Optimal joint estimation of multiple Rabi frequencies
Hongzhen Chen Haidong Yuan

























hdyuan@mae.cuhk.edu.hk



Department of Mechanical and Automation Engineering,
The Chinese University of Hong Kong, Shatin, Hong Kong












We study the joint estimation of multiple Rabi frequencies in a multi-level system. By analytically identifying the optimal probe state and the optimal measurement, we obtain the highest precision limit for the joint estimation of multiple Rabi frequencies. We show that the joint estimation does not always outperform the separate estimation. There exist different regimes where the joint estimation can either outperform or underperform the separate estimation. We further show that if additional adaptive quantum controls are allowed then the advantage of the joint estimation over the separate estimation can be reestablished.







Optimal joint estimation of multiple Rabi frequencies
    Haidong Yuan
    December 30, 2023
=====================================================







§ INTRODUCTION

Quantum metrology, which makes use of quantum mechanical effects to achieve higher precision, is gaining increasing attention for its broad applications in many areas <cit.>, such as atomic clocks <cit.>, spectroscopy <cit.>, magnetometry <cit.>, gravitational-wave detection <cit.>, etc.
Much progress has been made in the single parameter quantum estimation <cit.>.
However, in many practical applications such as microscopy and imaging <cit.>, there are typically more than one parameter.
This puts a high demand on better understandings of multi-parameter quantum estimation.
There have been recent studies on joint estimation of multiple parameters in various specific settings, such as multiple phase estimation under commuting dynamics <cit.>, estimation of  multidimensional fields <cit.>, estimation of multiple parameters in quantum states <cit.> and unitary operators <cit.>, joint estimatin of the phase and decoherence rate <cit.>, as well as joint estimation of two decoherence rates <cit.>.
However, the optimal performance for multi-parameter quantum estimation is still not well understood<cit.>.

A distinct feature of multi-parameter quantum estimation is the tradeoff among the precisions of different parameters <cit.>, as the optimal probe states and the optimal measurements for different parameters are typically different. Such trade-offs are usually quantified by a particular figure of merit, typically a combination of the weighted precisions of different parameters, one can then optimize the probe state and the measurement according to the chosen figure of merit. This turns out to be a hard problem.
While general methods on the identification of the optimal probe states exist for the single-parameter quantum estimation<cit.>, the probe states for multi-parameter quantum estimation are typically chosen heuristically<cit.>, the optimal probe states are only identified for very few cases<cit.>.

In this article, we study the joint estimation of multiple Rabi frequencies. We start with the joint estimation of two Rabi frequencies in a three-level system. By explicitly optimizing the probe state and the measurement, we analytically obtain the highest precision for the joint estimation of two Rabi frequencies. We find that, in contrast to the expectation, the joint estimation does not always outperform the separate estimation. There exist different regimes that the joint estimation can either outperform or underperform the separate estimation. This enriches the understandings on the relationship between the joint and the separate estimation. We then consider adding optimal quantum controls in the scheme and show that with the adaptive quantum controls, the joint estimation can restore the advantage over the separate estimation.

The article is organized as follows: in Sec.<ref> we make a brief introduction of the basic tools for the multi-parameter quantum estimation; in Sec.<ref> we derive the precision limits for the joint estimation of two Rabi frequencies in a three-level system by explicitly obtaining the optimal probe state and the optimal measurement scheme; we then compare the joint estimation with the separate estimation in Sec.<ref>; in Sec.<ref> we show that with optimal adaptive controls the joint estimation can always outperform the separate estimation. This is then generalized to systems with more levels in Sec.<ref>; Sec.<ref> concludes the paper.



§ MULTI-PARAMETER QUANTUM ESTIMATION




To estimate a set of parameters, φ=(φ_1,...,φ_p), encoded in a quantum channel Λ_φ, one can prepare a probe state, ρ, and get the output state, ρ_φ=Λ_φ(ρ), which contains the unknown parameters.
With a set of Positive Operator Valued Measurements (POVMs), {M_x}_x∈Ω, the information of the parameters can then be extracted. From the probability distribution of the measurement results, p_φ(x)=Tr(ρ_φM_x), one can construct an estimator, φ̂(x)=(φ̂_1,...,φ̂_p)(x).
For locally unbiased estimator with <φ̂>=∑_x∈Ωφ̂(x) p_φ(x)=φ and d<φ̂>/dφ_i=1, ∀ i, the precision is bounded below by the Fisher information matrix <cit.> as

    Cov(φ̂)≥ℐ(φ)^-1,

where Cov(φ̂) is the covariance matrix with its ij-th entry as [Cov(φ̂)]_ij=<(φ̂_̂î-φ_i)(φ̂_̂ĵ-φ_j)>, and ℐ(φ) is the Fisher information matrix with its ij-th entry given by <cit.>

    [ℐ(φ)]_ij=∑_x∈Ω∂ln p_φ(x)/∂φ_i∂ln p_φ(x)/∂φ_j p_φ(x).










The Fisher information matrix can be further bounded by the quantum Fisher information matrix (QFIM), which leads to the quantum Cramér-Rao bound (QCRB)
<cit.> as

    Cov(φ̂)≥ℐ(φ)^-1≥ J(φ)^-1,

here the ij-th entry of the QFIM is given by

    [J(φ)]_ij=1/2Tr(ρ_φ{L_i,L_j}),

where {∙,∙} is the anticommutator, L_i is the symmetrical logarithm derivative (SLD) of the i-th parameter, which can be obtained by ∂_φ_iρ_φ=(L_i ρ_φ+ρ_φL_i)/2. For the multi-parameter quantum estimation, the quantum Cramer-Rao bound is in general not achievable. However, if it satisfies the weaker commutation condition as

    Tr(ρ_φ[L_i,L_j])=0,  ∀ i,j,

then the quantum Cramer-Rao bound can be saturated<cit.>.

One distinct feature of the multi-parameter quantum estimation is the trade-off among the precisions of different parameters, as the optimal probe state and the optimal measurement for different parameters are typically different. In general this is treated by adding a weight matrix, W, in the figure of merit as Tr(WCov(φ̂)), then one can identify the optimal probe state and the optimal measurement according to the figure of merit. In this article, we will take W=I and consider the figure of merit as Tr(Cov(φ̂)). From the QCRB, it is easy to get a lower bound as

    Tr(Cov(φ̂))≥ Tr( J(φ)^-1).























§ MULTI-PARAMETER ESTIMATION IN A THREE-LEVEL SYSTEM









We first consider a three-level system, with Λ or ladder shape linkage, interacting with two resonating radiation fields as depicted in FIG. <ref>.




The dynamics of the system is described by the Schrödinger equation

    iħd/dt|ψ⟩=H|ψ⟩,

where H is the Hamiltonian for the system and its interaction with the radiation fields.
Under the rotating-wave approximation (RWA), the Hamiltonian can be written as

    H(Ω)=1/2[   0 Ω_1   0; Ω_1   0 Ω_2;   0 Ω_2   0 ],

here Ω_1 and Ω_2 are the amplitudes of the fields which are the parameters we would like to estimate.







If we take the initial state as |ψ_in⟩, then the output state is given by |ψ_Ω⟩=U(Ω,t)|ψ_in⟩, where U(Ω,t)=e^-iH(Ω)t.
The SLD operators with respect to Ω_1 and Ω_2 can be computed as

    L_i=2(|∂_iψ_Ω⟩⟨ψ_Ω|+|ψ_Ω⟩⟨∂_iψ_Ω|),

where ∂_i denotes the partial derivative with respect to the i-th parameter Ω_i,

  |∂_iψ_Ω⟩=∂ U(Ω,t)/∂Ω_i|ψ_in⟩.

The quantum Fisher information matrix is then given by

    [J_Ω]_ij=   1/2Tr(|ψ_Ω⟩⟨ψ_Ω|{L_i,L_j})
    
        =   2(∂_iψ_Ω∂_jψ_Ω+∂_jψ_Ω∂_iψ_Ω)
       +4∂_iψ_Ωψ_Ω∂_jψ_Ωψ_Ω,

where [J_Ω]_ij is the ij-th entry of the quantum Fisher information matrix.
For pure states, the weaker conditions Eq.(<ref>) for saturating the QCRB is reduced to Im∂_iψ_Ω∂_jψ_Ω=0, ∀ i,j <cit.>.







To optimize the figure of merit, one needs to identify the optimal probe state and the optimal measurement. We will first identify the optimal probe state that has the minimal Tr(J(φ)^-1), then show the quantum Cramer-Rao bound given in Eq.(<ref>) can be saturated, ensuring the optimal figure of merit.

To ease the calculation, we will use the eigenvectors of the Hamiltonian as a basis. These eigenbasis |Φ_0⟩, |Φ_±⟩ and the corresponding eigenvalues Ω_0, Ω_± of H(Ω) are as following:

    Ω_0   : |Φ_0⟩=cosϑ|0⟩-sinϑ|2⟩, 
    Ω_+   : |Φ_+⟩=1/√(2)(sinϑ|0⟩+|1⟩+cosϑ|2⟩),
    Ω_-   : |Φ_-⟩=1/√(2)(sinϑ|0⟩-|1⟩+cosϑ|2⟩),

where Ω_0=0, Ω_±=±1/2√(Ω_1^2+Ω_2^2), ϑ is the mixing angle with tanϑ=Ω_1/Ω_2.



The initial probe state can be written as |ψ_in⟩=C_0|Φ_0⟩+C_+|Φ_+⟩+C_-|Φ_-⟩, where C_0∈ℝ, C_±∈ℂ are coefficients to be optimized.
Note that this is just a way to expand the initial probe state in the eigenbasis, and the initial probe state is fixed once it is chosen.
The output state can now be easily obtained as

    |ψ_Ω⟩   =e^-iH(Ω)t|ψ_in⟩=(∑_n=0,±e^-iΩ_n t|Φ_n⟩⟨Φ_n|)|ψ_in⟩
       =∑_n=0,±C_n e^-iΩ_n t|Φ_n⟩.


The partial derivative of the output state can then be computed as

    |∂_iψ_Ω⟩=   (∑_n=0,±∂(e^-iΩ_n t|Φ_n⟩⟨Φ_n|)/∂Ω_i)|ψ_in⟩
    
        =   ∑_n=0,± [(∂_iΦ_nψ_in-itC_n∂_iΩ_n) e^-iΩ_n t|Φ_n⟩
       +C_n e^-iΩ_n t|∂_iΦ_n⟩].

This can then be used to get the quantum Fisher information matrix via Eq.(<ref>) (see the appendix for detailed calculations). The quantum Fisher information matrix turns out to have different behaviours at different time points. There exist some specific time points at which the quantum Fisher information matrices are singular while at other time points the quantum Fisher information matrices are full rank.

Specifically at the time points where Ω_+t=2nπ for n∈ℕ, the quantum Fisher information matrix is singular, which takes the form

    J_Ω=   4t^2(1-C_0^2-(|C_+|^2-|C_-|^2)^2)×
       [         (∂_1Ω_+ )^2 (∂_1Ω_+ ) (∂_2Ω_+ ); (∂_2Ω_+ ) (∂_1Ω_+ )         (∂_2Ω_+ )^2 ],

Intuitively, at these time points U(Ω, t)=e^-iH(Ω) t=I, the dynamics is always the Identity operator as long as Ω_+, which corresponds to the norm of (Ω_1,Ω_2), takes the specific value, i.e., one can not tell the differences between different pairs of Ω_1 and Ω_2 as long as they have the same specific norm. This indicates that at these time points joint estimations of Ω_1 and Ω_2 are not possible.

At other time points, the entries of the quantum Fisher information matrix are given by(see the appendix for detailed calculations)

  
    [J_Ω]_ij
          =   2(∂_iψ_Ω∂_jψ_Ω+∂_jψ_Ω∂_iψ_Ω)+4∂_iψ_Ωψ_Ω∂_jψ_Ωψ_Ω
    
    
    
    
          =      (∂_iϑ ) (∂_jϑ )A+ (∂_iΩ_+ ) (∂_jΩ_+ )B+( (∂_iϑ ) (∂_jΩ_+ )+ (∂_jϑ ) (∂_iΩ_+ ))C,


here A=-8C_0^2 Im^2 M +4C_0^2|P|^2+2|M|^2, B=-4t^2Re^2(MN^*)/|P|^4+4t^2-4C_0^2 t^2, C=-4√(2)C_0 Im M Re(MN^*)/|P|^2t+2√(2)C_0 t Im N, with P=e^-iΩ_+t-1, M=C_+^* P^* + C_-^* P, N=C_+^* P^* - C_-^* P.
With this it is then straightforward to get

  
    Tr(J_Ω^-1)
          =   ( (∂_1ϑ ) (∂_1ϑ )+ (∂_2ϑ ) (∂_2ϑ ))A+2( (∂_1ϑ ) (∂_1Ω_+ )+ (∂_2ϑ ) (∂_2Ω_+ ))C+( (∂_1Ω_+ ) (∂_1Ω_+ )+ (∂_2Ω_+ ) (∂_2Ω_+ ))B/( (∂_1ϑ ) (∂_2Ω_+ )- (∂_2ϑ ) (∂_1Ω_+ ))^2(AB-C^2)
    
          =   (1/4Ω_+^2) A+ (1/4)B/(1/16Ω_+^2)(AB-C^2)=4A/AB-C^2+4Ω_+^2 B/AB-C^2.


In the appendix, we show that the minimal value of Tr(J_Ω^-1) is achieved with the initial probe state taken as |ϕ_in⟩=P^*/√(2)|P||Φ_+⟩+P/√(2)|P||Φ_-⟩, with the corresponding minimal Tr(J_Ω^-1) as

    minTr(J_Ω^-1)=1/t^2+Ω_+^2/4sin^2(Ω_+t/2).

It can be checked directly that the weaker commutative conditions Im∂_iψ_Ω∂_jψ_Ω=0, ∀ i,j, hold in this case, and thus this minimal value can be saturated by some proper measurements. For example, one set of the measurements that saturates the quantum Cramer-Rao bound can be chosen as Π_1=|γ_1⟩⟨γ_1|, Π_2=|γ_2⟩⟨γ_2|, Π_3=|γ_3⟩⟨γ_3| and Π_4=I-Π_1-Π_2-Π_3, where

    |γ_1⟩=   |ψ_Ω⟩,
    |γ_2⟩=   |∂_1ψ_Ω⟩/√(∂_1ψ_Ω∂_1ψ_Ω),
    |γ_3⟩=   ∂_2ψ_Ωγ_2|γ_2⟩-|∂_2ψ_Ω⟩/√(∂_2ψ_Ω∂_2ψ_Ω-∂_2ψ_Ωγ_2^2).







§ COMPARISON BETWEEN THE JOINT ESTIMATION AND THE SEPARATE ESTIMATION

We compare the performances of the joint estimation with the separate estimation. The separate estimation is to estimate Ω_1 and Ω_2 separately. For each round of experiments, one can use the dynamical decoupling to remove one of the fields, and reduce the problem to the single parameter estimation. For example, by applying periodic π-pulses along the direction (|0⟩⟨1|+|1⟩⟨0|)/2, one can remove Ω_2 in the Hamiltonian and get the effective Hamiltonian as H=Ω_1 H_1=Ω_1(|0⟩⟨1|+|1⟩⟨0|)/2.
The quantum Fisher information (QFI) for the single parameter, which has been studied extensively <cit.>, is given by

    J(ρ(t))=4t^2(Tr(ρ_inH_1^2)-(Tr(ρ_inH_1)^2)),

where ρ_in is the input state.
The optimal input state that maximizes the quantum Fisher information is <cit.>

    |ψ_in⟩=1/√(2)(|λ_min⟩+|λ_max⟩),





where λ_min and λ_max are the minimal and maximal eigenvalues of H_1 with |λ_min⟩ and |λ_max⟩ denoting the corresponding eigenstates.
For H_1=(|0⟩⟨1|+|1⟩⟨0|)/2, the maximal QFI is J=t^2. Similarly, the maximal QFI for the separate estimation of Ω_2 is also J=t^2.

For comparison, we assume that the experiments are repeated m times for the estimation of each parameter in the separate estimation. For the joint estimation, the experiments are repeated 2m times so the total number of experiments are the same. The quantum Cramér-Rao bound for the joint estimation is given by

    Tr(Cov(Ω̂))≥1/2mTr(J_Ω^-1)
       = 1/2mt^2+Ω_+^2/8msin^2(Ω_+t/2).

For the separate estimation, the minimal variance given by the quantum Cramer-Rao bound is Var(Ω̂_i)≥1/mt^2, i∈{1,2}. The total variance is

    Tr(Cov(Ω̂))=Var(Ω̂_1)+Var(Ω̂_2)≥2/mt^2.


The performances of the two schemes can be seen in FIG.<ref>, which shows that in the short time regime the joint estimation outperforms the separate estimation while in the long time regime the joint estimation under-performs the separate estimation.




§ ESTIMATION WITH THE ADAPTIVE CONTROL





So far we considered the highest precision achievable by optimizing the probe state and the measurement, while assuming the dynamics is fixed as U(Ω, t)=e^-iH(Ω)t. In practice, however, additional controls can often be added during the evolutions as depicted in FIG. <ref>, where the total evolution time t has been divided into N intervals with dt=t/N and the adaptive controls U_1, U_2, ..., U_N are added sequentially.
For such control-enhanced sequential scheme, it has been shown that for any unitary evolution U(Ω,t)=e^-iH(Ω)t, the optimal controls are given by U_1=U_2=⋯=U_N=U^†(Ω,dt), where Ω is the true value of the parameter<cit.>. In practice, the true value is not known a priori, so the controls can only be taken as U_1=U_2=⋯=U_N=U^†(Ω̂,dt), with Ω̂ being the estimated value and need to be updated adaptively when more data are collected.

When the estimated value is close to the true value, the total evolution can be described as

    (U_1 U_dt)^N   =(e^iH(Ω̂)dte^-iH(Ω)dt)^N
       ≈ [(I+iH(Ω̂)dt)(I-iH(Ω)dt)]^N
       ≈ (I-i(H(Ω-Ω̂))dt)^N
       ≈ e^-iH(Ω-Ω̂)t,


Therefore, the effective Hamiltonian of the total dynamics is H(Ω-Ω̂).
While the estimator is being updated, the change of the total variance is depicted in FIG. <ref>.
It can be seen that the total variance is quite robust against the estimation error.
Asymptotically, when the estimated value converges to the true value, i.e., Ω̂=Ω, the adaptive control converges to the optimal control and the problem reduces to the joint estimation of Ω=(Ω_1,Ω_2) at the point (0,0). At this point the optimal probe state is |ψ_opt⟩=|1⟩, and the total variance can be obtained by taking the limit Ω_+→ 0 in Eq.(<ref>) as

    Tr(Cov(Ω̂))=1/mt^2,

which always has a 2-fold improvement over the separate estimation.
By taking the limit (Ω_1,Ω_2)→ (0,0) in Eq.(<ref>), one can verify that the optimal measurement saturating the QCRB can be chosen as ℳ≡{|0⟩⟨0|,|1⟩⟨1|,|2⟩⟨2|}.




When ΔΩ≡(ΔΩ_1,ΔΩ_2)=(Ω_1-Ω̂_1,Ω_2-Ω̂_2) are not zero, the dynamics has the effective Hamiltonian H(ΔΩ), the total variance can be directly obtained from Eq.(<ref>) as

    Tr(Cov(Ω̂))
    
        =   1/2mt^2+ΔΩ_1^2+ΔΩ_2^2/8msin^2(√(ΔΩ_1^2+ΔΩ_2^2)t/2)
    
        =   1/mt^2+ΔΩ_1^2+ΔΩ_2^2/24m+𝒪(ΔΩ^4).








To see how quickly Ω̂ approaches its true value under the adaptive control, we perform a numerical simulation of the adaptive estimation as follows:

  
  * Choose an initial estimation, either randomly or based on some prior knowledge, as (Ω̂_1^(0),Ω̂_2^(0)), and construct the control, U^†(Ω̂^(0),dt).
  
  * Choose the initial state as |1⟩, and let the system evolve for a period of time t, perform the projective measurement, ℳ={|0⟩⟨0|,|1⟩⟨1|,|2⟩⟨2|}, on the state. Repeat the step for k=30 times
  
  * Based on all the collected measurement results, use the maximum likelihood estimator to update the estimation to (Ω̂_1^(1),Ω̂_2^(1)),  then update the control based on the new estimation.
  
  * Repeat Steps 2-3. 


The probe state and the measurement we choose in Step 2 is the optimal probe state and optimal measurement under the optimally controlled scheme. When the estimator approaches the true value, the adaptive scheme converges to the optimal scheme.
It can be seen from FIG. <ref>(a) and (b) that after just a few iterations, the estimators are already close to the true value. FIG. <ref>(c) shows that just after a couple of iteration, the controlled scheme already outperforms the uncontrolled schemes, and after a few iterations the precision under the adaptive control already close to the highest precision achievable under the optimally controlled scheme.

















§ EXTENSION TO MULTIPLE LEVELS

The advantage of the joint estimation with adaptive controls can be extended to systems with more levels.
Considering the system with l+1 levels as shown in FIG.<ref>, we would like to estimate l Rabi frequencies between |0⟩ and the other l energy eigenstates.
In this case the Hamiltonian can be written as H_l(Ω)=∑_i=1^l Ω_i E_i, E_i=(|0⟩⟨i|+|i⟩⟨0|)/2, where the Rabi frequencies (Ω_1,Ω_2,...,Ω_l) are the parameters to be estimated. For separate estimation, the quantum Fisher information for each parameter is J=t^2. If the experiment is repeated m times for each parameter, the total variance becomes

    ∑_i=1^l Var(Ω̂_̂î)≥∑_i=1^l 1/mt^2=l/mt^2.



For the joint estimation, by using the optimal adaptive control U_1=U_2=⋯=U_N=e^iH_l(Ω̂)dt as described in FIG.<ref> and choosing the initial state as |ψ_in⟩=|0⟩, we can obtain the quantum Fisher information matrix as [J_Ω]_ij=t^2δ_ij.
In this case, the weaker commutative condition holds, and thus the quantum Cramer-Rao bound can be saturated. For comparison with the separate estimation, suppose the experiments are repeated for the same total number of times, which is lm times, then the total variance for the joint estimation is

    ∑_i=1^l Var(Ω̂_̂î)=Tr(Cov(Ω̂)≥1/lmTr(J^-1)=1/mt^2,

which has an l-fold improvement over the separate estimation. Furthermore, if we consider the highest precision achievable for each parameter with the total number of lm experiments, it should be Var(Ω̂_̂î)≥1/lmt^2. It is generally believed that the highest precision can not be achievable simultaneously for all l parameters as the optimal probe state and the optimal measurement for different parameters are typically incompatible. However, from Eq.(<ref>) we can see that in this case the joint estimation under the adaptive control actually achieves the highest precision for all l parameters simultaneously. No tradeoffs among the precisions of different parameter need to be made. Intuitively, it can be understood that for the estimation of each parameter Ω_i, the state |0⟩ is the optimal state. Without adaptive controls, however, this state is only optimal at the beginning. It will evolve to different states which are no longer optimal. The adaptive controls, on the other hand, can keep the state at the optimal point, thus achieves the highest precision for all parameters simultaneously.








§ CONCLUSION


We have explicitly obtained the optimal probe states and measurements for the joint estimation of multiple Rabi frequencies, and compared with the separate estimation.
These results show that without adaptive controls there exist some time points at which the two Rabi frequencies in a three-level system can not be jointly estimated. There also exist different regimes where joint estimation can either outperform or under-perform separate estimation. However, with additional adaptive controls, the joint estimation can always outperform the separate estimation.
This adds concrete results to the literature on the optimal performance of multi-parameter quantum estimation.






§ OPTIMAL PROBE STATE FOR THE JOINT ESTIMATION OF TWO RABI FREQUENCIES


 We write the pure input state in the eigenbasis of the Hamiltonian H as
 |ψ_in⟩=C_0|Φ_0⟩+C_+|Φ_+⟩+C_-|Φ_-⟩, where C_0∈ℝ, C_±∈ℂ are coefficients to be optimized.
 The evolution of the input state is described by the unitary operator U=e^-iH(Ω)t. Therefore the output state is
 
    |ψ_Ω⟩=e^-iH(Ω)t|ψ_in⟩=(∑_n=0,±e^-iΩ_n t|Φ_n⟩⟨Φ_n|)|ψ_in⟩.


 The partial derivative of the state with respect to the i-th parameter Ω_i is given by
 
    |∂_iψ_Ω⟩=   (∑_n=0,±∂(e^-iΩ_n t|Φ_n⟩⟨Φ_n|)/∂Ω_i)|ψ_in⟩
    
         =   ∑_n=0,± (∂_iΦ_nψ_in-itC_n∂_iΩ_n) e^-iΩ_n t|Φ_n⟩
       +C_n e^-iΩ_n t|∂_iΦ_n⟩,

 Note that
 
    |∂_iΦ_0⟩=   -1/√(2) (∂_iϑ )(|Φ_+⟩+|Φ_-⟩)
    |∂_iΦ_±⟩=   1/√(2) (∂_iϑ )|Φ_0⟩,

 hence
 
    ∂_iΦ_0ψ_in=   -1/√(2) (∂_iϑ )(C_++C_-)
    ∂_iΦ_±ψ_in=   1/√(2) (∂_iϑ )C_0.

 Substituting (<ref>) back to (<ref>), we can get the explicit form of |∂_iψ_Ω⟩ expanded by the eigenbasis of H:
 
    |∂_iψ_Ω⟩=1/√(2) (∂_iϑ )[C_+(e^-iΩ_+ t-1)+C_-(e^-iΩ_- t-1)]|Φ_0⟩
       +[-itC_+ (∂_iΩ_+ ) e^-iΩ_+ t+1/√(2)C_0 (∂_iϑ )(e^-iΩ_+ t-1)]|Φ_+⟩
       +[-itC_- (∂_iΩ_- ) e^-iΩ_- t+1/√(2)C_0 (∂_iϑ )(e^-iΩ_- t-1)]|Φ_-⟩,

 where i=1,2, and
 
    ∂_1ϑ =∂ϑ/∂Ω_1=Ω_2/4Ω_+^2,   ∂_2ϑ =∂ϑ/∂Ω_2=-Ω_1/4Ω_+^2
    ∂_1Ω_+ =∂Ω_+/∂Ω_1=Ω_1/4Ω_+,  ∂_2Ω_+ =∂Ω_+/∂Ω_2=Ω_2/4Ω_+.




 Recall that the SLD operators for the pure state |ψ_Ω⟩ can be calculated by L_i=2(|∂_iψ_Ω⟩⟨ψ_Ω|+|ψ_Ω⟩⟨∂_iψ_Ω|), i=1,2. Thus the entries of SLD Fisher information matrix can be calculated by
 
    [J_Ω]_ij=   1/2Tr(|ψ_Ω⟩⟨ψ_Ω|{L_i,L_j})
    
         =   2(∂_iψ_Ω∂_jψ_Ω+∂_jψ_Ω∂_iψ_Ω)
       +4∂_iψ_Ωψ_Ω∂_jψ_Ωψ_Ω,


 Denote P=e^-iΩ_+t-1, M=C_+^* P^* + C_-^* P, N=C_+^* P^* - C_-^* P. First we consider when P= 0. In this case,
 
    ∂_iψ_Ωψ_Ω=it (∂_iΩ_+ )(|C_+|^2-|C_-|^2),

 
    ∂_iψ_Ω∂_jψ_Ω= (∂_iΩ_+ ) (∂_jΩ_+ )t^2(1-C_0^2).

 Substituting them to (<ref>), we can get the QFIM as
 
    J_Ω=   4t^2(1-C_0^2-(|C_+|^2-|C_-|^2)^2)×
       [          (∂_1 Ω_+ )^2 (∂_1 Ω_+ ) (∂_2 Ω_+ ); (∂_2 Ω_+ ) (∂_1 Ω_+ )          (∂_2 Ω_+ )^2 ].

 It can be easily seen that the QFIM is singular, indicating that we cannot estimate Ω_1 and Ω_2 simultaneously.


When P≠ 0, 







we have
 
    ∂_iψ_Ωψ_Ω=   i√(2) (∂_iϑ )C_0 Im[(C_+^* P^* + C_-^* P) ]
       +it (∂_iΩ_+ )(|C_+|^2-|C_-|^2)
    
         =   i√(2) (∂_iϑ )C_0 Im M + it (∂_iΩ_+ ) Re(MN^*)/|P|^2,

and
 
   
    ∂_iψ_Ω∂_jψ_Ω
           =     (∂_iΩ_+ ) (∂_jΩ_+ )t^2 +( (∂_iϑ ) (∂_jϑ )|P|^2- (∂_iΩ_+ ) (∂_jΩ_+ )t^2)C_0^2
           +1/2 (∂_iϑ ) (∂_jϑ )|C_+^* P^* + C_-^* P|^2
       +i1/√(2) (∂_iΩ_+ ) (∂_jϑ ) t C_0(C_-^*P-C_+^*P^*)
           +i1/√(2) (∂_jΩ_+ ) (∂_iϑ ) t C_0(C_+P-C_-P^*)
    
           =    (∂_iΩ_+ ) (∂_jΩ_+ )t^2 +( (∂_iϑ ) (∂_jϑ )|P|^2- (∂_iΩ_+ ) (∂_jΩ_+ )t^2)C_0^2
           +1/2 (∂_iϑ ) (∂_jϑ )|M|^2
       +i-1/√(2) (∂_iΩ_+ ) (∂_jϑ ) t C_0 N
           +i1/√(2) (∂_jΩ_+ ) (∂_iϑ ) t C_0 N^*.

 
where Rex and Imx denote the real and imaginary parts of x respectively. Substituting (<ref>) and (<ref>) back to (<ref>), we can get
 
   
    [J_Ω]_ij
           =   2(∂_iψ_Ω∂_jψ_Ω+∂_jψ_Ω∂_iψ_Ω)+4∂_iψ_Ωψ_Ω∂_jψ_Ωψ_Ω
    
           =     (∂_iϑ ) (∂_jϑ )(-8C_0^2 Im^2 M +4C_0^2|P|^2+2|M|^2)
           +  (∂_iΩ_+ ) (∂_jΩ_+ )(-4t^2Re^2(MN^*)/|P|^4+4t^2-4C_0^2 t^2)
       + ( (∂_iϑ ) (∂_jΩ_+ )+ (∂_jϑ ) (∂_iΩ_+ ))
           ( -4√(2)C_0 Im M Re(MN^*)/|P|^2t+2√(2)C_0 t Im N)
    
           =      (∂_iϑ ) (∂_jϑ )A+ (∂_iΩ_+ ) (∂_jΩ_+ )B+( (∂_iϑ ) (∂_jΩ_+ )+ (∂_jϑ ) (∂_iΩ_+ ))C,

 
 where  A=-8C_0^2 Im^2 M +4C_0^2|P|^2+2|M|^2, B=-4t^2Re^2(MN^*)/|P|^4+4t^2-4C_0^2 t^2, C=-4√(2)C_0 Im M Re(MN^*)/|P|^2t+2√(2)C_0 t Im N. It then straightforward to get
 
   
    Tr(J_Ω^-1)
           =   ( (∂_Pϑ ) (∂_Pϑ )+ (∂_Sϑ ) (∂_Sϑ ))A+2( (∂_Pϑ ) (∂_PΩ_+ )+ (∂_Sϑ ) (∂_SΩ_+ ))C+( (∂_PΩ_+ ) (∂_PΩ_+ )+ (∂_SΩ_+ ) (∂_SΩ_+ ))B/( (∂_Pϑ ) (∂_SΩ_+ )- (∂_Sϑ ) (∂_PΩ_+ ))^2(AB-C^2)
    
           =   (1/4Ω_+^2) A+ (1/4)B/(1/16Ω_+^2)(AB-C^2)=4A+4Ω_+^2 B/AB-C^2.

 

Since |M|^2+|N|^2=2|P|^2(1-C_0^2), we can let
 
    M=   √(2|P|^2(1-C_0^2)) e^iαcosθ,
    
         N=   √(2|P|^2(1-C_0^2)) e^iβsinθ.

Then
 
    A=   4|P|^2C_0^2+4|P|^2(1-C_0^2)cos^2θ
       -16|P|^2C_0^2(1-C_0^2)cos^2θsin^2α,
    
         B=   4t^2(1-C_0^2)-4t^2(1-C_0^2)^2sin^2 2θcos^2(α-β).

With this we are going to show that AAB-C^2≥14t^2 and BAB-C^2≥14|P|^2.

To show AAB-C^2≥14t^2, we note that
 
    A/AB-C^2-1/4t^2=4t^2A-(AB-C^2)/4t^2(AB-C^2)
       ≥A(4t^2-B)/4t^2(AB-C^2).

From the fact that the quantum Fisher information matrix is positive semidefinite, we can obtain AB-C^2≥ 0.
Also 
    A≥   4|P|^2C_0^2cos^2θ+4|P|^2(1-C_0^2)cos^2θ
       -16|P|^2C_0^2(1-C_0^2)cos^2θsin^2α
    ≥   4|P|^2cos^2θ-4|P|^2cos^2θsin^2α≥0,

 where we used the fact 4C_0^2(1-C_0^2)≤ 1, and
 
    4t^2-B= 4t^2C_0^2+4t^2(1-C_0^2)^2sin^2 2θcos^2(α-β)≥ 0.

Thus A/AB-C^2-1/4t^2≥ 0. The equality can be saturated when C=0 and B=4t^2.

To show BAB-C^2≥14|P|^2, we note that
 
    B/AB-C^2-1/4|P|^2=4|P|^2 B-(AB-C^2)/4|P|^2(AB-C^2)
       ≥B(4|P|^2 -A)/4|P|^2(AB-C^2).

 Since
 
    B≥ 4t^2(1-C_0^2)-4t^2(1-C_0^2)^2=4t^2C_0^2(1-C_0^2)≥ 0,

 and
 
    4|P|^2 -A
    
         =    4|P|^2(1-C_0^2)(1-cos^2θ)
       +16|P|^2C_0^2(1-C_0^2)cos^2θsin^2α
    ≥    0.

 Thus B/AB-C^2-1/4|P|^2≥ 0 and the equality can be saturated when C=0 and A=4|P|^2.

With A/AB-C^2≥14t^2 and B/AB-C^2≥1/4|P|^2, it is then easy to see that
 
    Tr(J_Ω^-1)=   4A+4Ω_+^2 B/AB-C^2
    ≥   1/t^2+Ω_+^2/|P|^2
    
         =   1/t^2+Ω_+^2/4sin^2(Ω_+t/2),

where the equality can be achieved with A=4|P|^2, B=4t^2 and C=0. It is straightforward to check that when the input state takes the form |ψ_in⟩=P^*/(√(2)|P|)|Φ_+⟩+P/(√(2)|P|)|Φ_-⟩, the equality is saturated. And thus it is the optimal probe state.
Since P=e^-iΩ_+t-1=-2isin(Ω_+t/2)e^-iΩ_+t/2, the optimal input state can also be written as
 
    |ψ_in⟩=   P^*/√(2)|P||Φ_+⟩+P/√(2)|P||Φ_-⟩
    
         =   e^iΩ_+t/2|Φ_+⟩-e^iΩ_-t/2|Φ_-⟩/√(2)

























































































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