Abstract
Quantum metrology aims at delivering new quantummechanical improvement to technologies of parameter estimations with precision bounded by the quantum CramérRao bound. The currently used quantum CramérRao bound was established with measurements of observables restricted to be Hermitian. This constrains the bound and limits the precision of parameter estimation. In this paper, we lift the constraint and derive a previously unknown quantum CramérRao bound. We find that the new bound can reach arbitrary small value with mixed states and it breaks the Heisenberg limit in some cases. We construct a setup to measure nonHermitian operators and discuss the saturation of the present bound. Two examples—the phase estimation with GreenbergerHorneZeilinger states of trapped ions and the adiabatic quantum parameter estimation with the nuclear magnetic resonance—are employed to demonstrate the theory. The present study might open a new research direction—nonHermitian quantum metrology.
1 Introduction
One among the fields advanced by quantum mechanics is metrology, which concerns the estimation of unknown physical parameters [1,2,3,4,5] aiming at improving the estimation beyond classical precision bounds. Higher estimation precision demands more resources. The tradeoff between the precision and the resources is constrained by the quantum CramérRao bound (QCRB), which states that the variance of the estimation is at least as high as the inverse of the quantum Fisher information [6,7,8,9,10,11,12,13]. To attain the highest achievable precision, one needs to optimize over the input state and the encoding Hamiltonian as well as the measurements performed at the output.
In the past decades since Yuen and Lax [14] who first proposed an idea to estimate a complex parameter by measuring nonHermitian observables, no progress has been made in the improvement of parameter estimation by optimizations of measurements invoking nonHermitian operators. The reason is probably that in quantum mechanics observables are represented by Hermitian operators. In the last two decades, the study of nonHermitian system and their unique properties have attracted fast growing interest [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29], this motivates the examination of parameter estimation involving nonHermitian operators. We should address that there are estimation protocols (or sensors) based on nonHermitian system recently [30,31,32,33,34,35,36,37,38], but all analyses so far are based on either the quantum Fisher information and quantum CramérRao bound with Hermitian operators or the properties of exceptional points. There is no study of quantum CramérRao bound and its consequent estimation theory based on measurements of nonHermitian operators.
In this paper, inspiring by the recent progress on the measurement of nonHermitian operators [39,40,41,42,43], we present a scheme to enhance the precision of parameter estimation by measurement of nonHermitian operators. To this goal, we first derive a previously unknown quantum CramérRao bound defined by a quantum Fisher information for nonHermitian operators. We find that the Fisher information is significantly increased due to the optimization over measurements including nonHermitian operators, in particular for systems in mixed states. This is in contrary to the results of the quantum Fisher information defined by symmetric logarithmic derivatives. Saturation of the bound is analyzed and the optimal measurement to attain the bounds is derived. We demonstrate the present theory with the phase estimation setup in trapped ions [5, 44] and the adiabatic quantum parameter estimation setup.
The remainder of this paper is organized as follows. In Section 2, we first give the RobertsonSchrödinger uncertainty relation for nonHermitian operators and extend the error propagation function from Hermitian systems [45, 46] to nonHermitian systems. Then, we present a bound for parameter estimation based on the quantum Fisher information invoking measurements of nonHermitian operators. An expression for the quantum Fisher information is derived and the optimal measurement to saturate the bound is given. In Section 3, we apply this theory to quantum parameter estimation with GHZ states of ions and show that mixed states can further enhance the sensitivity of phase estimation with measurements of nonHermitian operators. The optimal measurement for this setup is derived in details. In Section 4, we apply this theory to adiabatic quantum parameter estimation, taking the estimation of magnetic field on a nuclear magnetic resonance as an example. The measurement of the nonHermitian optimal operator is discussed in Section 5, and finally we conclude in Section 6.
2 Quantum CramérRao bound and quantum Fisher information of nonHermitian system
In this section, we first give the RobertsonSchrödinger uncertainty relation for nonHermitian operators. Then, the error propagation is extended from Hermitian to nonHermitian systems. Finally, we derive the nonHermitian quantum Fisher information based on the new error propagation.
2.1 RobertsonSchrödinger uncertainty relation for nonHermitian operators
In order to present a RobertsonSchrödinger uncertainty relation for nonHermitian operators that will be used to derive a nonHermitian quantum Fisher information with nonHermitian operators, let us consider two linear operators A and B, which can be either Hermitian or nonHermitian. Defining \(\Delta A=A\langle A\rangle\), \(\Delta B=B\langle B\rangle\) and
as the variance for operator \(O\,\) (\(O=A,B\)), the RobertsonSchrödinger uncertainty relation follows from the Schwarz inequality \(\langle F\vert F\rangle \langle G\vert G\rangle \ge \vert \langle F\vert G\rangle \vert ^2\) with \(\vert F\rangle =\Delta A\vert \Psi \rangle ,\) \(\vert G\rangle =\Delta B\vert \Psi \rangle\) and \(\vert \Psi \rangle\) being an arbitrary state of a system that,
Though the proof is performed for pure states, it is easy to find that the uncertainty relation holds valid for mixed states (see Appendix A).
2.2 Error propagation for nonHermitian systems
In order to derive a quantum Fisher information invoking measurements of nonHermitian operators, here, we extend the error propagation from Hermitian to nonHermitian systems. Take \(\rho =\rho (\theta )\) as the encoding state of estimation parameter \(\theta\), the expectation value of operator \(A^\dagger\) and its conjugate A would depend on the parameter. Let us denote \(\theta _i\) as \(i\)th measurement result for the estimation parameter \(\theta\). The fluctuation due to the quantum uncertainty is then,
where \(\bar{\theta }=\sum _j p_j\theta _j\), and \(p_j\) (\(j=1,2,3,...\)) stand for probabilities of obtaining \(\theta _j\). Expanding \(\langle A\rangle (\theta _j)\) around \(\bar{\theta }\) as
and keeping \(\langle \Delta A^\dagger \Delta A\rangle\) up to the second order in (\(\theta _j\bar{\theta }\)), we obtain
where \((\Delta \theta )^2=\sum _jp_j(\theta _j\bar{\theta })^2\). This is the error propagation [47] invoking measurement of nonHermitian operators.
2.3 NonHermitian quantum Fisher information
With the error propagation (3), we now derive the nonHermitian quantum Fisher information where we concern the estimation of the unknown parameter \(\theta\) encoded in \(\rho (\theta )\). Introducing an operator L, which is socalled right logarithmicderivative and defined by [14, 48]
we have
where the second equality holds for \(A=\gamma L\) with a constant \(\gamma\) (not necessary real), and \(\langle L\rangle =0\) has been used in the last equality implied by \(\text {Tr} \left( \frac{\partial \rho }{\partial \theta }\right) =0\). Substituting Eq. (4) into Eq. (5), we obtain
Here, \(F_{nH}\) is defined as nonHermitian quantum Fisher information.
The present derivation is based on the uncertainty relation, which has the following advantages. (i) It is easy to find an optimal measurement \(A^{opt}\) to saturate the bound,
with \(\gamma\) being a constant. And (ii) it provides an alternative understanding for the origin of the variance and bound. Namely, the variance and bound results from the uncertainty relation in contrast to its classical counterpart that is from statistics. In later discussions, we focus only on the saturation condition \(A^{opt}=\gamma L,\) since \(\langle L\rangle =\langle L^\dagger \rangle =0\).
In the following, we give the expression of the quantum Fisher information in terms of the eigenstates and eigenvalues of the encoding density matrix. We assume that the dimension of the Hilbert space is N and the state \(\rho (\theta )\) encoding the unknown parameter \(\theta\) may not be of full rank [49]. Its jth eigenvalue is denoted by \(p_j(\theta )\) and the corresponding eigenstate by \(\vert \phi _j(\theta )\rangle\). Namely, the density matrix \(\rho (\theta )\) can be decomposed as
Here, we assume j running from 1 to M and \(M\le N\). With these notations, the quantum Fisher information defined in Eq. (6) takes
Recalling Eq. (4), we have
Note that L should not be limited to the space spanned by the eigenstates of \(\rho (\theta )\). This means that \(L\vert \phi _i\rangle\) might be out of the space spanned by the eigenstates of \(\rho (\theta )\). To compute \(L_{ij}\), we need to know \((\partial _\theta \rho )_{ij}\). From Eq. (8) we have
leading to
Here \(\langle \partial _\theta \phi _i\vert \phi _j\rangle +\langle \phi _i\vert \partial _\theta \phi _j\rangle =0\) has been applied. Substituting Eq. (12) and Eq. (10) into Eq. (9), we have
where \(F_c=\sum _{i=1}^M\frac{(\partial _\theta p_i)^2}{p_i}\) is the socalled classical Fisher information, which would be zero for \(\theta\)independent \(p_i\). We will focus on this situation hereafter. The nonHermitian quantum Fisher information \(F_{nH}\) for pure state \(\vert \phi \rangle\) reduces to \(F_{nH}=\langle \partial _\theta \phi \vert \partial _\theta \phi \rangle \langle \partial _\theta \phi \vert \phi \rangle \langle \phi \vert \partial _\theta \phi \rangle ,\) which is 1/4 times smaller than the Hermitian quantum Fisher information \(F_H^p=4(\langle \partial _\theta \phi \vert \partial _\theta \phi \rangle \langle \partial _\theta \phi \vert \phi \rangle \langle \phi \vert \partial _\theta \phi \rangle )\). The situation, however, is different for mixed states. Consider an encoding mixed state \(\rho (\theta )=\sum _{j=1}^M p_j\vert \phi _j(\theta )\rangle \langle \phi _j(\theta )\vert\) with an eigenvalue, say, \(p_1\) very close to zero and all \(p_j\) independent of \(\theta\). Equation (13) shows that the terms with \(\frac{p_1^3+p_j^3}{2p_1p_j}\) dominate \(F_{nH},\) and it could approach an arbitrary large value depending on how small \(p_1\) might take in experiments. This makes the present quantum Fisher information different from the wellknown one defined through the symmetric logarithmic derivative \(\mathcal {L}\), which satisfies \(\frac{\partial \rho }{\partial \theta }=\frac{1}{2} (\mathcal {L}\rho +\rho \mathcal {L}).\) We note that the Hermitian quantum Fisher information \(F_H\) for a density matrix is always smaller than or equal to that for any eigenstate of the density matrix.
3 Quantum parameter estimation with GreenbergerHorneZeilinger (GHZ) states of several ions
Since the nonHermitian quantum Fisher information \(F_{nH}\) benefits from mixed states, we will focus on encoding the unknown parameter into mixed states in the undergoing example. Without loss of generality, we consider a type of simplest mixed states—there are only two eigenstates and the corresponding eigenvalues are \(p_1\) and \(p_2,\) respectively, where \(p_1+p_2=1\).
For mixed states of form \(\rho =\sum _{i=1}^2 p_i\vert \phi _i\rangle \langle \phi _i\vert\) with \(\theta\)independent \(p_i\), the Hermitian quantum Fisher information \(F_H\) [50,51,52] and nonHermitian quantum Fisher information \(F_{nH}\) reduce to,
Take GHZ states \(\vert \phi _{1,2}\rangle =\frac{1}{\sqrt{2}} (\vert a\rangle ^{\otimes N}\pm \vert b\rangle ^{\otimes N})\) as the two eigenstates and suppose that the parameter \(\theta\) is encoded into these states through 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 encoding states follow,
The rotation \(U(\theta )=e^{i\theta J_z}\) generates a relative phase \(N\theta\) between states \(\vert a\rangle ^{\otimes N}\) and \(\vert b\rangle ^{\otimes N}\). The GHZ states have been created with up to \(N=6\) \(^9Be^+\) ions [44] and \(N=14\) \(^{40}Ca^+\) ions [53] in a linear Paul trap. With these states, the encoded mixed state reads
and \(p_i\, (i=1,2)\) are independent of \(\theta .\)
It is easy to find that
Collecting these results, we obtain
The dependance of \(F_H\) and \(F_{nH}\) are shown in Fig. 1. We find that for mixed states the nonHermitian quantum Fisher information are always larger than the Hermitian quantum Fisher information, except the points \(p=0,0.5, 1.\) The quantum Fisher information of pure states \(F_H^p\) is larger than \(F_{nH}\) for \((2\sqrt{2})/4 . Especially, when the mixed state gets very close to pure states (\(p\rightarrow 0\) or \(p\rightarrow 1\)), \(F_{nH}\) has a very large value manifesting itself as a witness of transition from mixed states to pure states. This feature might be used to design sensors. It is worth addressing that at points \(p=0,1\), the state is pure. So, the quantum Fisher information should be calculated by the formula of pure states, which yield \(F_H=N^2\) and \(F_{nH}=0.25N^2.\) This suggests that a sensor working with mixed states very close to pure states is sensitive more than that working with pure states.
The variance \((\Delta \theta )^2\) of estimation can be given by the error propagation in Eq. (3), which is bounded by the quantum CramérRao bound defined through the quantum Fisher information as \((\Delta \theta )^2\ge \frac{1}{\nu F}, \quad F=F_H, \, F_{nH},\) where \(\nu\) is the number of repetitions of the experiment. Given a signal Hamiltonian and initial state, the bounds can be saturated by carefully chosen measurement described by operator A. Denoting \(A_1^{opt}\) as the optimal measurement and noticing Eqs. (4) and (8), we find that,
where p , \((1p)\) and \(\vert \phi _i\rangle\) are the eigenvalues and its corresponding eigenstates of \(\rho (\theta )\), see Eqs. (15) and (17).
Applying this optimal measurement \(A_1^{opt}\) to \((\Delta \theta )^2=\frac{\langle \Delta A_1^{opt \dagger }\Delta A_1^{opt}\rangle }{\partial \langle A_1^{opt\dagger }\rangle /\partial \theta \cdot \partial \langle A_1^{opt}\rangle /\partial \theta }\) (see Eq. (3)) and comparing it with numerical results, we find that the operator given in Eq. (18) indeed saturate the bound \(1/F_{nH}\), see Fig. 2. With this setup, measurement of nonHermitian operator would enhance the estimation precision for \(p<(2\sqrt{2})/4\) or \(p>(2+\sqrt{2})/4.\) The variance approaches to zero as the encoding states tend to pure states.
4 Parameter estimation on a nuclear magnetic resonance
Nuclear magnetic resonance (NMR) has long been at the forefront of precision tests of fundamental physics [54, 55], including the earliest efforts to search for a parity and timereversal violating permanent electric dipole moment of the neutron [56, 57] and recent applications in battery and sensors [58, 59].
Most recently, an experimental adiabatic quantum parameter estimation on the NMR quantum processor has been performed [60]. The authors demonstrated that with the optimized adiabatic path the precision can achieve the Heisenberg scaling. The physics behind this enhanced precision is the first order quantum phase transition and the speedup of the addibatic evolution in the sensor. This scheme possesses the advantages of easy implementation, robustness against decays, and tunable energy gaps. Here, we propose the other scheme with measuring nonHermitian operators and encoding unknown parameters in mixed states to enhance the precision of this setup.
The effective dimensionless Hamiltonian to describe the setup can be written as [60]
where \(B_z>0\) is the parameter to be estimated, \(B_x\) stands for the small transverse field assisting the estimation of \(B_z\), \(\sigma _0\) denotes the \(2\times 2\) identity operator, and \(\sigma _{x,z}\) are Pauli operators. The eigenstates of the effective Hamiltonian H can be written as
where \(\vert a\rangle =\vert 11\rangle\), \(\vert b\rangle =\frac{\vert 01\rangle +\vert 10\rangle }{\sqrt{2}}\) and \(\tan \theta =\frac{\sqrt{2}B_x}{1B_z}\) [61].
The quantum Fisher information of eigenstates \(\vert g\rangle \equiv \vert g(B_z)\rangle\) and \(\vert e\rangle \equiv \vert e(B_z)\rangle\) is equal as \(\langle \partial _{B_z} g\vert \partial _{B_z} g\rangle =\langle \partial _{B_z} e\vert \partial _{B_z} e \rangle\) and \(\langle g \vert \partial _{B_z} g \rangle =\langle e \vert \partial _{B_z} e \rangle =0\). With these considerations and the use of \(F_H^p=4(\langle \partial _{B_{z}}g\vert \partial _{B_{z}}g\rangle \vert \langle g\vert \partial _{B_{z}}g\rangle \vert ^{2})\) [60], the quantum Fisher information of eigenstates \(\vert g\rangle\) or \(\vert e\rangle\) reads,
where superscript p in \(F_{H}^p\) stands for the quantum Fisher information of pure states. The key point of Ref. [60] to enhance the estimation precision is that \(F_H^p\) has a large value when \(B_x\rightarrow 0\) near the critical point \(B_z \sim 1\), and the quantum fisher information scales as \(T^2\), where T is the cost—the time required for the adiabatic evolution.
The time cost T required for the adiabatic path is determined by the adiabatic condition, which makes T long, although the shortcut to adiabaticity can solve this difficulty in some circumstances. We here propose to encode the estimation parameter \(B_z\) into mixed states,
with \(B_z\)independent p. Considering \(\langle g\vert \partial _{B_z} g\rangle =\langle e\vert \partial _{B_z} e\rangle =0,\) we have
Straightforward calculation yields,
and
with
Substituting the last three equations into \(F_{nH}\), we obtain
It is always larger than the Hermitian quantum Fisher information of mixed state given by \(F_{H}=(2p1)^2F_H^p\). The dependence of quantum Fisher information on p and \(B_x\) is shown in Fig. 3. Note that \(F_{nH}>F_H\) and \(F_H^p>F_H\) holds for all p, while \(F_{nH}>F_H^p\) for \(p<\frac{1}{4}(2\sqrt{2})\) or \(p>\frac{1}{4}(2+\sqrt{2})\). In particular, \(F_{nH}\) has a very large value when the mixed states are very close to pure states. This feature provides a scheme for parameter estimation to enhance the precision. The estimation of \(B_z\) is bounded by \(1/F_{nH}\). To saturate the bound, we need to optimize the measurement A. It is easy to find that
Thus
From the error propagation function Eq. (3), we can find that the factor \(\frac{\gamma }{2} (12p)\frac{\partial \theta }{\partial B_z}\) can be dropped.
We perform numerically searching for the optimal measurement A. The result is shown in Fig. 4. We find that the variance of \(B_z\) becomes very small while p approaches to 0 as shown in Fig. 4 (a). This is reminiscent of sensors designed near quantum phase transition [60].
5 Measuring the nonHermitian optimal operator
In the last sections, we have derived an unknown quantum CramérRao bound defined by a quantum Fisher information for nonHermitian operators and demonstrated the theory with the phase estimation of trapped ions and a setup of adiabatic quantum parameter estimation. One may wonder how to measure the nonHermitian operators \(A_i^{opt}\, (i=1,2)\) to attain the bounds? Here, we follow the proposal in Refs. [39,40,41,42] to show it in experiment.
The expectation value of \(A_i^{opt}\,(i=1,2)\) in a quantum state \(\vert \phi _{in}\rangle\) is given by \(\langle \phi _{in}\vert A_i^{opt}\vert \phi _{in}\rangle\). It is complex in general and makes the nonHermitian operator \(A_i^{opt}\) unobservable in experiment. Nevertheless, recent studies shown that this obstacle can be overcome with the help of polar decomposition [62]. Namely, \(A_i^{opt} = UR\) with unitary operator U and Hermitian semidefinite operator R, \(R=\sqrt{(A_i^{opt})^\dagger A_i^{opt}}.\) This connects the average of nonHermitian operator \(A_i^{opt}\) with the weak value of Hermitian operator R as follows,
where \(\langle \phi \vert \equiv \langle \phi _{in} \vert U.\) It is wellknown that \(\frac{\langle \phi \vert R\vert \phi _{in}\rangle }{\langle \phi \vert \phi _{in}\rangle }\) is a weak value of the positivesemidefinite operator R, which can be measured directly in the weak measurement with successful probability depending on the post and preselection states [63]. However, there are other approaches to obtain the weak value via the weak measurement [40, 41] without postselection [42] in experiment. And the measurement of nonHermitian operator can be implemented with an interferometric technique [41]. The average of the nonHermitian operator can be determined through the intensity measured by a detector at the output without postselection, so that there are no concerns of successful probability. Next, before going into details of such a technique, we first find R and U to decompose the nonHermitian operator \(A^{opt}=UR\).
To be specific, let us consider the optimal operator \(A_2^{opt}\) given in Eq. (25). Straightforward derivation yields,
R can be rewritten as a project operator, \(R=\frac{1}{p} (\sigma _0+\frac{2p1}{1p}\vert e\rangle \langle e\vert )\), and U can be treated as a beam splitter. Both Hermitian operator R and unitary operator U depend on \(B_z\) and can be realized in experiments with the established \(B_z\) based on prior data [60, 64]. \(\sigma _0\) is the \(2\times 2\) identity operator. The schematic setup to measure the average of \(A_2^{opt}\) on state \(\rho (B_z)\) is illustrate in Fig. 5.
To simplify the notation, we write \(\vert g\rangle =\vert g(B_z)\rangle\) and \(\vert e\rangle =\vert e(B_z)\rangle\). To measure \(A^{opt}_2\), we first show that the average of \(A^{opt}_2\) in pure state \(\vert g\rangle\), \(\langle g\vert A^{opt}_2 \vert g\rangle \equiv \vert A^{opt}_2 \vert e^{i\zeta _g}\) can be readout from the intensity measured by the detector. Then, we extend this result to mixed state \(\rho (B_z) = p\vert g\rangle \langle g\vert +(1p)\vert e\rangle \langle e\vert .\) Namely, the average of \(A^{opt}_2\) on state \(\rho (B_z)\) can be read out from the intensity at the detector too.
In bases spanned by \(\vert a,1\rangle , \vert b,1\rangle ,\vert a,2\rangle , \vert b,2\rangle\) (\(\vert x,n\rangle \equiv \vert x\rangle \otimes \vert n\rangle , x=a,b; n=1,2\)) [65], these operation \(B_i (i=1 ,2), R, U^\dagger , e^{i\chi }\) can be represented by [47],
where \(R_{ab}=\langle a\vert R\vert b\rangle\), and similar notations hold for \(R_{aa}, R_{bb}, R_{ba}\) and \(U^\dagger _{xy}, x,y=a,b\). The input state that only occupies spatial state \(\vert 1\rangle\) can be written as,
The output state reads
Simple algebra yields,
The intensity the detector measures can be represented by
with
leading to
where \(\zeta _g\) is the argument of \(\langle g\vert A^{opt}_2 \vert g\rangle\), i.e., \(\zeta _g=\arg \langle g\vert A^{opt}_2 \vert g\rangle .\)
To measure the average of \(A^{opt}_2\) on mixed state \(\rho (B_z)=p\vert g\rangle \langle g\vert +(1p)\vert e\rangle \langle e\vert ,\) we replace the input state \(\vert g\rangle\) by \(\rho (B_z)\). The output state is then,
where \(Q =DB_2e^{i\chi }U^{\dagger }RB_1\). The intensity that the detector measures is
where \(I_g(\chi )=\vert \langle g\vert Q ^\dagger Q \vert g\rangle \vert ^2\) and \(I_e(\chi )=\vert \langle e\vert Q ^\dagger Q \vert e\rangle \vert ^2.\) Substituting Q into \(I_x\) (\(x=g,e\)), we find
and \(\langle x\vert A_2^{opt}\vert x\rangle =\vert \langle x\vert A_2^{opt}\vert x\rangle \vert e^{i\zeta _x}.\) In experimental implementations, the intensity \(I(\chi )\) together with \(\langle x\vert R^2\vert x\rangle\) (the average of Hermitian operator \(R^2\) on state \(\vert x\rangle\)) can determine the average of nonHermitian operator \(A_2^{opt},\) as both \(\vert \langle x\vert A\vert x\rangle \vert\) and \(\zeta _x\) can be inferred from the intensity \(I(\chi )\).
6 Conclusion
The framework of quantum mechanics in which observables are associated only with Hermitian operators constrains the optimization in quantum parameter estimation. Considering the fact that in the past two decades the nonHermitian physics has attracted fast growing interest in various research field, we have extended measurements from Hermitian operators to all operators including nonHermitian ones. With this extension, a previously unknown expression for quantum Fisher information has been derived and a new quantum CramérRao bound has been established. The saturation of the bound has been analyzed and the optimal measurement to attain the bounds have been given. The theory was elucidated with two experimentally feasible systems. In the example of phase estimation, we find that mixed states can further enhance the sensitivity of phase estimation with measurements of nonHermitian operators. While in the example of adiabatic quantum parameter estimation for the estimation of magnetic fields on a nuclear magnetic resonance, the variance of the magnetic field under estimation can reach arbitrary small values depending on the encoding states. A setup to measure nonHermitian operator was also proposed.
Availability of data and materials
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References

C. Helstrom, Minimum meansquared error of estimates in quantum statistics. Phys. Lett. A 25, 101 (1967)

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976)

A.S. Holevo, Probabilistic and Statistical Aspect of Quantum Theory (NorthHolland, 1982)

C.L. Degen, F. Reinhard, P. Cappellaro, Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017)

L. Pezzè, A. Smerzi, M.K. Oberthaler, R. Schmied, P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys. 90, 035005 (2018)

C.M. Caves, Quantummechanical noise in an interferometer. Phys. Rev. D 23, 1693 (1981)

A.W. Chin, S.F. Huelga, M.B. Plenio, Quantum metrology in nonMarkovian environments. Phys. Rev. Lett. 109, 233601 (2012)

R. DemkowiczDobrzański, M. Jarzyna, J. Kolodyński, Quantum limits in optical interferometry. Prog. Opt. 60, 345 (2015)

V. Giovannetti, S. Lloyd, L. Maccone, Advances in quantum metrology. Nat. Photonics 5, 222 (2011)

A. Luis, Breaking the weak Heisenberg limit. Phys. Rev. A 95, 032113 (2017)

K. Bai, Z. Peng, H.G. Luo, J.H. An, Retrieving ideal precision in noisy quantum optical metrology. Phys. Rev. Lett. 123, 040402 (2019)

R.A. Fisher, Theory of statistical estimation. Proc. Camb. Soc. 22, 700 (1925)

C.L. Matson, A. Haji, Biased CramérRao lower bound calculations for inequalityconstrained estimators. J. Opt. Soc. Am. A 23, 2702 (2006)

H.P. Yuen, M. Lax, Multipleparameter quantum estimation and measurement of nonselfadjoint observables. IEEE Trans. Inf. Theory 19, 740 (1973)

C.M. Bender, S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

C.M. Bender, D.C. Brody, H.F. Jones, B.K. Meister, Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98, 040403 (2007)

A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. VolatierRavat, V. Aimez, G.A. Siviloglou, D.N. Christodoulides, Observation of PTsymmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)

V.V. Konotop, J. Yang, D.A. Zezyulin, Nonlinear waves in PTsymmetric systems. Rev. Mod. Phys. 88, 035002 (2016)

L. Feng, R. ElGanainy, L. Ge, NonHermitian photonics based on paritytime symmetry. Nat. Photon. 11, 752 (2017)

R. ElGanainy, K.G. Makris, M. Khajavikhan, Z.H. Musslimani, S. Rotter, D.N. Christodoulides, NonHermitian physics and PT symmetry. Nat. Phys. 14, 11 (2018)

M.A. Miri, A. Alú, Exceptional points in optics and photonics. Science 363, eaar7709 (2019)

S.K. Özdemir, S. Rotter, F. Nori, L. Yang, Paritytime symmetry and exceptional points in photonics. Nat. Mater. 18, 783 (2019)

R. Hamazaki, K. Kawabata, M. Ueda, NonHermitian manybody localization. Phys. Rev. Lett. 123, 090603 (2019)

M.V. Berry, Physics of nonhermitian degeneracies. Czech. J. Phys. 54, 1039 (2004)

W.D. Heiss, The physics of exceptional points. J. Phys. A 45, 444016 (2012)

O. Sigwarth, C. Miniatura, Time reversal and reciprocity. AAPPS Bull 32, 23 (2022)

C. Wu, A. Fan, S.D. Liang, Complex Berry curvature and complex energy band structures in nonHermitian graphene model. AAPPS Bull 32, 39 (2022)

A. Sergi, K.G. Zloshchastiev, Quantum Entropy of Systems described by nonHermitian Hamiltonians. J Math Phys 3, 033102 (2016)

D. Li, C. Zheng, NonHermitian Generalization of Rényi Entropy. Entropy 24, 1563 (2022)

N. Moiseyev, NonHermitian Quantum Mechanics (Cambridge University Press, Cambridge, 2011)

D.C. Brody, E.M. Graefe, Information geometry of complex Hamiltonians and exceptional points. Entropy 15, 3361 (2013)

J. Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for singleparticle detection. Phys. Rev. Lett. 112, 203901 (2014)

Z.P. Liu, J. Zhang, S.K. Özdemir, B. Peng, H. Jing, X.Y. Lü, C.W. Li, L. Yang, F. Nori, Y.X. Liu, Metrology with PTsymmetric cavities: enhanced sensitivity near the PTphase transition. Phys. Rev. Lett. 117, 110802 (2016)

W. Chen, S.K. Özdemir, G. Zhao, J. Wiersig, L. Yang, Exceptional points enhance sensing in an optical microcavity. Nature 548, 192 (2017)

H. Hodaei, A.U. Hassan, S. Wittek, H. GarciaGracia, R. ElGanainy, D.N. Christodoulides, M. Khajavikhan, Enhanced sensitivity at higherorder exceptional points. Nature 548, 187 (2017)

P. Djorwe, Y. Pennec, B. DjafariRouhani, Exceptional point enhances sensitivity of optomechanical mass sensors. Phys. Rev. Applied 12, 024002 (2019)

X. Mao, G.Q. Qin, H. Yang, H. Zhang, M. Wang, G.L. Long, Enhanced sensitivity of optical gyroscope in a mechanical paritytimesymmetric system based on exceptional point. New J. Phys. 22, 093009 (2020)

Y. Chu, Y. Liu, H. Liu, J. Cai, Quantum sensing with a singleQubit pseudoHermitian system. Phys. Rev. Lett. 124, 020501 (2020)

A.K. Pati, U. Singh, U. Sinha, Measuring nonHermitian operators via weak values. Phys. Rev. A 92, 052120 (2015)

G. Nirala, S.N. Sahoo, A.K. Pati, U. Sinha, Measuring average of nonHermitian operator with weak value in a MachZehnder interferometer. Phys. Rev. A 99, 022111 (2019)

S.N. Sahoo, S. Chakraborti, A.K. Pati, U. Sinha, Quantum state interferography. Phys. Rev. Lett. 125, 123601 (2020)

A.A. Abbott, R. Silva, J. Wechs, N. Brunner, C. Branciard, Anomalous weak values without postselection. Quantum 3, 194 (2019)

C. Zheng, Universal quantum simulation of singlequbit nonunitary operators using duality quantum algorithm. Sci Rep 11, 3960 (2021)

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R.B. Blakestad, J. Chiaverini, D.B. Hume, W.M. Itano, J.D. Jost, C. Langer, R. Ozeri, R. Reichle, D.J. Wineland, Creation of a sixatom ‘Schrödinger cat’ state. Nature (London) 438, 639 (2005)

W. Zhong, X.M. Lu, X.X. Jing, X. Wang, Optimal condition for measurement observable via errorpropagation. J. Phys. A Math. Theor. 47, 385304 (2014)

M. Hotta, M. Ozawa, Quantum estimation by local observables. Phys. Rev. A 70, 022327 (2004)

J. Li, H. Liu, Z. Wang, X. Yi, CramérRao bound and quantum parameter estimation with nonHermitian systems. (2021) arXiv:2103.07099

C.W. Helstrom, CramérRao inequalities for operatorvalued measures in quantum mechanics. Int. J. Theor. Phys. 8, 361 (1973)

J. Liu, X.X. Jing, W. Zhong, X.G. Wang, Quantum Fisher information for density matrices with arbitrary ranks. Commun. Theor. Phys. 61, 45 (2014)

B.R. Frieden, Fisher information, disorder, and the equilibrium distributions of physics. Phys. Rev. A 41, 4265 (1990)

S.L. Braunstein, C.M. Caves, Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439 (1994)

J. Liu, H. Yuan, X.M. Lu, X. Wang, Quantum Fisher information matrix and multiparameter estimation. J. Phys. A: Math. Theor. 53, 023001 (2020)

T. Monz, P. Schindler, J.T. Barreiro, M. Chwalla, D. Nigg, W.A. Coish, M. Harlander, W. Hänsel, M. Hennrich, R. Blatt, 14Qubit Entanglement: Creation and Coherence. Phys. Rev. Lett. 106, 130506 (2011)

M. Safronova, D. Budker, D. DeMille, D.F. Jackson Kimball, A. Derevianko, C.W. Clark, Search for new physics with atoms and molecules. Rev. Mod. Phys. 90, 025008 (2018)

D. DeMille, J.M. Doyle, A.O. Sushkov, Probing the frontiers of particle physics with tabletopscale experiments. Science 357, 990 (2017)

E.M. Purcell, N.F. Ramsey, The Possibility of Electric Dipole Moments for Elementary Particles and Nuclei. Phys. Rev. 78, 807 (1950)

J.H. Smith, E.M. Purcell, N.F. Ramsey, Limit to the electric dipole moment of the neutron. Phys. Rev. 108, 120 (1957)

O. Pecher, J. CarreteroGonzsalez, K.J. Grinth, C.P. Grey, Materials’ methods: NMR in battery research. Chem. Mater. 29, 213 (2017)

K. S. Liu, A. Henning, M. W. Heindl, R. D. Allert, J. D. Bartl, I. D. Sharp, R. Rizzato, D. B. Bucher, Surface NMR using quantum sensors in diamond. (2021) arXiv: 2103.15955

R. Liu, Y. Chen, M. Jiang, X. Yang, Z. Wu, Y. Li, H. Yuan, X. Peng, J. Du, Experimental adiabatic quantum metrology with the Heisenberg scaling. (2021) arXiv: 2102.07056

J. Zhang, X. Peng, N. Rajendran, D. Suter, Detection of quantum critical points by a probe qubit. Phys. Rev. Lett. 100, 100501 (2008)

B.C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer, New York, 2015)

Y. Aharonov, D.Z. Albert, L. Vaidman, How the result of a measurement of a component of the spin of a spin1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)

A. Fujiwara, Strong consistency and asymptotic efficiency for adaptive quantum estimation problems. J. Phys. A 44, 079501 (2011)

G. Long, Collapsein and collapseout in partial measurement in quantum mechanics and its wise interpretation. Sci China Phys Mech Astron 64, 280321 (2021)
Acknowledgements
We thank Dorje Brody for first reading the manuscript and Xiaoming Lu for helpful discussions.
Funding
This work was supported by National Natural Science Foundation of China (NSFC) under Grants No. 12175033 and National Key R &D Program of China No. 2021YFE0193500.
Author information
Authors and Affiliations
Contributions
X. X. Yi developed the concept for the article. Jianning Li and X. X. Yi performed the specific calculation. Haodi Liu and Zhihai Wang checked the algebra and discussed the results. Finally, X. X. Yi and Jianning Li drafted the work and critically revised the work. All authors read and approved the final manuscript.
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A: The uncertainty relation Eq. (1) for mixed states
Appendix A: The uncertainty relation Eq. (1) for mixed states
Here, we show that the uncertainty relation Eq. (1) holds for mixed states. We prove this by introducing an ancilla a, such that a mixed state \(\rho =\sum _j q_j\vert \psi _j\rangle \langle \psi _j\vert\) can be purified to be
and the state of the system is obtained by tracing \(\vert \Psi ^\prime \rangle\) over the ancilla, \(\rho =\text {Tr}_a\vert \Psi ^\prime \rangle \langle \Psi ^\prime \vert .\) With this consideration, Eq. (1) can be straightforwardly extended to the composite system consisting of the system and the ancilla,
Here, \(I_a\) is the identity operator of ancilla a. Noticing \(\langle \Psi ^\prime \vert \Delta A^\dagger \otimes I_a \Delta A\otimes I_a \vert \Psi ^\prime \rangle =\text {Tr}(\rho \Delta A^\dagger \Delta A)\) and denoting \(\text {Tr}(\rho \Delta A^\dagger \Delta A)=\langle \Delta A^\dagger \Delta A \rangle\) with \(\text {Tr}\) representing the trace over the system, we finish the proof of the weaker uncertainly relation in the main text.