Exponential sensitivity revival of noisy non-Hermitian quantum sensing with two-photon drives

Unique properties of multimode non-Hermitian lattice dynamics can be utilized to construct exponentially sensitive sensors. However, the impact of noise remains unclear, which may severely degrade their sensitivity. We analytically characterize and highlight the impact of loss and gain on the sensitivity revival and stability of non-Hermitian sensors. Defying the general belief that the superiority of quantum sensing will vanish in the presence of loss, we find that by proactively tuning the loss, the exponential sensitivity can be surprisingly regained when the sensing dynamics is stable. Furthermore, we prove that gain is crucial to fully revive the ideally exponential sensitivity and to ensure the stability of non-Hermitian sensing by making a balanced loss and gain. Our paper opens a way to significantly enhance the sensitivity by proactively tuning the loss and gain, which may promote future quantum sensing and quantum engineering.

The pursuit of high sensitivity is a fundamental objective in developing sensing technology.
Recent progress has shown that the intriguing degeneracy property of NH systems can be employed to enhance the sensitivity of sensors operating at finely tuned exceptional points (EPs), where the coalesced eigenenergies have a diverging susceptibility to small perturbations [19][20][21][22][23][24][25][26].However, to assess the performance of sensors based on EPs, we should also take into account of the effect of the coalesced eigenstates, which may counteract the diverging susceptibility of eigenenergies [20,21].Other distinct properties of NH systems have also been harnessed to enhance the sensitivity of NH sensors, which do not necessarily work at EPs.As studied in [31], nonreciprocity [27][28][29][30][31][32] can be identified as a powerful resource for sensing, since it allows one to exceed the fundamental bounds constraining conventional, reciprocal sensors [31].Remarkably, a class of sensors having exponential sensitivity have been theoretically proposed [33][34][35][36][37].The drastic enhancements rely upon the strikingly anomalous sensitivity to the boundary conditions of NH systems.Furthermore, the implication of optimizing controllable parameters in attaining an exponential enhancement was investigated in [17,34].
In practical applications the existence of noise is unavoidable, which may severely degrade the performance, such as the stability and sensitivity of NH sensors.Since most sensing schemes are measured at equilibrium states, a stable sensing dynamics is a fundamental requirement for achieving these high precision sensing.Loss noise has an essential impact on the attainable sensitivity in quantum sensing.For conventional sensors, it is well-known that by using quantum strategies, the precision can be scaled as 1/N in terms of the number N of quantum resources for noiseless processes [38][39][40][41][42][43].However, it has been demonstrated in [44][45][46][47] that even a weak loss noise can quickly degrade the precision from 1/N to 1/ √ N , independently of the initial state of the probes and even regardless of the use of adaptive feedback.This is very frustrating and it has been a general belief that advantages of quantum sensing will soon vanish in the presence of loss.Gain has been demonstrated to be a necessary ingredient to have an enhanced signal power in NH sensing [31], whereas too much gain may result in an unstable sensing dynamics.
Since manipulating loss and gain has become feasible [48][49][50][51][52], it is of vital importance to understand the impact of loss and gain on the sensitivity and stability of NH sensing.Different from existing results, in this work, our aim is to find out whether we can achieve exponentially enhanced and stable NH quantum sensing by proactively tuning the loss and gain.In general, the loss and gain bring about two effects for the NH sensing dynamics: one is the diffusion noise that may be further amplified during the sensing and then severely degrade the sensitivity; the other one is the dissipative drift that may lead to system instability.
We find conditions to fully recover the ideally noiseless sensitivity for noisy NH quantum sensing in both the perturbation regime and the case beyond linear response.To be specific, we discover that the coupling of loss plays a pivotal role in obtaining exponential sensitivity revival in our setting.Counterintuitively, we find that by proactively tuning the loss couplings properly, an exponential signal-to-noise ratio (SNR) can be surprisingly regained when the sensing dynamics is stable.We further point out that balanced gain and loss is vital to fully recover the ideally noiseless sensitivity and to ensure the stability of the NH sensing dynamics.
We also analyze the robustness of the sensitivity under the designed loss and gain and provide a guideline on how to realize our strategy in practice through concrete examples.

II. SETUP OF NOISY NH SENSORS
In the absence of noise, an exponentially enhanced quantum sensing scheme was proposed in [33] based on NH lattice dynamics.In this work, we adopt the model in [33], and then further investigate the impact of loss and gain on the sensitivity and stability of NH sensing.
A generic multimode noisy NH setup is illustrated in Fig. 1(a).Consider a 1-dimensional array of N bosonic modes, and let âi denote the mode annihilation operator on the ith site.Our aim is to detect a small perturbation ϵ of a perturbation Hamiltonian ϵ V , where V is a system operator.In Fig. 1(a) and Fig. 1(b), V = â † N âN , and thus the aim in this case is to estimate a small change ϵ in the resonance frequency of the last site.A general measurement strategy is to couple mode 1 to an input-output waveguide with rate κ, and then inject a coherent drive with amplitude β at the resonant frequency of the mode.The reflected signal is measured by a homodyne detection [56] to infer ϵ [31][32][33][34].In the rotating frame set by the mode resonance frequency, the system Hamiltonian reads where ω depicts the hopping between neighbor modes and ∆ describes the nearest-neighbour two-photon drive [33,57].We assume that w > ∆ > 0. Up to now, this is the ideal model utilized in [33].
To fully account for the noise, we couple the modes to N Z loss and N Y gain baths, which are mutually independent.Without loss of generality, the coupling rates are described by the real matrices Z and Y , . . .(a) The setup consists of a 1-dimensional chain of N bosonic modes.The parameter to be detected is ϵ, which represents a small change in the resonance frequency of the last site.
To detect ϵ, a coherent drive β accompanied by quantum noise Bin is injected into the chain at mode 1 through an input-output waveguide with coupling rate κ.The reflected field is measured by homodyne detection.The modes are coupled via nearest neighbour hopping w and coherent twophoton drive ∆.To account for the noise, couplings between the modes and the loss/gain baths (blue solid/red dashed) are included.The coupling rate between the ith mode and the jth loss (gain) bath is described by Zij (Yij).(b) The nonreciprocal amplification between modes can be described by two N -site NH Hatano-Nelson chains [53][54][55] with effective hopping amplitude J and amplification factor A. For the top (bottom) X (P) chain, hopping to the right is a factor of e 2A larger (smaller) than hopping to the left.The last modes of the two chains are coupled due to the presence of small tunneling with amplitude ϵ, allowing the signal to be transmitted between the two chains.
respectively.The element Z ij (Y ij ) of the loss (gain) coupling matrix Z (Y ) describes the coupling rate between the ith mode and the jth loss (gain) bath.Using the standard input-output theory [58], the total effective Hamiltonian (see Appendix A for details) reads where Ĥκ describes the damping of mode 1 due to the coupling with the waveguide, while ĤG and ĤL describe the damping owing to the coupling with the gain and loss baths, respectively.The Heisenberg-Langevin equations (see Appendix A for details) read Here, Bin denotes the quantum noise entering from the waveguide, and Ĉin j ( Din j ) are quantum noises arising from the gain (loss) process of the baths.To ensure Markovian dynamics, Bin , Ĉin j and Din j are assumed to be quantum Gaussian white noise satisfying: where Q ∈ { Bin , Ĉin j , Din j }, and there are no correlations between different noise operators.Here, nth is the number of thermal quanta in the input field.Hereafter, we focus on the vacuum noise, namely, nth Q = 0. To see clearly how the signal is amplified, we turn to the picture of canonical quadratures xn and pn related with ân via ân = (x n + ip n )/ √ 2. Define quadrature vectors X = (x 1 , x2 , . . ., xN ) ⊤ and P = (p 1 , p2 , . . ., pN ) ⊤ , respectively.Then the Heisenberg-Langevin equations (see Appendix A for details) turn to Ẋ (5) Here, h X and h P represent the ideally noiseless dynamical matrices of the quadratures X and P, respectively, which read where J = √ w 2 − ∆ 2 denotes the hopping amplitude and the amplification factor A is defined via Due to h X and h P , we can find that for the top X (bottom P) chain in Fig. 1(b), hopping to the right is a factor of e 2A larger (smaller) than hopping to the left.The commutators with the perturbation Hamiltonian V are defined in an element-wise way, e.g., The coherent input vector and Ωin denotes the quantum noise vector (see Appendix A for details), whose elements are described by for i ∈ {1, 2, . . ., N }, with Bin = , and Din j = Din j,X +i Din .

III. SNR PER PHOTON
We now introduce the figure of merit that evaluates the performance of sensing.
From the input-output theory, the output field Bout (t) reads To estimate the perturbation ϵ, we should integrate the output field over a long time period [0, τ ].The corresponding temporal mode is defined by which is a canonical bosonic annihilation operator.For the perturbation ϵ which is the p-quadrature of the temporal output field B [33].
Let us first consider the case when ϵ is infinitesimal.Define the signal power in terms of the optimal observable M as and the noise power as Here, the average ⟨•⟩ ϵ represents the mean with the steady state whose dynamics is governed by Ĥ[ϵ].Since ϵ is infinitesimal, we can only consider the zeroth order of ϵ for the noise power.The SNR is defined by Since the dominant term of the SNR(ϵ) with respect to ϵ is the same as that of the quantum Fisher information when |β| ≫ 1 [31][32][33][34], below we use the SNR(ϵ) to evaluate the performance of NH sensors.
To make a fair comparison, the resources used in the measurement should be constrained.Following [33,34] we take the SNR per photon denoted by as the figure of merit, where the total average photon number is in the large-drive limit.Following the same reasoning as that of the noise power, only the zeroth order of ntot in ϵ is concerned.

IV. SNR OF NH SENSORS
We now derive the SNR for NH sensors.In the following, we take the perturbation Hamiltonian V in Eq. ( 5) as V = â † N âN , and let the number of modes N be odd.When N is even, the scaling of SNR in terms of A and N is the same as that when N is odd, except that the corresponding preceding multiplicative factors are different.
We derive the signal power S, noise power N , and the total average photon number ntot in the presence of loss and gain as follows (see Appendix B for details): with information matrices and In the absence of loss and gain, namely Z = 0 and Y = 0, it was demonstrated in [33] that SNR(ϵ) ∝ exp{2A(N −1)} implying that an exponentially enhanced sensitivity can be obtained.The key idea is illustrated in Fig. 1(b).To detect ϵ, a real drive is injected at site 1 to excite the X chain, then the wavepacket propagates rightwards.When it reaches the last site, the signal power grows with a factor of e 2A(N −1) .Then at site N , due to the perturbation, the wavepacket scatters off the boundary and changes to pN quadrature.It then propagates backwards to site 1 amplifying the signal.If the p-quadrature of the output field is measured, then a total amplification factor e 4A(N −1) of the signal power is obtained.While for the total average photon number, it amplifies only along one traversal of the chain obtaining an amplification factor of e 2A(N −1) .As for the noise power, for the ideal case of zero internal loss and gain, the noise power is the same as that of the input field, namely, N (0) = 1/2.Combining this with the amplification factors of the signal power and the total average photon number, the exponentially large factor e 2A(N −1) of SNR(ϵ) can be explained.
In the presence of loss and gain, owing to the nonreciprocal dynamics governed by (Q X ) −1 and (Q P ) −1 , the noise power may be significantly amplified in general, which satisfies N ∝ e 2A(N −1) , causing the vanishing of the ideally exponential sensitivity.In addition, from Eq. ( 5), the net noise matrix (Y Y ⊤ − ZZ ⊤ ) may cause the sensing dynamics to become unstable.This implies that noise may lead to at least one of the eigenvalues of the noisy NH dynamical matrix (h X + Y Y ⊤ − ZZ ⊤ ) or (h P + Y Y ⊤ − ZZ ⊤ ) to sit in the right half plane.In this case, Eq. ( 5) has no steady state and the expectation of some of the canonical quadratures will diverge to infinity, which is physically meaningless [59] (see Appendix C for details).

V. TUNING LOSS AND GAIN
We now present how to achieve exponentially enhanced and stable NH sensing by proactively tuning the loss and gain.
It is widely believed that introducing gain is necessary to address the sensitivity revival problem in the presence of loss.However, we find that the loss Z plays a pivotal role.To this end, consider the case where there is only loss and no gain, namely, Y = 0. Given the dynamical matrix h P , we prove that if the loss couplings Z can be tuned such that all its columns lie in the linear space spanned by the second column h P •2 through the last column h P •N of the dynamical matrix h P , then we can revive the ideally exponential sensitivity when the sensing dynamics is stable (see Appendix D for details).In short, to attain an exponential sensitivity, the loss coupling matrix Z should meet This finding is remarkable as it defies the general belief that even a weak loss will quickly lead to vanishing of quantum advantages in high precision sensing.To illustrate the above, consider a simplest 3-site NH sensor.In the ideal case, namely there is no loss and gain, the ideal SNR ∝ e 4A as depicted by the black line in Fig. 2(a).Assume now that there are two loss baths in total (N Z = 2) and consider two different loss couplings described by and respectively.It is clear that Z 1 and Z 2 do not obey (C1).With parameters α = 0.5, κ = 10, ω = 10 5 and J = ω 2e A e 2A +1 , we depict the SNR under Z 1 (dashed blue) and Z 2 (dotted green) in Fig. 2(a).Here, we only care about the amplification factor A's that make the dynamics stable.Note that the SNR under Z 1 approaches a constant as A increases, while under Z 2 the SNR grows as A increases, but is still much smaller than the ideal case.In Fig. 2(b), we illustrate the SNR under Z 1 with different α's.If α = 0, it is the ideal case (black).
To revive the ideally exponential sensitivity, for Z 1 we proactively add an exponentially small coupling Z 32 = e −A between the third mode and the second loss bath, while for Z 2 we proactively add an exponentially large coupling Z 12 = −e A between the first mode and the second bath.The resulting loss coupling matrix becomes which satisfies (C1).The SNR under Z (red circle) is illustrated in Fig. 2(a).It is clear that the ideally exponential sensitivity is regained in the stable region and the best SNR obtained under Z is much better than those attained under Z 1 and Z 2 .When tuning Z 1 to be Z, for different values of α, the range of A that makes the system stable is different, so is the best sensitivity that can be regained.In Fig. 2(b), we depict the corresponding best sensitivity that can be attained by colored dots on the ideal black line for different α's.
It is clear that, as α increases, the best sensitivity that can be attained decreases.
To address the instability problem and fully regain the ideally exponential sensitivity, we can further introduce gain, and tune Y such that it meets the balanced condition We prove that under (C1) and (C2), an exponential enhancement can be fully revived for noisy NH sensing, that is, SNR ∝ e 2A(N −1) (see Appendix E for details).In this case, the range of A in Fig. 2 can, in principle, be arbitrarily large, which of course depends on the parameters of the real setup.
We now consider the robustness of the sensitivity under (C1) and (C2).To this end, we can first reverse the roles played by the loss coupling matrices Z and Z 1 .Assume that when tuning the desired loss Z, we obtain Z 1 instead of Z, namely the desired loss rate Z 32 = e −A is set to be 0 in practice.From Fig. 2(a), it is clear that even though there is only an exponentially small imperfection, the sensitivity can be greatly reduced when A ∈ [2, 4.6].
Second, assuming it can be verified that to ensure a stable sensing dynamics, a necessary condition for γ is |γ| < κe −A(N −1) (see Appendix C for details).In this case, there is a striking tradeoff between the enhancement of the sensitivity and the exponential decrement of the robust stability.This owes to the fact that there is "no free lunch".For a sensor having exponential sensitivity, it must have an inherent highly nonlinear amplification mechanism, such as that described by h P .Thus, fine tunings at certain key points are inevitable.Otherwise, the residual noise may either be greatly amplified or lead to system instability, both of which will severely degrade the sensitivity.Using a similar analysis to the above, we can determine which coupling rates of loss and gain need to be finely tuned.In this way, we can adjust these couplings as well as possible to construct a noisy NH sensor with ultra-high sensitivity.
The feasibility of our approach is well supported by the current capability of engineering loss and gain in optics in a controlled manner [48][49][50].By proactively tuning the loss and gain couplings, our proposal opens a new way to significantly enhance the sensitivity in the presence of loss and gain.

VI. REGIME BEYOND LINEAR RESPONSE
We now relax the assumption that the perturbation is infinitesimal.When the parameter to be detected, ϵ 0 , is not infinitely small, not only the linear response of ϵ 0 , but all orders in ϵ 0 of the output field should be calculated.
Note that in the regime beyond linear response, to revive the ideal sensitivity, (C3) and (C4) are slightly stricter than (C1), which constrains the loss in linear response.Here, (C3) means that the columns of Z should reside in the linear space generated by the second column through the (N − 1)th column of the dynamical matrix h P , while (C4) implies that the columns of Z should be orthogonal to the N th row of the ideal information matrix (h X ) −1 .

VII. CONCLUSION
We have investigated the ideal sensitivity revival and the stability of noisy NH quantum sensing.We present a strategy to proactively tune the loss and gain couplings to construct a stable NH sensor achieving an exponential sensitivity.We find that the loss is key to revive the sensitivity, and that balanced gain and loss are crucial to fully regain the ideal sensitivity and to ensure a stable NH sensor, no matter if the parameter is infinitesimal or in the regime beyond linear response.We also point out that to design a noisy sensor with ultra-high sensitivity, fine tunings are inevitable at certain key points.Our proposal opens a new way to enhance the sensitivity of noisy sensors by proactively tuning the loss and gain, and may have potential applications in quantum sensing and quantum engineering.

APPENDICES
To make the paper self-contained, this appendices are organized as follows.We first describe the total Hamiltonian of the sensor and derive the Heisenberg-Langevin equations in Appendix A .Then we derive the signal-to-noise ratio (SNR) per photon in Appendix B. The real matrix Z (Y ) describes the coupling between the system and the loss (gain) bath.In Appendix C we derive the necessary conditions to ensure the stability of the dynamics if the loss and gain are unbalanced.In Appendix D we prove that when the gain coupling Y = 0, the loss coupling Z satisfies condition (C1), and if the dynamics is stable, the signal power S, noise power N , and the total average photon number ntot (0) are the same as those of the ideal noise-free case.The SNR per photon under conditions (C1) and (C2) are given in Appendix E. In Appendix F we calculate the SNR per photon in the regime beyond linear response.The calculations of the elements of H[ϵ] −1 and H[ϵ 0 ] −1 are shown in Appendix G and Appendix H, respectively.annihilation operator of the jth loss bath mode with wave number k.The real matrix Z (Y ) depicts the coupling between the system and the loss (gain) bath.
The Heisenberg equations of motion for the cavity modes and the field modes are The solutions of the last three equations in Eq. (A10) are bk = e −ik(t−t0) bk (t Substituting Eq. (A11) into the first equation of Eq. (A10) yields where we have defined and used the equations dke −ik(t−t ′ ) = 2πδ(t−t ′ ) and To ensure the Markovian nature of the entire dynamics, Bin , Ĉin j and Din j are assumed to be quantum Gaussian white noise: ), and ⟨Q(t)Q(t ′ )⟩ = 0, where Q ∈ { Bin , Ĉin j , Din j }, and there are no correlations between different noise operators.Here, nth Q is the number of thermal quanta in the input field.Therefore, the Heisenberg-Langevin equations can be expressed as To see how the signal is amplified, it is better to turn to the picture of canonical quadratures xn and pn defined via ân = xn+i pn √ 2 .Then the corresponding Heisenberg-Langevin equations in terms of xn and pn read By defining Bin = , Ĉin j = Ĉin j,X +i Ĉin , Din j = Din j,X +i Din , and let J = √ w 2 − ∆ 2 and exp{2A} = w+∆ w−∆ , the above equation can be described by Z n,j Din j,P .

(A16)
By defining the quadrature vectors X = (x 1 , x2 , . . ., xN ) ⊤ and P = (p 1 , p2 , . . ., pN ) ⊤ , we can convert the Heisenberg-Langevin equations into a compact form: Here, the dynamical matrices h X and h P are and the quantum noise vectors Ωin are Appendix B: Derivations of the SNR per photon In this appendix, we calculate the signal power, noise power, and the total average photon number when ϵ is infinitesimal.
According to the Heisenberg-Langevin equations and the definition of the perturbation Hamiltonian ϵ V = ϵâ † N âN , we have (B1) Since the system is stable, for sufficiently long time, xn and pn can be described as where Using Dyson's equation and keeping it up to the first order in ϵ, we have The detailed calculation of the elements of H[ϵ] −1 can be found in Appendix G.
It can be computed that the steady state of mode 1 is where we have used From the definition of M, we have where {p i } are the roots of the characteristic equation (C3) or the poles of the transfer function (C2), and {K i } depend on the initial condition and the zero locations of the transfer function (C2).Here, we have assumed that all the roots of Eq. (C3) are distinct for simplicity.If any poles are repeated, the corresponding coefficient K i in Eq. (C4) must include a polynomial in t, but the conclusion is the same.
The system is stable if and only if (necessary and sufficient condition) every term in Eq. (C4) goes to zero as t → ∞: This will happen if all the eigenvalues of the system matrix A are strictly in the left half plane, where Re{p i } < 0. If the system has any poles in the right half plane, then y(t) → ∞, it is unstable.Thus, we can determine the stability of a system by determining whether all the eigenvalues of the system matrix A sit in the left half plane.A well-known tool to do this is the Routh's stability criterion (e.g., see Sec. 3.6.3 of [56] of the main text).We now turn back to our work to employ Routh's stability criterion to derive the necessary conditions of the elements of (Y Y ⊤ − ZZ ⊤ ) to ensure the stability of the system.Case 1:

𝜅 𝛽
The noisy setup is illustrated in Fig. 3(a), where there is an effective coupling γ between the first and the (N − 1)-th modes.The characteristic polynomial of (h X + Y Y ⊤ − ZZ ⊤ ) can be calculated in a recursive way as where and [x] is the integer function which returns the largest integer not larger than x.
According to Routh's stability criterion, a necessary (but not sufficient) condition for stability is that all the coefficients of λ are positive.It can be computed that the coefficient of the first order with respect to λ in Eq. (C5) is It can be verified that when e −A(N −2) is sufficiently small, to ensure Eq. (C7) be positive, γ should satisfy γ 1 < γ < γ 2 , where Thus, if e −A(N −2) is sufficiently small, to ensure the stability of (h X + Y Y ⊤ − ZZ ⊤ ), a necessary condition for γ is After a similar analysis with the characteristic polynomial of (h P + Y Y ⊤ − ZZ ⊤ ), it can be verified that if e −A(N −2) is sufficiently small, to ensure the stability of (h P + Y Y ⊤ − ZZ ⊤ ), a necessary condition is According to Routh's stability criterion, a necessary condition for (h X + Y Y ⊤ − ZZ ⊤ ) to be stable is that all the coefficients of λ of the characteristic polynomial are positive.Thus, from c 0 > 0, we have Then the elements of (h X ) −1 are related to the elements of h −1 as (h X ) −1 i,j = h −1 i,j e A(i−j) ; (G5) while the elements of (h P ) −1 relate to the elements of h −1 as (h P ) −1 i,j = h −1 i,j e A(j−i) .(G6) We now introduce how to compute the elements of h −1 .According to the definition of h, we have where

(G8)
Simplifying the above recursive formula, we have The other elements can be computed in a similar way.In this section, we calculate the elements of H[ϵ 0 ] −1 under (C2).Since the parameter ϵ 0 is not infinitesimal, we have to consider all orders of ϵ 0 . Define

FIG. 1 .
FIG. 1. (Color online) A general multimode noisy NH setup.(a)The setup consists of a 1-dimensional chain of N bosonic modes.The parameter to be detected is ϵ, which represents a small change in the resonance frequency of the last site.To detect ϵ, a coherent drive β accompanied by quantum noise Bin is injected into the chain at mode 1 through an input-output waveguide with coupling rate κ.The reflected field is measured by homodyne detection.The modes are coupled via nearest neighbour hopping w and coherent twophoton drive ∆.To account for the noise, couplings between the modes and the loss/gain baths (blue solid/red dashed) are included.The coupling rate between the ith mode and the jth loss (gain) bath is described by Zij (Yij).(b) The nonreciprocal amplification between modes can be described by two N -site NH Hatano-Nelson chains[53][54][55] with effective hopping amplitude J and amplification factor A. For the top (bottom) X (P) chain, hopping to the right is a factor of e 2A larger (smaller) than hopping to the left.The last modes of the two chains are coupled due to the presence of small tunneling with amplitude ϵ, allowing the signal to be transmitted between the two chains.

FIG. 2 .
FIG. 2. (Color online)The performance log( SNR τ ϵ 2 ) versus A under different loss couplings and α's.The range of A is identified to guarantee a stable dynamics.(a) α = 0.5.Dashed blue: loss Z1, dotted green: loss Z2, red circle: loss Z which satisfies (C1), black line: the ideal case and the tuned case where both (C1) and (C2) are met.(b) For loss Z1, black solid: α = 0, blue dashed: α = 0.5, magenta dotdashed: α = 1, green dotted: α = 2.The best sensitivity that can be revived under loss Z for different values of α is shown by the colored dots on the ideal black solid line.
ACKNOWLEDGMENTS L.B. acknowledges the support of the Fundamental Research Funds for the Central Universities of China (No. 3122023QD23).B.Q. acknowledges the support of the National Natural Science Foundation of China (No. 61773370).D.D. acknowledges the support of the Australian Research Council Future Fellowship funding scheme under Project FT220100656.F.N. is supported in part by: Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP), and the Moonshot R&D Grant Number JPMJMS2061], the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Office of Naval Research (ONR) Global (via Grant No. N62909-23-1-2074).

FIG. 3 .
FIG. 3. A schematic of an N -site non-Hermitian setup.(a)There is an effective coupling γ between the first and the (N − 1)-th modes.(b) There is an effective coupling γ between the first and the N -th modes.

Appendix H:
Calculation of the elements of H[ϵ0] −1