As we will show
later, a non-Hermitian $A$ will lead a different Cramér-Rao bound.

Let us start to discuss a special case that $A$ is Hermitian. For Hermitian
$A$, the last line of Eq. (S9) becomes,

$\displaystyle|\langle A^{\dagger}L\rangle-\langle A^{\dagger}\rangle\langle
L\rangle|^{2}$ $\displaystyle=$ $\displaystyle|\frac{1}{2}\langle
AL+L^{\dagger}A\rangle-\langle A\rangle\langle L\rangle+\frac{1}{2}\langle
AL-L^{\dagger}A\rangle|^{2}$ (S10) $\displaystyle\geq$
$\displaystyle\left(\frac{\langle AL+L^{\dagger}A\rangle}{2}\right)^{2}.$

To have the last inequality, we have ignored $\frac{1}{2}\langle
AL-L^{\dagger}A\rangle$, which is pure imaginary, while term $\langle
AL+L^{\dagger}A\rangle$ is real. By carefully choosing $L$, we can have
$\langle L\rangle=0.$ Eq. (S9) reduces to Eq.(3) in the main text, where $L$
satisfies Eq.(4).

When $A$ is not Hermitian but with $\langle L\rangle=0$, we still obtain Eq.
(3), but in this case, $L$ satisfies Eq. (5). The proof is almost the same as
that for Eq. (4).

Discussions are in order. Either

$\frac{\partial\rho}{\partial\theta}=\frac{1}{2}(L^{(1)}\rho+\rho
L^{(1)\dagger}),$ (S11)

or

$\frac{\partial\rho}{\partial\theta}=L^{(2)}\rho=\rho L^{(2)\dagger}.$ (S12)

guarantee $\langle L\rangle=0.$ For Eq.(S11), as we will discuss below,
$L^{\dagger}=Le^{i\beta}$ is assumed. This assumption together with
$\text{Tr}\left(\frac{\partial\rho}{\partial\theta}\right)=0$ leads to
$\langle L\rangle=0.$ While for Eq.(S12), it is obvious that $\langle
L\rangle=0$ and $\langle L^{\dagger}\rangle=0,$ since
$\text{Tr}\left(\frac{\partial\rho}{\partial\theta}\right)=0$.

##  S2 S2. non-Hermitian quantum Fisher information

We present a detailed calculations for the Fisher information $F_{nH}^{(1)}$
and $F_{nH}^{(2)}$ defined in Eq. (3). We first calculate $F_{nH}^{(1)}.$
Without lose of generality, we assume the dimension of the Hilbert space is
$N$, while the state of the system $\rho$ may be not of full rank. It has
positive eigenvalues $p_{j}$ required by quantum theory, and the corresponding
eigenstates are denoted by $|\phi_{j}\rangle$, i.e.,

$\rho=\sum_{j=1}^{M}p_{j}|\phi_{j}\rangle\langle\phi_{j}|.$ (S13)

Here we assume $j$ runs from $j=1$ to $j=M$, $M\leq N$. With these notations,
the Fishier information defined in Eq.(3) takes,

$\displaystyle F_{nH}^{(1)}$ $\displaystyle=$ $\displaystyle\langle
L^{(1)}L^{(1)}\rangle
e^{i\beta}=\sum_{i=1}^{M}p_{i}\langle\phi_{i}|L^{(1)}L^{(1)}|\phi_{i}\rangle
e^{i\beta}=\sum_{j=1}^{N}\sum_{i=1}^{M}p_{i}\langle\phi_{i}|L^{(1)}|\phi_{j}\rangle\langle\phi_{j}|L^{(1)}|\phi_{i}\rangle
e^{i\beta}$ (S14) $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\sum_{i=1}^{M}p_{i}(L^{(1)})_{ij}(L^{(1)})_{ji}e^{i\beta}$

Recalling Eq. (4) in the main text that,

$\frac{\partial\rho}{\partial\theta}=\frac{1}{2}(L^{(1)}\rho+\rho
L^{(1)\dagger}),$ (S15)

and assuming $L^{(1)\dagger}=L^{(1)}e^{i\beta}$, we have

$L_{ij}^{(1)}=\langle\phi_{i}|L^{(1)}|\phi_{j}\rangle=2\frac{(\partial_{\theta}\rho)_{ij}}{p_{j}+p_{i}e^{i\beta}}.$
(S16)

As addressed in the main text, $L^{(1)}$ should not be limited to the space
spanned by the eigenstates of $\rho$, however for $j>M$ or $i>M$,
$L_{ij}^{(1)}$ can not established by Eq. (S16). We will return to this point
later. To compute $L_{ij}^{(1)}$, we need to know
$(\partial_{\theta}\rho)_{ij}$. Starting from Eq.(S13), we have

$\displaystyle\partial_{\theta}\rho=\sum_{i}\partial_{\theta}p_{i}|\phi_{i}\rangle\langle\phi_{i}|+\sum_{i}p_{i}|\partial_{\theta}\phi_{i}\rangle\langle\phi_{i}|+\sum_{i}p_{i}|\phi_{i}\rangle\langle\partial_{\theta}\phi_{i}|,$
(S17)

leading to

$\displaystyle(\partial_{\theta}\rho)_{ij}$ $\displaystyle=$
$\displaystyle\partial_{\theta}p_{i}\delta_{ij}+p_{i}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle+p_{j}\langle\phi_{i}|\partial_{\theta}\phi_{j}\rangle$
(S18) $\displaystyle=$
$\displaystyle\partial_{\theta}p_{i}\delta_{ij}+(p_{i}-p_{j})\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle.$

Here,
$\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle+\langle\phi_{i}|\partial_{\theta}\phi_{j}\rangle=0$
has been applied in the derivation. Substituting Eq.(S18) and Eq. (S16) into
Eq. (S14), we have

$\displaystyle F_{nH}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{N}4p_{i}\frac{(\partial_{\theta}p_{i})^{2}\delta_{ij}}{p_{i}^{2}+p_{j}^{2}+2p_{i}p_{j}\cos\beta}+\sum_{i=1}^{M}\sum_{j=1}^{N}4p_{i}\frac{(p_{i}-p_{j})^{2}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle}{p_{i}^{2}+p_{j}^{2}+2p_{i}p_{j}\cos\beta}$
(S19) $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\frac{2(\partial_{\theta}p_{i})^{2}}{p_{i}(1+\cos\beta)}+\sum_{i=1}^{M}\sum_{j=1}^{M}4p_{i}\frac{(p_{i}-p_{j})^{2}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle}{p_{i}^{2}+p_{j}^{2}+2p_{i}p_{j}\cos\beta}+\sum_{i=1}^{M}\sum_{j=M+1}^{N}4p_{i}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\frac{2(\partial_{\theta}p_{i})^{2}}{p_{i}(1+\cos\beta)}+\sum_{i=1}^{M}\sum_{j=1}^{M}4p_{i}\frac{(p_{i}-p_{j})^{2}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle}{p_{i}^{2}+p_{j}^{2}+2p_{i}p_{j}\cos\beta}$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{M}4p_{i}\langle\partial_{\theta}\phi_{i}|\left(\frac{}{}I-|\phi_{j}\rangle\langle\phi_{j}|\frac{}{}\right)|\partial_{\theta}\phi_{i}\rangle,$

where $I$ is the identity operator of the Hilbert space, i.e.,
$I=\sum_{i=1}^{N}|\phi_{i}\rangle\langle\phi_{i}|$. Note that the sum runs
from 1 to $N$. Finally, we obtain Eq. (8) in the main text,

$\displaystyle
F_{nH}^{(1)}=\sum_{i=1}^{M}\frac{2(\partial_{\theta}p_{i})^{2}}{p_{i}(1+\cos\beta)}+\sum_{i=1}^{M}4p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle+\sum_{i=1}^{M}\sum_{j=1}^{M}\left(\frac{4p_{i}(p_{i}-p_{j})^{2}}{p_{i}^{2}+p_{j}^{2}+2p_{i}p_{j}\cos\beta}-4p_{i}\right)\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle.$

$F_{nH}^{(2)}$ can be derived by the same technique as employed in the
derivation of $F_{nH}^{(1)}$. Note that
$L^{(2)\dagger}_{ij}=\frac{(\partial_{\theta}\rho)_{ij}}{p_{i}}$, and
$L_{ji}^{(2)}=\frac{(\partial_{\theta}\rho)_{ji}}{p_{i}}$ for $F_{nH}^{(2)}$.

##  S3 S3. examples

Detailed calculation for the non-Hermitian quantum Fisher information
$F_{nH}^{(1)}$ and $F_{nH}^{(2)}$ will be presented below. For mixed states of
form $\rho=\sum_{i=1}^{2}p_{i}|\phi_{i}\rangle\langle\phi_{i}|$ with
$\theta$-independent $p_{i}$, the Fisher information $F_{H}$, $F_{nH}^{(1)}$
and $F_{nH}^{(2)}$ reduce to,

$\displaystyle F_{H}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\left(4p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle-4p_{i}|\langle\phi_{i}|\partial_{\theta}\phi_{i}\rangle|^{2}\right)-16p_{1}p_{2}|\langle\phi_{1}|\partial_{\theta}\phi_{2}\rangle|^{2},$
$\displaystyle F_{nH}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}4p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle,\,(\beta=\pi),$
$\displaystyle F_{nH}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\left(p_{i}\langle\partial_{\theta}\phi_{i}|\partial_{\theta}\phi_{i}\rangle-
p_{i}|\langle\phi_{i}|\partial_{\theta}\phi_{i}\rangle|^{2}\right)+\left(\frac{p_{2}(1-3p_{1})}{p_{1}}+\frac{p_{1}(1-3p_{2})}{p_{2}}\right)|\langle\phi_{1}|\partial_{\theta}\phi_{2}\rangle|^{2}.$
(S21)

In the first example, the state before encoding is

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}p_{i}|\phi_{i}\rangle\langle\phi_{i}|,$
$\displaystyle|\phi_{1,2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|a\rangle^{\otimes N}\pm|b\rangle^{\otimes
N}).$ (S22)

The signal Hamiltonian used to encode the parameter into the state is $J_{z}$,
i.e., the encoding operation is given by $U(\theta)=e^{-i\theta J_{z}}$. The
states that carry the information of the parameter is then

$\displaystyle\rho(\theta)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}p_{i}|\phi_{i}(\theta)\rangle\langle\phi_{i}(\theta)|,$
$\displaystyle|\phi_{1,2}(\theta)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(e^{\frac{i}{2}N\theta}|a\rangle^{\otimes
N}\pm e^{-\frac{i}{2}N\theta}|b\rangle^{\otimes N}).$ (S23)

It is easy to find that

$\displaystyle\langle\partial_{\theta}\phi_{1}|\partial_{\theta}\phi_{1}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{4}N^{2},$
$\displaystyle\langle\partial_{\theta}\phi_{2}|\partial_{\theta}\phi_{2}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{4}N^{2},$
$\displaystyle\langle\partial_{\theta}\phi_{1}|\phi_{1}\rangle=\langle\partial_{\theta}\phi_{2}|\phi_{2}\rangle$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle|\langle\partial_{\theta}\phi_{1}|\phi_{2}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4}N^{2}.$ (S24)

Finally, we obtain Eq.(12) of the main text.

In the second example, the state before encoding is

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}p_{i}|\phi_{i}\rangle\langle\phi_{i}|,$
$\displaystyle|\phi_{1}\rangle$ $\displaystyle=$
$\displaystyle|r\alpha\rangle_{00}\equiv S(r)\otimes D(\alpha)|00\rangle,$
$\displaystyle|\phi_{2}\rangle$ $\displaystyle=$
$\displaystyle|r\alpha\rangle_{01}\equiv S(r)\otimes
D(\alpha)b^{\dagger}|00\rangle,$ (S25)

where $|00\rangle$ denotes that both modes $a$ and $b$ are in its vacuum
state. $S(r)$ and $D(\alpha)$ was defined in the main text, i.e.,
$S(r)=\exp(\frac{r^{*}}{2}a^{2}-\frac{r}{2}(a^{\dagger})^{2})$,
$D(\alpha)=\exp(\alpha a^{\dagger}-\alpha^{*}a)$, and we will set
$r=|r|e^{i2\phi}$. The parameter is encoded into the state after passing
through a beam splitter represented by $B_{\pi}$. The state encoding the
parameter $\theta$ is then

$\displaystyle\rho(\theta)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}p_{i}|\phi_{i}(\theta)\rangle\langle\phi_{i}(\theta)|,$
$\displaystyle|\phi_{1,2}(\theta)\rangle$ $\displaystyle=$ $\displaystyle
U(\theta)BS(r)D(\alpha)|0\rangle_{a}\otimes|X_{1,2}\rangle_{b},$ (S26)

where $|X_{1}\rangle_{b}=|0\rangle_{b}$,
$|X_{2}\rangle_{b}=b^{\dagger}|0\rangle_{b},$ and $p_{1}+p_{2}=1.$ Further, we
find

$\displaystyle\langle\partial_{\theta}\phi_{1}|\partial_{\theta}\phi_{1}\rangle$
$\displaystyle=$ $\displaystyle|\alpha|^{4}+|\alpha|^{2},$
$\displaystyle\langle\partial_{\theta}\phi_{2}|\partial_{\theta}\phi_{2}\rangle$
$\displaystyle=$ $\displaystyle|\alpha|^{4}+5|\alpha|^{2}+1,$
$\displaystyle|\langle\partial_{\theta}\phi_{1}|\phi_{1}\rangle|^{2}$
$\displaystyle=$ $\displaystyle|\alpha|^{4},$
$\displaystyle|\langle\partial_{\theta}\phi_{2}|\phi_{2}\rangle|^{2}$
$\displaystyle=$ $\displaystyle(1+|\alpha|^{2})^{2},$
$\displaystyle|\langle\partial_{\theta}\phi_{1}|\phi_{2}\rangle|^{2}$
$\displaystyle=$ $\displaystyle|\alpha|^{2}.$ (S27)

Collecting these results, we arrive at

$\displaystyle F_{H}$ $\displaystyle=$ $\displaystyle
4|\alpha|^{2}(4p^{2}-6p+3),$ $\displaystyle F_{nH}^{(1)}$ $\displaystyle=$
$\displaystyle 4|\alpha|^{4}+(20-16p)|\alpha|^{2}+4(1-p),$ $\displaystyle
F_{nH}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{2p^{3}-2p^{2}+1}{p(1-p)}|\alpha|^{2}.$ (S28)

This is Eq.(14) in the main text. The dependence of the Fisher information on
the photon number is depicted in Fig.S1.

Figure S1: Fisher information as a function of photon number. $p=0.3$. This is
the numerical simulation for Eq.(14) in the main text.

The Hermitian Fisher information for pure state is also shown for comparison
with the others. We can clearly find that for large photon number,
$F_{nH}^{(1)}$ scales dominantly with $|\alpha|^{4}=N^{2}$, indicating the
breakdown of the Heisenberg limit.

##  S4 S4. non-Hermitian signal Hamiltonian

In the following, we will consider a situation that the signal Hamiltonian is
not Hermitian for $F_{H}$. Comparison is carried out between the Hermitian
quantum Fisher information $F_{H}$ with Hermitian signal Hamiltonian and the
Hermitian quantum Fisher information with non-Hermitian signal Hamiltonian
(will be denoted by $F_{H}^{nhs}$), and only the case of pure states is
considered.

For pure state $|\phi\rangle=|\phi(\theta)\rangle$, $F_{nH}^{(1)}=F_{H}$, so
we do not take $F_{nH}^{(1)}$ into comparison. For pure states, $F_{H}^{nhs}$
and $F_{H}$ takes,

$\displaystyle F_{H}$ $\displaystyle=$ $\displaystyle
4\langle\partial_{\theta}\phi|\partial_{\theta}\phi\rangle-4\langle\partial_{\theta}\phi|\phi\rangle\langle\phi|\partial_{\theta}\phi\rangle,$
(S29) $\displaystyle F_{H}^{nhs}$ $\displaystyle=$ $\displaystyle
4\left(M^{2}\langle\partial_{\theta}\phi^{\prime}|\partial_{\theta}\phi^{\prime}\rangle-\frac{1}{M^{2}}(\partial_{\theta}M)^{2}-\left|\frac{\partial_{\theta}M}{M}+M^{2}\langle\phi^{\prime}|\partial_{\theta}\phi^{\prime}\rangle\right|^{2}\right),$
(S30)

where $|\phi^{\prime}\rangle$ denotes the unnormalized encoding state and $M$
is its normalization constant, namely,
$\langle\phi^{\prime}|\phi^{\prime}\rangle=\frac{1}{M^{2}}$. We might prove
Eq.(S30) by substitution of $|\phi\rangle=M|\phi^{\prime}\rangle$ into
$F_{H}=4\langle\partial_{\theta}\phi|\partial_{\theta}\phi\rangle-4\langle\partial_{\theta}\phi|\phi\rangle\langle\phi|\partial_{\theta}\phi\rangle$,
and noticing

$\displaystyle\langle\partial_{\theta}\phi|\partial_{\theta}\phi\rangle$
$\displaystyle=$ $\displaystyle
M^{2}\langle\partial_{\theta}\phi^{\prime}|\partial_{\theta}\phi^{\prime}\rangle-\frac{(\partial_{\theta}M)^{2}}{M^{2}},$
$\displaystyle\langle\phi|\partial_{\theta}\phi\rangle$ $\displaystyle=$
$\displaystyle\frac{\partial_{\theta}M}{M}+M^{2}\langle\phi^{\prime}|\partial_{\theta}\phi^{\prime}\rangle.$
(S31)

In the first example, the signal Hamiltonian $J_{z}$ is replaced with
$J_{z}(1-i\gamma)$, the encoding state would have the same form as in Eq.(S3)
but with $\bar{N}=N(1-i\gamma)$ instead of $N$. With these knowledge, we have

$\displaystyle F_{H}$ $\displaystyle=$ $\displaystyle N^{2},$ $\displaystyle
F_{H}^{nhs}$ $\displaystyle=$ $\displaystyle{|\bar{N}|^{2}}(1-\tanh^{2}(\gamma
N\theta)),$ (S32)

which is Eq.(15) in main text.

Figure S2: Fisher information as a function of ion number. This is the
numerical simulation for Eq.(15) in the main text. The parameters chosen are
$\theta=0.25\pi$ and $\gamma=0.2$.

One might wonder why the signal Hamiltonian takes $J_{z}(1-i\gamma)$ instead
of $J_{z}$. This can be understood as follows. Consider the rotation
$U(\theta)=e^{-i\theta J_{z}}$ applied to encode the parameter $\theta$ into
the state. This rotation can be realized in trapped ionsmonz11  through an
effective Hamiltonian $H_{signal}$,

$H_{signal}=\frac{\hbar\Omega}{2}J_{z},$ (S33)

where $J_{z}=\sum_{l}s_{z}^{(l)}$ is a collective spin of those trapped ions,
$\Omega$ is the Rabi frequency of each ion or the magnetic field $B_{z}$ to
which the ion coupled. The time evolution operator
$U(T)=e^{-\frac{i}{\hbar}H_{singal}T}\equiv e^{-i\theta J_{z}}$ plays the role
of the rotation operator. Here the accumulated phase
$\theta=\frac{\Omega}{2}T.$

There is no quantum system being completely isolated from its surroundings
including the trapped ions. A system that is coupled with its environment is
called open system. A complete description of an open quantum system requires
the inclusion of the environment. As a result of the interaction with the
environment, the dynamics of the open system is governed by the master
equationgardiner04 ,

$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[H_{signal},\rho]+\cal{L}(\rho),$
$\displaystyle\cal{L}(\rho)$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}\sum_{l}(2s_{-}^{(l)}\rho s_{+}^{(l)}-\rho
s_{+}^{(l)}s_{-}^{(l)}-s_{+}^{(l)}s_{-}^{(l)}\rho).$ (S34)

Consider a very short encoding time, terms with $s^{(l)}_{-}\rho s^{(l)}_{+}$
can be ignored. And the dynamics of the open system is governed by an
effective Hamiltonian

$H_{eff}=H_{signal}-\frac{i\hbar\kappa}{2}\sum_{l}s^{(l)}_{+}s^{(l)}_{-}.$
(S35)

Consider each ion being modelled by a two-level system, the terms
$\sum_{l}s^{(l)}_{+}s^{(l)}_{-}$ would contribute to $H_{signal}$ as $\sim
J_{z}$. Namely, in this open system

$\bar{H}_{singal}\simeq J_{z}(1-i\gamma).$ (S36)

Here $\gamma\sim\kappa$ stands for the decay rate of collective spin $J_{z}$.

For the example with Mach-Zehnder interferometer, we take
$\bar{H}=a^{\dagger}a(1-i\gamma)$ to replace $H$ in the Hermitian encoding
case. The calculation is involved but straightforward, it shows that

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\exp\left[\frac{}{}|\alpha|^{2}(1-e^{-2\gamma\theta})\right],$
$\displaystyle\langle\partial_{\theta}\phi^{\prime}|\partial_{\theta}\phi^{\prime}\rangle$
$\displaystyle=$
$\displaystyle\frac{1+\gamma^{2}}{M^{2}}|\alpha|^{2}e^{-2\gamma\theta}(1+|\alpha|^{2}e^{-2\gamma\theta}),$
$\displaystyle\langle\partial_{\theta}\phi^{\prime}|\phi^{\prime}\rangle$
$\displaystyle=$
$\displaystyle(i-\gamma)\frac{|\alpha|^{2}}{M^{2}}e^{-2\gamma\theta}.$ (S37)

Substituting Eq.(S4) into Eq.(S30), we obtain Eq.(16) in the main text.

##  S5 S5. results with the other beam splitter

In Sec.S3, we employed a beam splitter,
$B_{\pi}=e^{-\frac{i}{2}\pi(ab^{\dagger}+a^{\dagger}b)}$, in a standard Mach-
Zehnder setup to study the Fisher information. With beam splitter $B_{\pi}$,
the mode $b$ (or mode $a$) is simply transmitted to another mode $a$ (or mode
$b$), so there is no interferometer at all, the Fisher information is however
not zero jarzyna12 . In the following, we will employ the other beam splitter,
$B_{\frac{\pi}{2}}=e^{-\frac{i}{4}\pi(ab^{\dagger}+a^{\dagger}b)}$, to study
the Fisher information. Involved but straightforward calculations show that,

$\displaystyle\langle\partial_{\theta}\phi_{2}|\partial_{\theta}\phi_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\partial_{\theta}\phi_{1}|\partial_{\theta}\phi_{1}\rangle+\sinh^{2}|r|+|\alpha|^{2}+\frac{1}{2},$
$\displaystyle\langle\partial_{\theta}\phi_{1}|\partial_{\theta}\phi_{1}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(|\alpha|^{4}+2|\alpha|^{2}+4|\alpha|^{2}\sinh^{2}|r|+\sinh^{4}|r|+2\sinh^{2}|r|\cosh^{2}|r|+\sinh^{2}|r|\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\Re(\alpha^{2}e^{-2i\phi})\cosh|r|\sinh|r|,$
$\displaystyle|\langle\phi_{1}|\partial_{\theta}\phi_{1}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(\sinh^{2}|r|+|\alpha|^{2})^{2},$
$\displaystyle|\langle\phi_{1}|\partial_{\theta}\phi_{2}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4}|\alpha|^{2},$
$\displaystyle|\langle\phi_{2}|\partial_{\theta}\phi_{2}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(\sinh^{2}|r|+|\alpha|^{2}+1)^{2}.$
(S38)

Here $\Re(...)$ stands for the real part of $(...)$. Substituting Eq.(S5) into
Eqs(S3), we obtain the Fisher information of states with beam splitter
$B_{\frac{\pi}{2}}$. Numerical results are presented in Fig. S3. We find that
$F_{nH}^{(2)}<F_{H}<F_{nH}^{(1)}$ except $p\rightarrow 0,1$. This is similar
to the results in Fig. 2 of the main text.

Figure S3: Fisher information as a function of $|\alpha|^{2}$. This is the
numerical simulation for Fisher information with beam splitter
$B_{\frac{\pi}{2}}$. The parameters chosen are $\phi=0.25\pi$ and $|r|=2$. The
surf figure just below $F_{H}$ (mesh figure) is for the Hermitian quantum
information in pure state $|\phi_{1}\rangle$.

##  S6 S6. Saturation of non-Hermitian quantum Cramér-Rao bounds
$1/F_{nH}^{(1)}$ and $1/F_{nH}^{(2)}$

The variance $(\Delta\theta)^{2}$ of the estimated parameter $\theta$ is given
by the error propagation Eq.(2) of the main text, it is bounded by the quantum
Cramér-Rao bound (QCRB) defined through the quantum Fisher information as

$(\Delta\theta)^{2}\geq\frac{1}{F},\quad
F=F_{H},\,F_{nH}^{(1)},\,F_{nH}^{(2)}.$

Given a signal Hamiltonian and initial state, the bounds can be saturated by
carefully chosen measurements characterized by variable (operator) $A$. In the
following, we will show that the bound given by $F_{nH}^{(2)}$ can be
saturated, while $F_{nH}^{(1)}$ can not. The optimal measurement $A^{opt}$ to
saturate the bound $\frac{1}{F_{nH}^{(2)}}$ is also given.

Recalling the Cauchy-Schwarz inequality, $\langle F|F\rangle\langle
G|G\rangle\geq|\langle F|G\rangle|^{2}$ and noticing the equality holds if and
only if $|F\rangle=C|G\rangle,$ ($C$ is a constant) we claim that to saturate
the bounds, $A^{opt}=\Gamma L$ ($\Gamma$ is a constant). In other words, to
reach the lower Cramér-Rao bound, the optimal measurement variable $A^{opt}$
is required to be proportional to the symmetric logarithmic derivative $L$.

In Eq. (S10), a term $\langle A^{\dagger}L-L^{\dagger}A\rangle$ had been
ignored in order to have the last inequality. For the equality to hold,
$(A^{opt})^{\dagger}L=L^{\dagger}A^{opt}$ is required, this together with
$A^{opt}=\Gamma L$ require that $\Gamma$ is real. This means that to saturate
the bounds, $A^{opt}$ and $L$ must be Hermitian or non-Hermitian
simultaneously. This suggests that bound $1/F_{nH}^{(1)}$ can not be
saturated, as $A$ is Hermitian but $L$ is not as we stated in the main text.
In addition, $\beta=\pi$ can not be met as discussed in the main text.

As to the bound given by $1/F_{nH}^{(2)}$, all requirements for saturation are
met, and the optimal measurement $A^{opt}$ can be expressed in terms of the
encoding state $\rho(\theta)$. Firstly,
$(A^{opt})^{\dagger}L=L^{\dagger}A^{opt}$ and $A^{opt}=\Gamma L$ can be met
simultaneously, and there are no contradictions with the requirement in the
main text. Secondly, from $A^{opt}=\Gamma L$ and Eq. (S12), we find that

$\displaystyle A^{opt}_{ij}$ $\displaystyle\equiv$
$\displaystyle\langle\phi_{i}|A^{opt}|\phi_{j}\rangle=\Gamma L_{ij},$
$\displaystyle L_{ij}$ $\displaystyle=$
$\displaystyle\frac{(\partial_{\theta}\rho)_{ij}}{p_{j}},$ (S39)

where, $p_{i}$ and $|\phi_{i}\rangle$ are the eigenvalues and its
corresponding eigenstates of the encoding state $\rho(\theta)$, respectively.

To be specific, we assume $p_{i}$ to be $\theta$-independent as we did in the
main text, then

$\displaystyle(\partial_{\theta}\rho)_{ij}=p_{i}\langle\partial_{\theta}\phi_{i}|\phi_{j}\rangle+p_{j}\langle\phi_{j}|\partial_{\theta}\phi_{i}\rangle.$
(S40)

We apply these results to the first example, and find that the operators given
in Eq. (S6) indeed saturate the bound $1/F_{nH}^{(2)}$, see Fig. S4. The
saturation of bound $1/F_{H}$ is also presented for comparison, see Fig. S5.

Figure S4: Bounds $1/F_{H}$ (red-dashed), $1/F_{nH}^{(1)}$ (blue-dotted) and
$1/F_{nH}^{(2)}$ (blue-solid) as well as the variance of the estimated
parameter $(\Delta\theta)^{2}$ (green-circles) versus $p$. $p$ is defined in
Eq. (S3), by which we plot the bounds. The variance (green-circles) is
calculated by randomly generating variable $A$ and computing
$(\Delta\theta)^{2}$ with Eq. (S8). Figure S5: The same as Fig. S4. The
difference is that the optimal $A^{opt}$ here is for the saturation of
$1/F_{H}$.

We would like to notice that for $0.4<p<0.6$, the bounds $1/F_{nH}^{(2)}$ and
$1/F_{H}$ are almost same, while $1/F_{nH}^{(1)}$ remains unchanged. See Fig.
S6. For $p>0.6$ the bounds and the variance behave similarly with respect the
those in the region of $p<0.4$.

Figure S6: The same as Fig. S4. The bounds $1/F_{H}$ (red-solid) and
$1/F_{nH}^{(2)}$ (blue-dashed) coincide, while $1/F_{nH}^{(1)}$ is very small
with respect to $1/F_{H}$ and $1/F_{nH}^{(2)}$.

From Figures S4, S5 and S6 we find that the bound $1/F_{nH}^{(1)}$ plays no
role yet in constraint on the parameter estimation. To show the role that
$1/F_{nH}^{(1)}$ plays, we examine the Mach-Zehnder setup with $B_{\pi}$
splitter. The results are presented in Fig. S7. The circles in the region
enclosed by the red-dashed, blue-solid and the blue-dashed lines (i.e., the
region with dotted ellipse) break the constraint by $1/F_{H}$ and
$1/F_{nH}^{(2)}$ while they are above the lower bound given by
$1/F_{nH}^{(1)}$.

Figure S7: The bounds $1/F_{H}$ (red-dashed), $1/F_{nH}^{(2)}$ (blue-solid)
and $1/F_{nH}^{(1)}$ (blue-dotted) as well as the variance
$(\Delta\theta)^{2}$ (green-circles) as a function of $p$. This figure shows
the relation between the bounds and the variance in the second example, i.e.,
the Mach-Zehnder setup. We can find that bound $1/F_{nH}^{(1)}$ can not be
saturated, however it really provides a lower bound for the variance
$(\Delta\theta)^{2}$.

##  S7 S7. Measuring the average of a non-Hermitian operator

The key point of parameter estimation with non-Hermitian systems is to measure
the average of non-Hermitian operator $A$. In this section, we follow the
proposal in Refs pati15 ; nirala19 ; abbott19 ; sahoo20  to show how to
measure the non-Hermitian operator in our case.

Let us consider a non-Hermitian operator $A$. The expectation value of $A$ in
a quantum state $|\phi_{in}\rangle$ given by
$\langle\phi_{in}|A|\phi_{in}\rangle$ is complex in general. This makes the
non-Hermitian operator $A$ unobservable in experiments. Nevertheless, recent
studies shown that this obstacle can be overcome with the help of polar
decomposition hall15 , which states that given any operator $A$, we can always
decompose $A$ as $A=UR$ with $U$ being an unitary operator and $R$ a Hermitian
semidefinite operator, $R=\sqrt{A^{\dagger}A}.$ This bridges the average of
non-Hermitian operator $A$ and the weak value of Hermitian operator $R$ as
follows,

$\displaystyle\langle\phi_{in}|A|\phi_{in}\rangle=\langle\phi_{in}|UR|\phi_{in}\rangle=\frac{\langle\phi|R|\phi_{in}\rangle}{\langle\phi|\phi_{in}\rangle}\langle\phi|\phi_{in}\rangle,$
(S41)

where $\langle\phi|\equiv\langle\phi_{in}|U.$ It is Well-known that
$\frac{\langle\phi|R|\phi_{in}\rangle}{\langle\phi|\phi_{in}\rangle}$ is a
weak value of the positive-semidefinite operator $R$, which can be measured
directly in the weak measurement aharonov88 .

In fact, the weak measurement is not necessary for the weak value and the
average of non-Hermitian operator $A$—they can be given in a single-shot
measurement with an interferometric technique reported in sahoo20 . Before
going into details of such a technique, we first show how to decompose the
non-Hermitian operator $A$ in our first model.

To simplify the representation, let us consider a single trapped ion. For such
a system, any non-Hermitian operator $A$ can be written as,

$A=a\sigma_{+}+b\sigma_{-}+c\sigma_{z}+d,$ (S42)

where $a,b,c,d$ are complex parameters in general, and
$\sigma_{+}=|a\rangle\langle b|=(\sigma_{-})^{\dagger},$
$\sigma_{z}=|a\rangle\langle a|-|b\rangle\langle b|.$ It is easy to find that

$A^{\dagger}A=A_{x}\sigma_{x}+A_{y}\sigma_{y}+A_{z}\sigma_{z}+A_{0},$ (S43)

where $A_{j},\,j=x,y,z,0$ can be written in terms of $a,b,c,d$ and are all
real since $A^{\dagger}A$ is Hermitian. Namely,
$A_{x}=\Re(b^{*}d-b^{*}c+ac^{*}+ad^{*}),$
$A_{y}=-\Im(b^{*}d-b^{*}c+ac^{*}+ad^{*}),$ $A_{z}=2{\cal
R}(c^{*}d)+\frac{1}{2}(|b|^{2}-|a|^{2}),$
$A_{0}=\frac{1}{2}(|a|^{2}+|b|^{2})+|c|^{2}+|d|^{2}.$ $\Re(...)$ and
$\Im(...)$ denote the real and imaginary part of $(...).$ Simple algebra
yields the eigenstates and the corresponding eigenvalues of $A^{\dagger}A$,

$\displaystyle|E^{A}_{+}\rangle$ $\displaystyle=$
$\displaystyle\cos\frac{\Theta}{2}e^{-i\Xi}|a\rangle+\sin\frac{\Theta}{2}|b\rangle,$
$\displaystyle|E^{A}_{-}\rangle$ $\displaystyle=$
$\displaystyle\sin\frac{\Theta}{2}e^{-i\Xi}|a\rangle-\cos\frac{\Theta}{2}|b\rangle,$
$\displaystyle E^{A}_{\pm}$ $\displaystyle=$ $\displaystyle
A_{0}\pm\sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}.$ (S44)

Here, $\tan\Xi=\frac{A_{y}}{A_{x}},$ and
$\cos\Theta=\frac{A_{z}}{\sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}}.$ Then we have,
$A^{\dagger}A=\sum_{j=+,-}E^{A}_{j}|E^{A}_{j}\rangle\langle E^{A}_{j}|,$
leading to the decomposition,

$R=\sqrt{A^{\dagger}A}=\sum_{j=+,-}\sqrt{E^{A}_{j}}|E^{A}_{j}\rangle\langle
E^{A}_{j}|.$ (S45)

Note that $E^{A}_{\pm}\geq 0$ as $A^{\dagger}A$ is Hermitian and positive.
This observation together with $A=UR$ leads to

$U=A\sum_{j=+,-}\frac{1}{\sqrt{E^{A}_{j}}}|E^{A}_{j}\rangle\langle
E^{A}_{j}|.$ (S46)

With the decomposition $A=UR$, Eq. (S45) and Eq. (S46), we now go to the
measurement of non-Hermitian operator $A$. The schematic setup is illustrated
in Fig. S8. The average of $A$ in state $|\phi_{in}\rangle$,
$\langle\phi_{in}|A|\phi_{in}\rangle\equiv|A|e^{i\zeta}$ can be readout from
the average intensity measured by the detector.

Figure S8: Schematic setup for measurement of non-Hermitian operator $A=UR.$
$B_{1}$ and $B_{2}$ are 50:50 Hadamard-type beam splitters, which split the
spatial modes representing by $|1\rangle$ and $|2\rangle$. $e^{i\chi}$ is a
phase shifter that introduces a relative phase $\chi$ between the two arms,
and we measure the intensity at the detector as a function of $\chi$.

One point deserved to be addressed is that this set up can work for both
polarized photons and two-level atoms. So, we would not specify which
interferometer it is (Mach-Zehnder or Ramsey). In bases spanned by
$|a,1\rangle,|b,1\rangle,|a,2\rangle,|b,2\rangle$
($|x,n\rangle\equiv|x\rangle\otimes|n\rangle,x=a,b;n=1,2$), these operation
$B_{i}(i=1,2),R,U^{\dagger},e^{i\chi}$ can be written as

$\displaystyle B_{1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}1&0&i&0\\\
0&1&0&i\\\ i&0&1&0\\\ 0&i&0&1\\\ \end{array}\right),\quad
R=\left(\begin{array}[]{cccc}R_{aa}&R_{ab}&0&0\\\ R_{ba}&R_{bb}&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right),$ (S55) $\displaystyle e^{i\chi}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&e^{i\chi}&0\\\ 0&0&0&e^{i\chi}\\\ \end{array}\right),\quad
U^{\dagger}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&U^{\dagger}_{aa}&U^{\dagger}_{ab}\\\
0&0&U^{\dagger}_{ba}&U^{\dagger}_{bb}\\\ \end{array}\right),$ (S64)

where $R_{ab}=\langle a|R|b\rangle$, and similar denotations hold for
$R_{aa},R_{bb},R_{ba}$ and $U^{\dagger}_{xy},x,y=a,b.$ With the same bases,
the input state that only occupies spatial state $|1\rangle$ takes,

$\displaystyle|\phi_{in}\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\phi_{a}\\\ \phi_{b}\\\ 0\\\ 0\\\
\end{array}\right),\phi_{a}\equiv\langle
a|\phi_{in}\rangle,\,\phi_{b}\equiv\langle b|\phi_{in}\rangle.$ (S69)

The output state reads

$|\phi_{out}\rangle=B_{2}e^{i\chi}U^{\dagger}RB_{1}|\phi_{in}\rangle,$ (S70)

simple algebra yields,

$\displaystyle|\phi_{out}\rangle$ $\displaystyle=\frac{1}{2}$
$\displaystyle\left(\begin{array}[]{c}R_{aa}\phi_{a}+R_{ab}\phi_{b}-(U^{\dagger}_{aa}\phi_{a}+U^{\dagger}_{ab}\phi_{b})e^{i\chi}\\\
R_{ba}\phi_{a}+R_{bb}\phi_{b}-(U^{\dagger}_{ba}\phi_{a}+U^{\dagger}_{bb}\phi_{b})e^{i\chi}\\\
i[R_{aa}\phi_{a}+R_{ab}\phi_{b}+(U^{\dagger}_{aa}\phi_{a}+U^{\dagger}_{ab}\phi_{b})e^{i\chi}]\\\
i[R_{ba}\phi_{a}+R_{bb}\phi_{b}+(U^{\dagger}_{ba}\phi_{a}+U^{\dagger}_{bb}\phi_{b})e^{i\chi}]\\\
\end{array}\right).$ (S75)

The intensity the detector measures can be represented by

$I(\chi)=|\langle\phi_{D}|\phi_{D}\rangle|^{2},\quad|\phi_{D}\rangle\equiv
D|\phi_{out}\rangle,$ (S76)

with

$D=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right),$ (S77)

leading to

$I(\chi)=\frac{1}{4}\left(\frac{}{}1+\langle\phi_{in}|R^{2}|\phi_{in}\rangle+2|\langle\phi_{in}|A|\phi_{in}\rangle|\cos(\chi-\zeta)\right),$
(S78)

where $\zeta$ is the argument of $\langle\phi_{in}|A|\phi_{in}\rangle$, i.e.,
$\zeta=\arg\langle\phi_{in}|A|\phi_{in}\rangle.$ In experiment, the intensity
$I(\chi)$ together with $\langle\phi_{in}|R^{2}|\phi_{in}\rangle$ which is the
average of Hermitian operator $R^{2}$ can determine the average of non-
Hermitian operator $A,$ as both $|\langle\phi_{in}|A|\phi_{in}\rangle|$ and
$\zeta$ can be inferred from the intensity $I(\chi)$.

## References

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