Quantum Parameter Estimation: From Experimental Design to Constructive Algorithm

Transcription

1 Commun. Theor. Phys. 68 ( Vol. 68, No. 5, November 1, 017 Quantum Parameter Estimation: From Experimental Design to Constructive Algorithm Le Yang ( 杨乐, 1, Xi Chen ( 陈希, 1 Ming Zhang ( 张明, 1, and Hong-Yi Dai ( 戴宏毅,3 1 College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha , China College of Science, National University of Defense Technology, Changsha , China 3 Interdisciplinary Center for Quantum Information, National University of Defense Technology, Changsha , China (Received June 0, 017; revised manuscript received August 4, 017 Abstract In this paper we design the following two-step scheme to estimate the model parameter ω 0 of the quantum system: first we utilize the Fisher information with respect to an intermediate variable v = cos(ω 0 t to determine an optimal initial state and to seek optimal parameters of the POVM measurement operators; second we explore how to estimate ω 0 from v by choosing t when a priori information knowledge of ω 0 is available. Our optimal initial state can achieve the maximum quantum Fisher information. The formulation of the optimal time t is obtained and the complete algorithm for parameter estimation is presented. We further explore how the lower bound of the estimation deviation depends on the a priori information of the model. PACS numbers: f, Ta DOI: / /68/5/641 Key words: Fisher information, parameter estimation, optimal initial state, optimal measurement parameters 1 Introduction With the development of quantum theory and technology, enhancing the precision of measurement with quantum methods is gradually playing an important role in precise measurement. 1] Quantum parameter estimation is a process of parameter estimation, which utilises the quantum theory and technology to improve the precision of the parameters to be estimated in the classical system, or implementing the related technical methods in the quantum system. ] The core goal of quantum parameter estimation is to improve the precision of parameter estimation. The process of quantum parameter estimation is usually composed of several stages: 3] initial state preparation, dynamic process evolution, measurement and data processing. The initial quantum state is often considered as a probe. The parameter to be estimated is usually loaded into the system through a dynamic evolution process. Then measurements are made on the quantum state and the parameter is inferred from the measurement results. The optimal initial state and quantum measurement are designed and an appropriate data processing method is selected so as to optimize the precision of parameter estimation. The Fisher information characterizes the amount of information of estimating the parameter in mathematical statistics. It should be distinguished from the classic information in communication field as it is just an indicator of characterizing the ability for parameter estimation rather than real information. The more the Fisher information is, the more accurate parameter we get. The quantum Fisher information (QFI is an indicator of the minimum error of unbiased parameter estimation. It is the quantization of the Fisher information in quantum system with the optimal POVM measurements. In the single parameter estimation, the quantum Fisher information gives the highest precision of the parameter to be evaluated. In the multi-parameter case, the lower bound of the quantum Fisher information matrix in Cramer Rao theory is not always a tight bound and this is still a challenging problem. 4] For the single parameter estimation, the parameter estimation precision and the Fisher information have the following relationship: V θ 1 F θ, (1 where θ is the parameter to be estimated, V θ and F θ are the corresponding variance and Fisher information of θ. Quantum entanglement is needed in general metrological tasks to achieve a high precision, 5 11] while exception exists such as highly entangled pure states may not be beneficial for metrology. 1] In addition to the entangled state, the squeezed states of photon and atom can also be utilized to improve the measurement precision ] Specially Ref. 16] presents an enlightening quantum state Supported by the National Natural Science Foundation of China under Grant Nos , , and kongfuyale@16.com Corresponding author, zhangming@nudt.edu.cn c 017 Chinese Physical Society and IOP Publishing Ltd

2 64 Communications in Theoretical Physics Vol. 68 engineering algorithm to find a number of novel quantum states which can be used for probe states and beat the shot noise limit. Different parameterization process may be benefical such as parallel estimation strategies. 3,17 18] Welldesigned measurement operators and schemes are also necessary to help achieve high precision such as the POVM measurement corresponding to the SLD operator of the system, 19] adaptive measurement, 0 6] and so on. Quantum control is also introduced into quantum metrology to enhance the precision. 7 9] However for practical problems, it is not easy to implement optimal measurement operators. 30] Therefore it is of practical significance to explore the optimal preparation of the initial state with fixed measurement operators and the parameter optimization of the measurement operators. In this paper the POVM measurement operators have been given as constraints, then the optimal initial state is found and the optimal coefficients are defined. The traditional method to optimize the probe state is performed via (QFI, 31 35] while we get the optimal initial state which can achieve the maximum QFI in another way. Specially the parameter to be estimated can be got if there is a priori information and the optimal evolution time is determined. The rest of the paper is organized as follows. Section is the description of the problem. The prototype of the model comes from the atomic interferometer. The state is simplified to a single qubit rather than an ensemble for simplicity. In Sec. 3 we get the optimal initial state with given POVM measurement operators and get the optimal measurement operators. We achieve that by calculating the Fisher information of the intermediate variable which includes the parameter to be estimated. Then the optimal evolution time is determined and a parameter estimation algorithm requested the a priori information is shown. With this algorithm the exact value of the parameter is got. Section 4 is the conclusion. Description of the Parameter Estimation Process Considering a closed system with a single parameter to be estimated as follows: i d ψ(t = H ψ(t, ( dt where we set = 1, H = (1/σ z ω 0, σ z is the Pauli-Z operator and ω 0 is the parameter to be estimated. The initial state of the closed system is: ψ 0 = cos θ 0 + sin θ 1. (3 The density matrix evolution of the system will satisfy: ρ = Uρ 0 U cos θ e iω0t cos θ = sin θ ] e iω0t cos θ sin θ sin θ, (4 where e i ω0t ] 0 U = exp( iht = 0 e i ω, 0t cos θ cos θ ρ 0 = ψ 0 ψ 0 = sin θ ] cos θ sin θ sin θ and t is the evolution time. After evolution time t, the state is measured with POVM operators E 1 and E. The forms of the POVM operators are given as follows: ] ] a b 1 a b E 1 =, E =, (5 b 1 a b a where the parameters a and b are to be determined. These operators satisfy the completeness and positive semi-definiteness of POVM operators. With completeness, E 1 + E = I, (6 With positive semi- where I is the identify operator. definiteness, a(1 a b 0, (1 aa ( b 0. (7 Fig. 1 Three stages of optimal experimental design and parameter estimation. The three stages of the parameter estimation problem in this paper are shown in Fig. 1: (i Finding the optimal initial state in terms of Fisher information for the given a and b; (ii Finding the optimal POVM operators by adjusting a and b; (iii When we have prior information about ω 0, the algorithm estimating ω 0 is shown by using the aforementioned optimal initial state and optimal POVM operators. It should be noticed that it is the intermediate variable v = cos(ω 0 t, (8 for which the first two stages constitute the optimal experimental scheme is designed, rather than the parameter ω 0 to be estimated directly. Without a priori information, we can only firstly estimate the intermediate variable v. Since cosine is a periodic function, with an definite evolution time t, different estimates of ω 0 are corresponding to the same v and t so that we can not determine the unique ω 0. When we have a priori information of ω 0, we can adjust the time t so that the ω 0 t is in a certain period of the cosine function and then determine the unique ω 0 value.

3 No. 5 Communications in Theoretical Physics Result and Discussion We can only estimate the intermediate variable v = cos(ω 0 t when we have no a priori information of the parameter ω 0. The optimal initial state expression and the optimal measurement operator parameters can be obtained through calculating the Fisher information of v. Under the condition that the times of measurement are not limited, we can obtain an arbitrary estimation precision of v by adjusting the evolution time t. When we know the a priori information of ω 0, we can get the estimated value of the parameter ω 0 whose error precision is optimal by selecting the evolution time t satisfying certain conditions. The specific discussion is as follows. 3.1 Optimal Experimental Design When we have no a priori information about ω 0, we try to find the optimal initial state and the optimal a and b in Eq. (5, which makes the Fisher information of v = cos(ω 0 t achieve the maximum. We have a lemma as follows. Lemma 1 For the parameter estimation process given by Eqs. (, (3 and (5, the initial state ψ 0 is optimal when its polar angle θ satisfies tan θ = a(1 a (a 1bv. (9 Proof Supposing P 1 and P are the probability of the corresponding results obtained by the operator E 1 and E for the density matrix ρ, so that P 1 = tr(e 1 ρ = sin θ + a cos θ + bv sin θ, (10 P = tr(e ρ = cos θ a cos θ bv sin θ. (11 The Fisher information of v is: F v = i ( v P i /P i = (b sin θ 1/4 (a 1/ cos θ + bv sin θ]. (1 It can be transformed as follows: b F v = a(1 a1/tan θ + ((1 ab/a(1 av] + (a(1 a b v /4a(1 a, (13 where sin θ 0 and sin θ = tan θ/(1 + tan θ is used during the derivation. With 1 (1 ab ] + tan θ a(1 a v 0, and a(1 a 0, when a(1 a tan θ = (a 1bv, 1 (1 ab ] a(1 a + tan θ a(1 a v = 0, so that the denominator of the right side in Eq. (13 is minimum. At this time, F v = b (a(1 a b v /4a(1 a = 4ab (1 a a(1 a b v. (14 If the evolution time t can be adjusted without restriction so that the times of measurement is not limited, the value of v can achieve 1. At this moment, F v = 4ab (1 a a(1 a b v F opt = 4ab (1 a a(1 a b. (15 F opt in Eq. (15 is the optimal Fisher information for v. Thus, Eq. (9 is the optimal one that the polar angle of the initial state satisfies. Further we easily get the global optimal parameters θ = π/, a = 1/ and b = ±1/ for the parameter estimation of the intermediate variable v so that the error precision of v achieves the optimal. The reason is as follows. From Eq. (7, we get the range of a and b: a 0, 1], b 1, 1 ]. (16 With Eqs. (7 and (15, when a = 1/ and b = ±1/, the denominator of the Fisher information becomes 0 so that the Fisher information achieve and this means the error precision of v achieves the optimal. At this time, θ = π/ with Eq. (9. Remark The traditional method to optimize the probe state is performed via QFI. Although we get the optimal initial state by determining the parameters a and b of the POVM measurement operators rather than QFI, the optimal initial state can also achieve the maximum QFI. The reason is as follows. In Refs ], a method to design the initial state is presented when the initial state is pure and the parameterization is unitary in a two-dimensional quantum system: F pure = 4 H in, where F pure is the QFI for the parameter to be estimated, H := i( φ U U, U is a unitary matrix and φ is the parameter to be estimated. Denote H i = ψ i H ψ i ψ i H ψ i is the variance of H on the i-th eigenstate of the initial

4 644 Communications in Theoretical Physics Vol. 68 density matrix ρ 0, where ψ i satisfies the spectral decomposition of ρ 0, ρ 0 = Σ M i=1 p i ψ i ψ i. In our paper, ρ 0 = ψ 0 ψ 0 so that H in = ψ 0 H ψ 0 ψ 0 H ψ 0. Through some calculations by the equation of the QFI, we get F pure = sin θ. In our paper, the optimal parameter of the initial state in Eq. (3 is θ = π/, and it is obvious that θ = π/ makes the QFI achieve the maximum. In general, Fisher information will become quantum Fisher information when the POVM measurement operators are optimal. However it does not mean that the POVM measurement operators in our paper are the optimal ones to help us get the quantum Fisher information because the forms of the measurement operators are already fixed and designing the optimal POVM measurement operators for the quantum system is another problem. 3. Estimation of ω 0 Supposing we know the a priori information of ω 0 as ω 0 ω ω, ω + ω], (17 and the evolution time t is selected to satisfy ω 0 t kπ, kπ + π], (18 where k N so that the only ω 0 can be determined from the estimation of v. In practice, the system is usually required to evolve for a period of time, marked as t M. Theorem 1 ω 0 can be estimated from v = cos(ω 0 t if we choose the maximum evolution time as ( ω ω t max = floor ω ] π ω, (19 where the floor function is a function that takes as input a real number x and gives an output the greatest integer floor(x that is less than or equal to x. Proof We choose t as the definite evolution time t so that the following formulas are satisfied: t t M, ωt = k 0 π + π, ωt π, (0 where k 0 is a defined integer. It is obvious that t t M because the system must evolve at least for the period time t M. The selected evolution time should make sure ω 0 t kπ, kπ +π] and ωt π/ so that the ω 0 corresponding the estimated value is in one single period of the cosine function and the only ω 0 can be determined from the estimation of v = cos(ω 0 t. From Eq. (0 we have k 0 π + π/ = t t M i.e. k 0 ω t M 1 ω π 4, ω ω π/ k 0 π + π/ = 1 4k (1 The Fisher information of ω 0 is: F ω0 = ( ω0 P i /P i i bt sin θ sin(ω 0 t] = 1/4 (a 1/ cos θ + b sin θ cos(ω 0 t]. ( With the optimal conditions in Lemma 1, Eq. ( becomes F ω0 = t(sin ω 0t] 1 cos(ω 0 t] = t. (3 As Eq. (3 shows, the more the evolution time t is, the more Fisher information of ω 0 is and the error is less. We explore the maximum evolution time t so as to achieve the optimal precision. From Eq. (1, the following stands: ω ω k 0 4 ω. We choose ( ω ω k0 = floor, (4 4 ω and the greatest evolution time is ( t max = k0 + 1 ( ω ω π ω = floor ω ] π ω. (5 With the optimal design in Lemma 1, Eq. (10 minus Eq. (11 will become P 1 P = sin θ ] + a cos θ + b sin θ cos(ω 0t cos θ ] a cos θ b sin θ cos(ω 0t = (a 1 cos θ + b sin θ cos(ω 0 t, (6 where a = 1/, θ = π/ and we choose b = 1/. Thus, the estimated value of ω 0 is ˆω 0 = 1 t arccos(p 1 P + k0π], (7 max where k0 = floor(( ω ω/4 ω and ( ω ω t max = floor ω ] π ω. Remark Here the complete estimation steps with a priori information of the parameter to be estimated. Supposing the a priori information is: ω 0 ω ω, ω + ω]. The steps of estimation algorithm are given as follows: (i Preparing the initial state with Eq.(3 and the optimal parameter θ = π/ in Lemma 1: ψ 0 = cos θ 0 + sin θ 1 = 1 (

5 No. 5 Communications in Theoretical Physics 645 The parameter ω 0 to be estimated is loaded into the system with the Hamiltonian H = (1/ω 0 σ z during the evolution process. (ii After a period evolution time with Eq. (19, performing measurement for the density matrix ρ with measurement operators E 1 and E in Eq. (5 and the optimal a = 1/ and b = ±1/ in Lemma 1. Repeating the experiments N times. Counting the results corresponding to E 1 and E, noted as N 1 and N. Calculating the frequencies of N 1 and N and marking them as P 1 and P. (iii Estimating ω 0 with Eq. (7. It should be noticed that the frequencies in step (iii become the probabilities in Eqs. (10 and (11 corresponding the results of E 1 and E according to the Large Number Theorem in ideal situation. To analyze the effectiveness of our algorithm, we have the following formula D ω0 ω, (8 π where D ω0 is the deviation of the parameter to be estimated and ω is the variation range of the a priori information. According to Eqs. (1, (19 and (3, we have D ω0 = V ω0 1 = 1 Fω0 t. max Let K = ω/ ω, and suppose the sequence {c n }, n N, c n = 1/(4n 3, when K (c n+1, c n ], 1 1 ] 1 ] k0 = floor 4( K 1 = 4 (4n 3 1 = n 1, then ω D ω0 ( ω t max = ω k0 + 1 π ω ( = K (n n 3 π 4n 3 = π. π = 4n 3 Kπ Therefore, Eq. (8 holds, meaning that the lower bound of the parameter estimation deviation is determined by ω. Four typical values of ω/ ω are selected to examine the Fisher information, the variance, the deviation of ω 0 and D ω0 / ω in Table 1. An example is given as follows. Setting ω = 100 as the dimensionless standard value. Supposing the true value of ω 0 is 98 and ω = 5. Estimating ω 0 with (i (iii steps in the algorithm. Table 1 A priori information and the values of some critical parameter estimation indexes corresponding to 4 typical points of ω/ ω. ω/ ω 0% 10% 1% 0.1% Fisher information Variance Deviation If the process (i (iii is repeated N = 10 4 times and the number of the results corresponding to E 1 and E are N 1 = 758 and N = 47, P 1 = and P = k0 = 4 and t max = 17π/00 from Eqs. (4 and (5 are also needed to estimate ω 0 with Eq. (7 in step (iii. The estimation value of ω 0 is ˆω 0 = 1 t arccos(p 1 P + k0π] = max The performance indexes for the parameter ω 0 estimation are as follows: the Fisher information F ω0 equals , the variance V ω0 of ω 0 is 14.00, the corresponding standard deviation D ω0 equals Conclusion In this paper, we discuss the parameter ω 0 estimation for one qubit in a closed quantum system in two steps. Firstly, Eq. (9 is obtained based on the intermediate variable v = cos(ω 0 t, and the parameter values in this equation are determined, including the optimal initial state and the POVM measurement operators. The optimal initial state can achieve the maximum quantum Fisher information. Secondly, the evolution time is determined by the Eq. (19 through v under a priori information about the range of ω 0. The relationship of the parameter estimation deviation and the a priori information is D ω0 (/π ω. References 1] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306 ( ] G. Y. Xiang and G. C. Guo, Chin. Phys. B ( ] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 96 (011. 4] K. Matsumoto, J. Phys. A: Math. Gen. 35 ( ] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature (London 49 ( ] P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gas-

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