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

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,

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

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

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

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 ,

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,

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,

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,

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 ,

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 $\beta\rightarrow\pi$,

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 ,

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

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,

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 ,

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 $p=0,0.5,1.$

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

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 ,

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,

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 .

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 $H$,

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

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.

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.