Optimal condition for measurement observable via error-propagation
OOptimal condition for measurement observable viaerror-propagation
Wei Zhong , Xiao-Ming Lu , Xiao-Xing Jing and XiaoguangWang Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University,Hangzhou 310027, China. Centre for Quantum Technologies, National University of Singapore, 3 ScienceDrive 2, Singapore 117543, Singapore.E-mail: [email protected]
Abstract.
Propagation of error is a widely used estimation tool in experiments,where the estimation precision of the parameter depends on the fluctuation of thephysical observable. Thus which observable is chosen will greatly affect the estimationsensitivity. Here we study the optimal observable for the ultimate sensitivity boundedby the quantum Cram´er-Rao theorem in parameter estimation. By invoking theSchr¨odinger-Robertson uncertainty relation, we derive the necessary and sufficientcondition for the optimal observables saturating the ultimate sensitivity for singleparameter estimate. By applying this condition to Greenberg-Horne-Zeilinger states,we obtain the general expression of the optimal observable for separable measurementsto achieve the Heisenberg-limit precision and show that it is closely related to theparity measurement. However, Jose et al [Phys. Rev. A , 022330 (2013)] haveclaimed that the Heisenberg limit may not be obtained via separable measurements.We show this claim is incorrect. Submitted to:
J. Phys. A: Math. Gen.
PACS numbers: 06.20.Dk, 42.50.St, 03.65.Ta, 03.67.-a, a r X i v : . [ qu a n t - ph ] O c t ptimal condition for measurement observable via error-propagation
1. Introduction
An essential task in quantum parameter estimation is to suppress the fundamental boundon measurement precision imposed by quantum mechanics. Various quantum strategieshave been developed to enhance the accuracy of the parameter estimation, which areclosely related to some practical applications, such as the Ramsey spectroscopies, atomicclocks, and the gravitational wave detection [1, 2, 3, 4, 5, 6, 7, 8]. Two approaches incommon use for high-precision measurements are the parallel protocol with correlatedmulti-probes [9] and multi-round protocol with a single probe [10, 11]. Most recently,some novel methods, like environment-assisted metrology [12] and enhanced metrologyby quantum error correction [13, 14, 15, 16], were raised to achieve high precision inrealistic experiments.Rather than engineering the sensitivity-enhanced strategies, we concentrate on theproblem of how to attain the maximal sensitivity in realistic experiments. In general,a noiseless procedure of the quantum single parameter estimation can be abstractlymodeled by four steps (see figure 1): (i) preparing the input state ρ in , (ii) parameterizingit under the evolution of the parameter-dependent Hamiltonian, for instance, a unitaryevolution U ϕ with ϕ the parameter to be estimated, (iii) performing measurements of theobservable ˆ O on the output state ρ ϕ , (iv) finally estimating the value of the parameterfrom the estimator ϕ est as a function of the outcomes of the measurements.From estimation theory, the estimation precision is statistically measured by theunits-corrected mean-square deviation of the estimator ϕ est from the true value ϕ [17, 18], ( δϕ ) := (cid:68)(cid:16) ϕ est | ∂ ϕ (cid:104) ϕ est (cid:105) av | − ϕ (cid:17) (cid:69) av , (1)where the brackets (cid:104) (cid:105) av denote statistical average and the derivative ∂ ϕ (cid:104) ϕ est (cid:105) ≡ ∂ (cid:104) ϕ est (cid:105) /∂ϕ removes the local difference in the “units” of ϕ est and ϕ . Whatever isthe measurement scheme employed, the ultimate limit to the precision of the unbiased input state parametrization estimator (cid:1847) (cid:3101) (cid:2025) (cid:2919)(cid:2924) (cid:2030) (cid:2915)(cid:2929)(cid:2930) (cid:2025) (cid:3101) (cid:3)(cid:2281) (cid:3553) output state measurement Figure 1: The schematic representation of a general scheme of (noiseless) quantumparameter estimation is composed of four components: input state ρ in ,parametrization process U ϕ , measurements ˆ O , and estimator ϕ est . Here, weconcentrate on the part in shadow to find the optimal ˆ O attaining the highestsensitivity to the parameter ϕ in ρ ϕ . ptimal condition for measurement observable via error-propagation δϕ ) ≥ ( υ F ϕ ) − , (2)where υ is the repetitions of the experiment and F ϕ is the quantum Fisher information(QFI) (see equation (7) for definition), which measures the statistical distinguishabilityof the parameter in quantum states. This bound is asymptotically achieved for large υ under optimal measurements followed by the maximum likelihood estimator [17, 18, 19,20]. On the other hand, it is well-known that error-propagation is a widely acceptabletheory in experiments [1, 3, 9, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Withthis theory, to estimate the parameter ϕ is reduced to measuring the average valueof a physical observable ˆ O . After repeating the experiment υ times, the real accessibleprecision on ϕ is given by the error-propagation formula as follows [1, 3, 9, 21, 22, 23, 24],( δϕ ) := 1 υ (cid:104) (∆ ˆ O ) (cid:105)| ∂ ϕ (cid:104) ˆ O(cid:105)| , (3)where ∆ ˆ O = ˆ O − (cid:104) ˆ O(cid:105) and (cid:104) ˆ O(cid:105) = Tr( ρ ϕ ˆ O ). Note that the two estimation errors definedin equations (1) and (3) are closely related.To show the relationship between the two kinds of the estimation errors, ( δϕ ) and ( δϕ ) , we introduce ∆ ϕ est := ϕ est − (cid:104) ϕ est (cid:105) av . Then, it is easy to show that [17]( δϕ ) = (cid:104) (∆ ϕ est ) (cid:105) av | ∂ ϕ (cid:104) ϕ est (cid:105) av | + (cid:68) ϕ est | ∂ ϕ (cid:104) ϕ est (cid:105) av | − ϕ (cid:69) . (4)When viewing the arithmetic mean of the measurement outcomes of ˆ O over repetitionsof the experiment as the estimator in the quantum setting, one has in general ( δϕ ) ≥ ( δϕ ) ≥ ( υ F ϕ ) − by noting that (cid:104) (∆ ϕ est ) (cid:105) = (cid:104) (∆ ˆ O ) (cid:105) /υ for sufficiently large υ according to the central limit theorem [34] and comparison of the two definitions ofthe errors given by equations (1) and (3). In such situation, ( δϕ ) and ( δϕ ) have thesame QCRB, and the saturation of the former implies that of the latter.The formula equation (3) indicates that the fluctuation of the observable ˆ O propagates to the estimated values of the parameter ϕ . This means that what kindsobservable ˆ O employed directly affects the estimating precision of the parameter ϕ . Thepurpose of this paper is to address the question of with which kind of observable doesthe estimation error given by equation (3) achieve the QCRB given by equation (2).In this paper, we derive the necessary and sufficient (N&S) condition for theoptimal observable saturating the QCRB for the single parameter estimation by usingthe Schr¨odinger-Robertson uncertainty relation (SRUR). We then apply this conditionto GHZ states and find the general form of the optimal observable for separablemeasurements to achieve the Heisenberg-limit sensitivity (i.e., 1 /N ). Moreover, wediscuss the relation between the optimal separable observable and parity measurements.However, Jose et al. , in a recent work [35], made a contradictory conclusion with respectto the above result. They claimed that separable measurements are impossible to go ptimal condition for measurement observable via error-propagation / √ N ) for any entangled states. To clarify this issue,we revisit the method in [35] and show the causes for this inconsistency.This paper is structured as follows. In section 2, we first briefly review the singleparameter estimation and obtain the N&S condition for the optimal observable. Insection 3, we give an application of this condition to obtain the optimal separableobservables for GHZ states to saturate the Heisenberg-limit precision. In section 4, wefurther elucidate the reasons for contradiction between the result given in [35] and ours.At last, a conclusion is given in section 5.
2. N&S condition for optimal observable in single parameter estimation
We start by a brief review of quantum single parameter estimation via the generalestimator. Consider a parametric family of density matrices ρ ϕ containing an unknownparameter ϕ to be estimated. Suppose that the general quantum measurementperformed on ρ ϕ is characterized by a positive-operator-valued measure ˆ M := { ˆ M x } with x the results of measurement. The value of the parameter is inferred via anestimator ϕ est , which maps the measurement outcomes to the estimated value. Afterrepeating the experiment υ times, the standard estimation error ( δϕ ) in equation (1)is bounded from below as( δϕ ) ≥ ( υF ϕ ) − , (5)where F ϕ := (cid:88) x p ϕ ( x )[ ∂ ϕ ln p ϕ ( x )] (6)is the (classical) Fisher information of the measurement-induced probability distribution p ϕ ( x ) = Tr( ρ ϕ ˆ M x ). The maximization over all POVMs gives rise to the so-called QFI,which is defined by F ϕ := Tr( ρ ϕ ˆ L ϕ ) . (7)Hence, a more tighter bound of equation (5) is given by equation (2). Here ˆ L ϕ isthe symmetric logarithmic derivative (SLD) operator, which is a Hermitian operatordetermined by ∂ ϕ ρ ϕ = 12 [ ρ ϕ , ˆ L ϕ ] + (8)with [ · , · ] + denoting the anti-commutator, see reference [17]. It is remarkable that ˆ L ϕ may not be uniquely determined by equation (8) when ρ ϕ is not of full rank [36].However, in general the value of the parameter ϕ may not be directly measured.The most general method of estimating the value of ϕ in practice involves measurementscorresponding to a physical observable ˆ O which is generally ϕ -independent. In suchcases, the estimation error is given by the error-propagation formula equation (3),in which the fluctuations on the observable ˆ O propagate to the uncertainty in theestimation of ϕ . In the following, we follow Hotta and Ozawa [24] to derive the achievablelower bound of the estimation error ( δϕ ) by using the SRUR. ptimal condition for measurement observable via error-propagation X, ˆ Y must obey the following inequality (cid:104) (∆ ˆ X ) (cid:105)(cid:104) (∆ ˆ Y ) (cid:105) ≥ |(cid:104) [ ˆ X, ˆ Y ] (cid:105)| + 14 (cid:104) [∆ ˆ X, ∆ ˆ Y ] + (cid:105) , (9)where [ · , · ] denotes the commutator. The SRUR follows from the Schwarz inequality forthe Hilbert-Schmidt inner product, and naturally reduces to the Heisenberg uncertaintyrelation under the condition (cid:104) [∆ ˆ X, ∆ ˆ Y ] + (cid:105) = 0. By substituting ˆ X ( ˆ Y ) with ˆ O ( ˆ L ϕ ) andutilizing F ϕ = (cid:104) ˆ L ϕ (cid:105) = (cid:104) (∆ ˆ L ϕ ) (cid:105) , (10)as a result of (cid:104) ˆ L ϕ (cid:105) = 2 ∂ θ Tr( ρ ϕ ) = 0, equation (9) becomes (cid:104) (∆ ˆ O ) (cid:105) F ϕ ≥ |(cid:104) [ ˆ O , ˆ L ϕ ] (cid:105)| + 14 (cid:104) [ ˆ O , ˆ L ϕ ] + (cid:105) . (11)Moreover, since the observable operator ˆ O is independent of ϕ , we have (cid:104) [ ˆ O , ˆ L ϕ ] + (cid:105) = Tr([ ˆ O , ˆ L ϕ ] + ρ ϕ )= Tr( ˆ O [ ˆ L ϕ , ρ ϕ ] + )= 2 ∂ ϕ (cid:104) ˆ O(cid:105) , (12)where the second equality is obtained by employing the cyclic property of the traceoperation, and the third equality is due to the SLD equation (8). Provided that (cid:104) ˆ O(cid:105) isnonzero, combining equations (3), (11) and (12) yields( δϕ ) ≥ υ F ϕ (cid:16) |(cid:104) [ ˆ O , ˆ L ϕ ] (cid:105)| (cid:104) [ ˆ O , ˆ L ϕ ] + (cid:105) (cid:17) (13)= 1 υ F ϕ (cid:104) (cid:16) Im (cid:104) ˆ O ˆ L ϕ (cid:105) Re (cid:104) ˆ O ˆ L ϕ (cid:105) (cid:17) (cid:105) (14) ≥ ( υ F ϕ ) − . (15)The bound in equation (13) describes the achievable sensitivity of ϕ when employingan observable ˆ O . The bound in equation (15) gives the highest precision for ϕ for theoptimal observable ˆ O opt , which coincides with the QCRB in equation (2). It is shownthat the estimation error ( δϕ ) achieves the QCRB only when the two equalities inequations (13) and (15) hold simultaneously.Below, we consider the attainability of the above bounds and give the N&S conditionfor optimal observables. From the N&S condition for equality in the SRUR, the equalityin equation (13) holds if and only if∆ ˆ O√ ρ ϕ = α ˆ L ϕ √ ρ ϕ (16)is satisfied with a nonzero complex number α . Note that we restrict here α (cid:54) = 0 atthe request of (cid:104) [ ˆ O , ˆ L ϕ ] + (cid:105) (cid:54) = 0 in the denominator of equation (13). Furthermore, theequality in equation (15) holds if and only ifIm (cid:104) ˆ O ˆ L ϕ (cid:105) = 0 . (17) ptimal condition for measurement observable via error-propagation α to be a nonzeroreal number, i.e.,∆ ˆ O√ ρ ϕ = α ˆ L ϕ √ ρ ϕ with α ∈ R \{ } . (18)This is the of the optimal observable for density matrix ρ ϕ . It implies that the estimationerror achieves the QCRB given by the QFI for ρ ϕ only when the observable that wechoose satisfies equation (18). This is the main result of the paper. For pure states ρ ϕ = | ψ ϕ (cid:105)(cid:104) ψ ϕ | , the condition (18) is equivalent to∆ ˆ O| ψ ϕ (cid:105) = α ˆ L ϕ | ψ ϕ (cid:105) with α ∈ R \{ } . (19)If we assume that the parameter ϕ here is imprinted via a unitary operation [9],i.e., ρ ϕ = exp( − i ˆ Gϕ ) ρ in exp( i ˆ Gϕ ) with ˆ G the generator, associating with the equality ∂ ϕ ρ ϕ = − i [ ˆ G, ρ ϕ ], then condition (19) further reduces to∆ ˆ O| ψ ϕ (cid:105) = − iα ∆ ˆ G | ψ ϕ (cid:105) with α ∈ R \{ } . (20)This condition was alternatively obtained in Ref. [31]. It is deserved to note that theirproof is only valid in the case of unitary parametrization for pure states, and cannot begeneralized to obtain the condition (18).Here, we discuss the relations between the saturation of the QCRB with respectto ( δϕ ) and that with respect to ( δϕ ) . Following Braunstein and Caves [17], thesaturation of the QCRB with respect to the error ( δϕ ) can be separated as thesaturation of a classical Cram´er-Rao bound (CCRB) equation (5) and finding an optimalmeasurement attaining the QFI. The CCRB can always be asymptotically achievedby the maximum likelihood estimator, so whether the QCRB can be asymptoticallysaturated is determined by whether the measurement attains the QFI. The N&Scondition for the optimal measurement attaining the QFI reads [17] (cid:113) ˆ M x √ ρ ϕ = u x (cid:113) ˆ M x ˆ L ϕ √ ρ ϕ , (21)where { ˆ M x } denotes the POVM of the measurement and u x are real numbers. Inthe following, we show that the N&S condition (18) for the saturation of the QCRBwith respect to ( δϕ ) identifies an optimal measurement attaining QFI. Let ˆ O opt bethe optimal observable satisfying Eq. (18) and P x the eigenprojectors of ˆ O opt with theeigenvalues x . Left multiplying P x on both sides of Eq. (18), it is easy to see that { P x } is the optimal measurement attaining the QFI. That is to say, the projectivemeasurement { P x } followed by the maximum likelihood estimator of the measurementoutcomes saturate the QCRB with respect to the standard estimation error ( δϕ ) .
3. Optimal separable observable for GHZ states
Below, we apply the N&S condition to show the general optimal observable for GHZstates. Let us specifically consider an experimentally realizable Ramsey interferometryto estimate the transition frequency ω of the two-level atoms loaded in the ion trap[1, 2]. The Hamiltonian of the system with N atoms is ˆ H = ( ω/ (cid:80) Ni =1 ˆ σ iz where ˆ σ iz is ptimal condition for measurement observable via error-propagation i th particle. In this setup, the measurements are limitedto be performed separately on each atom. The observable operator may be describedas a tensor product of Hermitian matrices ˆ O = ˆ O ⊗ N q with ˆ O q = a I + a · ˆ σ dependentof four real coefficients { a , a , a , a } , where I is the identity matrix of dimension 2.Suppose that the input state is the maximally entangled states, i.e., GHZ states,which provides the Heisenberg-limit-scaling sensitivity of frequency estimation in theabsence of noise [1, 9, 39]. Under the time evolution ˆ U = exp ( − i ˆ Ht ), the output statecan be represented as | ψ GHZ ( ϕ ) (cid:105) = 1 √ | (cid:105) ⊗ N + e iNϕ | (cid:105) ⊗ N ) , (22)up to an irrelevant global phase with ϕ = ωt . Here, we adopt the standard notationwhere | (cid:105) and | (cid:105) are the eigenvectors of σ z corresponding to eigenvalues +1 and −
1, respectively. To determine the optimal separable observable ˆ O , we need tofind the solutions of the coefficients { a , a , a , a } to satisfy equation (19). Withˆ L ϕ = 2 ∂ ϕ ( | ψ ϕ (cid:105)(cid:104) ψ ϕ | ) for pure states, the SLD operator for the state of equation (22)is given by ˆ L ϕ = − iN e − iNϕ ( | (cid:105)(cid:104) | ) ⊗ N + iN e iNϕ ( | (cid:105)(cid:104) | ) ⊗ N . (23)We find that equation (19) is always satisfied for a = a = 0 and arbitrary realnumber a , a that do not vanish simultaneously. Therefore, the general expressionof the optimal separable observable is given byˆ O opt = ( a ˆ σ x + a ˆ σ y ) ⊗ N , (24)which is independent of the parameter ϕ , i.e., globally optimal in the whole range ofthe parameter. It is easy to check that such observables saturate the Heisenberg-limitsensitivity. Actually, according to the error-propagation formula equation (3), we have δϕ GHZ = 1 √ υ (cid:113) (cid:104) ˆ O (cid:105) − (cid:104) ˆ O opt (cid:105) | ∂ ϕ (cid:104) ˆ O opt (cid:105)| = 1 √ υN , (25)as a result of (cid:104) ˆ O opt (cid:105) = Re[ e − iNϕ ( a + ia ) N ] , (26) (cid:104) ˆ O (cid:105) = ( a + a ) N . (27)When setting a = 1 , a = a = a = 0, the optimal observable in equation (24) reducesto ˆ σ ⊗ Nx , as given in [9]. Note that here measuring the observable ˆ σ ⊗ Nx fails to attain theHeisenberg limit for the cases of ϕ = kπ/N, ( k ∈ Z ) in which equation (25) becomessingular. Besides, we note that measuring the spin observable ˆ σ ⊗ Ny also fail in thesecases when N is even, and it is useful except for the cases of ϕ = (2 k + 1) π/ N, ( k ∈ Z )when N is odd.We next show that the optimal observable in the form of equation (24) is closelyrelated to the parity measurement proposed originally by Bollinger et al [3]. As iswell known, in the standard Ramsey interferometry, there are generally two Ramsey ptimal condition for measurement observable via error-propagation ϕ ), andmeasurements often take place after the second pulse [1, 3]. Here the action of the pulseis modeled by a π/ y axis, i.e., R y [ π ] = exp[ − i ( π ) ˆ J y ],and the measurement observable is denoted as the operator ˆ O f . With equation (24),one has ˆ O f = R † y (cid:104) π (cid:105) ˆ O R y (cid:104) π (cid:105) = ( a ˆ σ z + a ˆ σ y ) ⊗ N . (28)When setting a = 1 , a = 0, equation (28) reduces toˆ O f = ˆ σ ⊗ Nz ≡ ( − j − ˆ J z (29)with j = N/
2, which is the so-called parity measurement [3]. It is shown that onlya parity measurement is necessary for the optimal estimate of the phase parameter ϕ for GHZ states, and it is more experimentally feasible than the detection strategy, asdiscussed in [9], that applies local operations and classical communication.
4. Further discussions
However, in a recent work [35], it was pointed out that the separable measurement (therestricted readout procedure) might not be possible to go beyond the shot-noise limiteven for arbitrary entangled states. It seems that this conclusion is inconsistent withours in the above discussion. In what follows, we clarify this issue by revisiting themethod in [35] and showing the causes for this inconsistency.For simplicity, let us consider the two-qubit parametric GHZ state | ψ (2)GHZ ( ϕ ) (cid:105) = 1 √ | (cid:105) + e iϕ | (cid:105) ) . (30)Following Ref. [35], we restrict the separable measurement to be the projectivemeasurements {| + (cid:105)(cid:104) + | , |−(cid:105)(cid:104)−|} for each qubit with |±(cid:105) = 1 √ | (cid:105) ± | (cid:105) ) . (31)According to the condition of equation (21), whether the above restricted measurementpresented by equation (31) is the optimal measurement saturating the QCRB can betested by asking whether or not the operators of the formˆ K = λ ++ | + + (cid:105)(cid:104) + + | + λ + − | + −(cid:105)(cid:104) + − | + λ − + | − + (cid:105)(cid:104)− + | + λ −− | − −(cid:105)(cid:104)− − | (32)can be the SLD operator for the state of equation (30). By domenstrating that for thestate in equation (30) with ϕ = 0, there is no solution of the SLD equation (8) for thecoefficients { λ ++ , λ + − , λ − + , λ −− } in equation (32), the authors in Ref. [35] claimed thatthe projective measurement about {| + + (cid:105) , | + −(cid:105) , | − + (cid:105) , | − −(cid:105)} is not the optimalmeasurement for the state of equation (30).However, as we showed in the Sec. 3, σ x ⊗ σ x is an optimal observable saturatingthe QCRB with respect to ( δϕ ) for the states (30). Although the estimation error ptimal condition for measurement observable via error-propagation δϕ ) , a contradiction still arises, as the projectivemeasurement of σ x ⊗ σ x attains the QFI of states (30) (see the end in Sec. 2) so that {| + + (cid:105) , | + −(cid:105) , | − + (cid:105) , | − −(cid:105)} is the optimal measurement regarding the estimationerror ( δϕ ) . Below, we shall show that actually for any other point except for ϕ = kπ/ , ( k ∈ Z ) in the range of the parameter, there do exist the SLD operatorin form of equations (32).First, note that the SLD operator for the non-full-rank density matrices is notuniquely determined, but ˆ L ϕ ρ ϕ (or ˆ L ϕ | ψ ϕ (cid:105) for pure state) is uniquely determined.Second, from equation (23), we seeˆ L ϕ = − ie − iϕ | (cid:105)(cid:104) | + 2 ie iϕ | (cid:105)(cid:104) | . (33)is a SLD operator for the state of equation (30). Third, since ˆ L ϕ | ψ ϕ (cid:105) is uniquelydetermined, then if ˆ K is the SLD operator for | ψ ϕ (cid:105) if and only ifˆ L ϕ | ψ ϕ (cid:105) = ˆ K | ψ ϕ (cid:105) (34)is satisfied. Thus, substituting equations (30), (32) and (33) into equation (34), weobtain the solutions for the coefficients as λ ++ = λ −− = − ϕ, λ + − = λ − + = 2 cot ϕ. (35)The above solutions are singular for ϕ = kπ/ , ( k ∈ Z ), which coincide with the resultsdiscussed below equation (27). Note that here the ϕ = 0 , ( k = 0) case is just consideredin Ref. [35]. Whilst, for a general value of the parameter except those singular points,the restricted separable measurement considered here indeed saturate the Heisenberg-limit-scaling sensitivity for the parametric state of equation (30). Moreover, it is easy tocheck that the same results of Eq. (35) can be obtained when restricting the separablemeasurement to be the projective measurements {| + (cid:105) y (cid:104) + | , |−(cid:105) y (cid:104)−|} for each qubit with |±(cid:105) y = 1 √ | (cid:105) ± i | (cid:105) ) (36)the eigenvectors of σ y . This is coincided with the result shown below equation (27) thatmeasuring the observable σ ⊗ Ny fails to attain Heisenberg limit for the ϕ = kπ/N, ( k ∈ Z )cases when N is even.
5. Conclusion
We have addressed the optimization problem of measurements for achieving the ultimatesensitivity determined by the QCRB. From the propagation of error, we derive theN&S condition of the optimal observables for single parameter estimate by using theSRUR. As an application of this condition, we examine the optimal observables forGHZ states to achieve the ultimate sensitivity at the Heisenberg limit. We consider anexperimentally feasible case that the observable operators are restricted to separablyacting on the subsystem. We then find the general expression of the optimal separableobservable by applying the N&S condition, and show that it is exactly equivalent tothe parity measurement when applying a π/ et al in ptimal condition for measurement observable via error-propagation only for some particular valuesof the parameter. Our results may be helpful for further investigation of the quantummetrology. Acknowledgments
We would like to thank Dr. Heng-Na Xiong and Dr. Qing-Shou Tan for helpfuldiscussions. We also thank the second referee for constructive suggestions. This workis supported by the NFRPC with Grant No. 2012CB921602, the NSFC with GrantsNo. 11025527 and No. 10935010, and National Research Foundation and Ministry ofEducation, Singapore, with Grant No. WBS: R-710-000-008-271.
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