Quantum metrology with unitary parametrization processes
aa r X i v : . [ qu a n t - ph ] J a n Quantum metrology with unitary parametrization processes
Jing Liu, Xiao-Xing Jing, and Xiaoguang Wang
1, 2, ∗ Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China
Quantum Fisher information is a central quantity in quantum metrology. We discuss an alter-native representation of quantum Fisher information for unitary parametrization processes. In thisrepresentation, all information of parametrization transformation, i.e., the entire dynamical infor-mation, is totally involved in a Hermitian operator H . Utilizing this representation, quantum Fisherinformation is only determined by H and the initial state. Furthermore, H can be expressed in anexpanded form. The highlights of this form is that it can bring great convenience during the calcu-lation for the Hamiltonians owning recursive commutations with their partial derivative. We applythis representation in a collective spin system and show the specific expression of H . For a simplecase, a spin-half system, the quantum Fisher information is given and the optimal states to accessmaximum quantum Fisher information are found. Moreover, for an exponential form initial state, ananalytical expression of quantum Fisher information by H operator is provided. The multiparameterquantum metrology is also considered and discussed utilizing this representation. PACS numbers: 03.67.-a, 03.65.Ta, 06.20.-f.
How to precisely measure the values of physical quantities, such as the phases of light in interferometers, magneticstrength, gravity and so on, is always an important topic in physics. Obtaining high-precision values of these quantitieswill not only bring an obvious advantage in applied sciences, including the atomic clocks, physical geography, civilnavigation and even military industry, but also accelerate the development of fundamental theories. One vivid exampleis the search for gravitational waves. Quantum metrology is such a field attempting to find optimal methods to offerhighest precision of a parameter that under estimation. In recently decades, many protocols and strategies have beenproposed and realized to improve the precisions of various parameters [1–24]. Some of them can even approach to theHeisenberg limit, a limit given by the quantum mechanics, showing the power of quantum metrology.Quantum Fisher information is important in quantum metrology because it depicts the theoretical lowest bound ofthe parameter’s variance according to Cramér-Rao inequality [25, 26]. The quantum Fisher information for parameter α is defined as F = Tr( ρL ) , where ρ is a density matrix dependent on α and L is the symmetric logarithmic derivative(SLD) operator and determined by the equation ∂ α ρ = ( ρL + Lρ ) / . For a multiparameter system, the counterpartof quantum Fisher information is called quantum Fisher information matrix F , of which the element is defined as F αβ = Tr( ρ { L α , L β } ) , where L α , L β are the SLD operators for parameters α and β , respectively.Recently, it has been found [27] that quantum Fisher information can be expressed in an alternative representation,that all information of parametrization process in quantum Fisher information is involved in a Hermitian operator H .This operator characterizes the dynamical property of the parametrization process, and totally independent of theselection of initial states. Utilizing this representation, the quantum Fisher information is only determined by H andthe initial state.In this report, we give a general expression of quantum Fisher information and quantum Fisher information matrixutilizing H operator. For a unitary parametrization process, H can be expressed in an expanded form. This formis particularly useful when the Hamiltonian owns a recursive commutation relation with its derivative on parameterestimation. We calculate the specific expression of H in a collective spin system, and provide an analytical expressionof quantum Fisher information in a spin-half system for any initial state. Based on this expression, all optimal statesto access maximum quantum Fisher information are found in this system. Furthermore, considering this spin-halfsystem as a multiparameter system, the quantum Fisher information matrix, can be easily obtained by the knownform of H in single parameter estimations. On the other hand, inspired by a recent work [28], for an exponential forminitial state, we provide an analytical expression of quantum Fisher information using H operator. A demonstrationwith a spin thermal initial state is given in this scenario. The maximum quantum Fisher information and the optimalcondition are also discussed. ResultsQuantum Fisher information with H operator. For a general unitary parametrization transformation, the ∗ Electronic address: [email protected] parametrized state ρ ( α ) is expressed by ρ ( α ) = U ( α ) ρ U † ( α ) , where ρ is a state independent of α . In this paper,since the parameter α is only brought by U ( α ) , not the initial state ρ , we use U instead of U ( α ) for short. Denotethe spectral decomposition of ρ as ρ = P Mi =1 p i | ψ i ih ψ i | , where p i and | ψ i i are the i th eigenvalue and eigenstate of ρ and M is the dimension of the support of ρ . It is easy to see that p i and U | ψ i i are the corresponding eigenvalueand eigenstate of ρ ( α ) , respectively. The quantum Fisher information for ρ ( α ) can then be expressed by [29, 30] F = M X i =1 p i h ∆ Hi i − X i = j p i p j p i + p j |h ψ i |H| ψ j i| , (1)where H := i ( ∂ α U † ) U (2)is a Hermitian operator since the equality ( ∂ α U † ) U = − U † ( ∂ α U ) . Meanwhile, h ∆ Hi i = h ψ i |H | ψ i i − h ψ i |H| ψ i i (3)is the variance of H on the i th eigenstate of ρ . When ∂ α U commutes with U , H can be explained as the generator ofthe parametrization transformation [27]. The expression (1) of quantum Fisher information is not just a formalizedrepresentation. The operator H is only determined by the parametrization process, that is the dynamics of thesystem or the device. For a known dynamical process of a parameter, i.e., known system’s Hamiltonian, H is a settledoperator and can be obtained in principle. In this representation, the calculation of quantum Fisher information isseparated into two parts: the diagonalization of initial state and calculation of H . For a general 2-dimensional state,the quantum Fisher information reduces to F qubit = 4 (cid:0) ρ − (cid:1) h ∆ Hi . (4)The subscript of the variance can be chosen as 1 or 2 as any Hermitian operator’s variances on two orthonormal statesare equivalent in 2-dimentional Hilbert space. For a pure state, the quantum Fisher information can be easily obtainfrom Eq. (4) with taking the purity Tr ρ = 1 and the variance on that pure state, i.e., [27] F pure = 4 h ∆ Hi in . (5)Namely, the quantum Fisher information is proportional to the variance of H on the initial state. In this scenario,denote the initial state ρ = | ψ ih ψ | , the quantum Fisher information can be rewritten into F pure = h ψ | L | ψ i ,with the effective SLD operator L eff = i H , | ψ ih ψ | ] . (6)For a well applied form of parametrization transformation U = exp( − itH α ) [27], where ¯ h has been set as 1 inPlanck unit, and being aware of the equation ∂ α e A = ˆ e sA ( ∂ α A ) e (1 − s ) A ds, (7) H can then be expressed by H = − ˆ t e isH α ( ∂ α H α ) e − isH α ds. (8)Defining a superoperator A × as A × ( · ) := [ A, · ] , H can be written in an expanded form H = i ∞ X n =0 f n H × nα ( ∂ α H α ) , (9)where the coefficient f n = ( it ) n +1 ( n + 1)! . (10)In many real problems, the recursive commutations in Eq. (9) can either repeat or terminate [28], indicating ananalytical expression of H . Thus, this representation of quantum Fisher information would be very useful in theseproblems. For the simplest case that H α = αH , all terms vanish but the first one, then H = − tH . When [ H α , ∂ α H α ] = C , with C a constant matrix or proportional to H α , only the first and second terms remain. In this case, H reducesto − t ( ∂ α H α + itC/ . A more interesting case is that [ H α , ∂ α H α ] = c∂ α H α , with c a nonzero constant number, then H can be written in the form H = ic [exp( itc ) − ∂ α H α . (11)In the following we give an example to exhibit Eq. (9). Consider the interaction Hamiltonian of a collective spinsystem in a magnetic field H θ = B (cos θJ x + sin θJ z ) = BJ n , (12)where J n = n · J with n = (cos θ, , sin θ ) T and J = ( J x , J y , J z ) T . B is the amplitude of the external magneticfield and θ is the angle between the field and the collective spin. Here J i = P k σ ( k ) i / for i = x, y, z with σ ( k ) i thePauli matrix for k th spin. Taking θ as the parameter under estimation, H can be expressed by H = 2 (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) Bt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) J n , (13)where J n = n · J with the vector n = µ (cid:18) cos (cid:18) Bt (cid:19) sin θ, − sin (cid:18) Bt (cid:19) , − cos (cid:18) Bt (cid:19) cos θ (cid:19) T , where µ = sgn(sin( Bt/ is the sign function and n is normalized.The operator H for Hamiltonian (12) may be also available to be solved using the procedure in Ref. [27], inthe (2j+1)-dimensional eigenspace of H θ (j is the total spin). In principle, the eigenstates of H θ can be found byrotating the Dicke state into the same direction of H θ . However, even one can analytically obtain all the eigenvaluesand eigenvectors, it still requires a large amount of calculations to obtain H , especially when the spin numbers aretremendous. Comparably, utilizing Eq. (9), it only takes a few steps of calculation, which can be found in the method.This is a major advantage of the expanded form of H .Utilizing Eq. (13), one can immediately obtain the form of H for a spin-half system H qubit = (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) Bt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n · σ , (14)with σ = ( σ x , σ y , σ z ) T , which was also discussed in the Hamiltonian eigenbasis in Ref. [27]. For any 2-dimensionalstate, based on Eq. (4), the quantum Fisher information can be expressed by F θ = 4 sin (cid:18) Bt (cid:19) | r in | (cid:2) − ( n · r e ) (cid:3) , (15)where r in = ( h σ x i , h σ y i , h σ z i ) T is the Bloch vector of the initial state ρ and r e is the Bloch vector of any eigenstate of ρ . For pure states, there is r e = r in and | r in | = 1 . Since the Bloch vector of a 2-dimensional state satisfies | r in | ≤ ,it can be found that the maximum value of Eq. (15) is F max θ = 4 sin (cid:18) Bt (cid:19) , (16)which can be saturated when | r in | = 1 and n · r in = 0 , namely, the optimal state to access maximum quantum Fisherinformation here is a pure state perpendicular to n , as shown in Fig. 1. In this figure, the yellow sphere representsthe Bloch sphere and the blue arrow represents the vector n . It can be found that all states on the joint ring ofthe green plane and surface of Bloch sphere can access the maximum quantum Fisher information, i.e., all states onthis ring are optimal states. One simple example is r opt = n , and another one is the superposition state of twoeigenstates of H [22, 27].Alternatively, B could be the parameter that under estimation. In the spin-half case, with respect to B , H B = − t n · σ / , then the quantum Fisher information can be expressed by F B = t | r in | (cid:2) − ( n · r e ) (cid:3) . (17)The optimal states to access the maximum value F max B = t are the pure states vertical to n . Exponential form initial state.
For an exponential form initial state ρ = exp( G ) , the parametrized state reads ρ α = U ρ U † = exp( U G U † ) . (18)Recently, Jiang [28] studied the quantum Fisher information for exponential states and gave a general form of SLDoperator. In his theory, the SLD operator can be expanded as L = ∞ X n =0 g n G × n ( ∂ α G ) , (19)where the coefficient g n = 4(2 n +2 − B n +2 ( n + 2)! (20)for even n and g n vanishes for odd n . Here B n +2 is the ( n + 2) th Bernoulli number and in our case, G = U G U † .Through some straightforward calculation, the derivative of G on α reads ∂ α G = − iU (cid:0) G × H (cid:1) U † . (21)Based on this equation, the n th order term in Eq. (19) is G × n ( ∂ α G ) = − iU (cid:2) ( G × ) n +1 H (cid:3) U † , (22)where H is given by Eq. (9). Generally, it is known that the quantum Fisher information reads F = Tr( U ρ U † L ) = Tr (cid:0) ρ L (cid:1) , (23)where the effective SLD operator L eff = U † LU . The effective SLD operator for pure states is already shown in Eq. (6).Substituting Eq. (22) into Eq. (19), the effective SLD operator can be expanded as L eff = − i ∞ X n =0 g n ( G × ) n +1 H . (24)In most mixed states cases, to obtain quantum Fisher information, the diagonalization of initial state is inevitable,which is the reason why the usual form of quantum Fisher information is expressed in the eigenbasis of density matrix.Thus, it is worth to study the expression of effective SLD operator and quantum Fisher information in the eigenbasisof G . We denote the i th eigenvalue and eigenstate of G as a i and | φ i i , and in the eigenbasis of G , the element of G × n H satisfies the recursion relation [ G × n H ] ij = ( a i − a j )[( G × ) n − H ] ij , (25)where [ · ] ij := h φ i | · | φ j i . Utilizing this recursive equation, a general formula of n th order term can be obtained, (cid:2) G × n H (cid:3) ij = ( a i − a j ) n H ij . (26)Substituting above equation into the expression of L eff and being aware of the equality ∞ X n =0 g n ( a i − a j ) n +1 = 2 tanh (cid:18) a i − a j (cid:19) , (27)the element of effective SLD operator in Eq. (24) can be written as [ L eff ] ij = − i (cid:18) a i − a j (cid:19) H ij . (28)Based on the equation F = Tr( e G L ) , the quantum Fisher information in the eigenbasis of G can finally beexpressed by F = X i>j e a i + e a j ) tanh (cid:18) a i − a j (cid:19) |H ij | . (29)This is one of the main results in this paper. In some real problems, the eigenspace of G could be find easily. Forinstance, the eigenspace of a bosonic thermal state is the Fock space. Thus, as long as the formula of H in Fock spaceis established, the quantum Fisher information can be obtained from Eq. (29).Now we exhibit Eq. (29) with a spin-half thermal state. The initial state is taken as ρ = 1 Z exp ( − βσ z ) , = exp ( − βσ z − ln Z ) , (30)where β = 1 / ( k b T ) with k b the Boltzmann constant and T the temperature. In Planck unit, k b = 1 . The partitionfunction reads Z = Tr[exp( − βσ z )] = 2 cosh β . In this case, G = − βσ z − ln z . Denoting the eigenstates of σ z as | i and | i , i.e., σ z = | ih |− | ih | , the eigenvalues of G read a = − βσ z − ln z and a = βσ z − ln z . The parametrizationprocess is still taken as H θ = B n · σ / with θ the parameter under estimation, indicating that H = (cid:12)(cid:12) sin (cid:0) Bt (cid:1)(cid:12)(cid:12) n · σ , then the squared norm of the off-diagonal element of H in the eigenbasis of σ z reads |H | = sin (cid:18) Bt (cid:19)(cid:20) − cos θ cos (cid:18) Bt (cid:19)(cid:21) . (31)Immediately, the quantum Fisher information can be obtained from Eq. (29) as F T = 4 tanh β sin (cid:18) Bt (cid:19)(cid:20) − cos θ cos (cid:18) Bt (cid:19)(cid:21) . (32)The maximum value of above expression is obtained at Bt = (4 k + 1) π for k = 0 , , ... and F maxT = 4 tanh ( β ) . (33)From this equation, one can see that the value of maximum quantum Fisher information is only affected by thetemperature. With the increase of temperature, the maximum value reduces. In the other hand, quantum Fisherinformation in Eq. (32) is related to Bt and θ . Fig. 2 shows the quantum Fisher information as a function of Bt and θ . The values of Bt and θ are both within [0 , π ] in the plot. The temperature is set as T = 1 here. From this figure,it can be found that the maximum quantum Fisher information is robust for θ since it is always obtained at Bt = π for any value of θ . Furthermore, this optimal condition of Bt is independent of temperature. With respect to Bt ,there is a large regime near Bt = π in which the quantum Fisher information’s value can surpass , indicating thatthe quantum Fisher information can be still very robust and near its maximum value even when Bt is hard to setexactly at π . Multiparameter processes.
For a multiparameter system, the element of quantum Fisher information matrix inRef. [30] can also be written with H operator, F αβ = M X i =1 p i cov i ( H α , H β ) − X i = j p i p j p i + p j Re( h ψ i |H α | ψ j ih ψ j |H β | ψ i i ) , (34)where U is dependent on a series of parameters α , β and so on, and H m = i ( ∂ m U † ) U, (35)with the index m = α, β, ... . The covariance matrix on the i th eigenstate of initial state is defined as cov i ( H α , H β ) := 12 h ψ i |{H α , H β }| ψ i i − h ψ i |H α | ψ i ih ψ i |H β | ψ i i , with {· , ·} the anti-commutation. For a single qubit system, Eq. (34) reduces to F qubit ,αβ = 4 (cid:0) ρ − (cid:1) cov ( H α , H β ) . (36)Similarly with the single-parameter scenario, the subscript in Eq. (36) can be chosen as 1 or 2 since the covariance fortwo Hermitian operators are the same on two orthonormal states in 2-dimensional Hilbert space. From this equation,the element of quantum Fisher information matrix for pure states can be immediately obtained as F pure ,αβ = 4cov in ( H α , H β ) , (37)namely, for pure states, the element of quantum Fisher information matrix is actually the covariance between two H operators on the initial state. When the total Hamiltonian can be written as P i α i H i and [ H i , H j ] = 0 for any i , j ,above equation can reduce to the covariance between H i and H j [31]. For the diagonal elements, they are exactly thequantum Fisher information for the corresponding parameters.For multiparamter estimations, the Cramér-Rao bound cannot always be achieved. In the scenario of pure states,the condition of this bound to be tight is Im h ψ out | L α L β | ψ out i = 0 , ∀ α, β [32, 33]. Here | ψ out i is dependent on theparameter under estimation. In the unitary parametrization, | ψ out i = U | ψ i and this condition can be rewritten into Im h ψ | L α eff L β eff | ψ i = 0 , ∀ α, β . Here L α ( β )eff = U † L α ( β ) U is the effective SLD operator for parameter α ( β ) . UtilizingEq. (6), this condition can be expressed in the form of H operator, h ψ | [ H α , H β ] | ψ i = 0 , ∀ α, β. (38)In other word, h ψ |H α H β | ψ i needs to be a real number for any α and β . When H α commutes with H β for any α and β , above condition can always be satisfied for any initial state.Generally, for the unitary parametrization process, the element of quantum Fisher information matrix can beexpressed by F = Tr( ρ { L α , L β } ) = Tr( ρ { L α eff , L β eff } ) . From the definition equation of SLD, one can see that L α eff satisfies the equation ∂ θ ρ = U { ρ , L eff } U † / . The quantum Fisher information matrix has more than one definitions.One alternative candidate is using the so-called Right Logarithmic Derivative (RLD) [26, 34, 35], which is defined as ∂ α ρ = ρR α , with R α the RLD. The element of RLD quantum Fisher information matrix can be written as J αβ = Tr (cid:16) ρR α R † β (cid:17) = Tr (cid:16) ρ R α eff R β † eff (cid:17) , (39)where the effective RLD reads R α ( β )eff = U † R α ( β ) U . For a unitary parametrization process, assuming the initial statehas nonzero determinant, R α eff can be expressed by H α and the initial state ρ , i.e., R α eff = i (cid:0) ρ − H α ρ − H α (cid:1) . (40)With this equation, the element of RLD quantum Fisher information matrix can be expressed by J αβ = Tr( H α ρ H β ρ − − H β H α ρ + H α H β ρ ) . (41)When the parametrization process is displacement, this equation can reduces to the corresponding form in Ref. [35].For pure states, the element reads J pure ,αβ = Tr [( ∂ α ρ ) ( ∂ β ρ )] = F pure ,αβ / . Recently, Genoni et al. [35] proposeda most informative Cramér-Rao bound for the total variance of all parameters under estimation. From the relationbetween J pure ,αβ and F pure ,αβ , one can see that Tr F − is always larger than Tr J − , namely, the SLD Cramér-Raobound is always more informative than the RLD counterpart in this scenario.We still consider the spin-half system with the Hamiltonian H = B n · σ / . Take both B and θ as the parametersunder estimations. First, based on aforementioned calculation, the H operator for B and θ read H B = − t n · σ , (42) H θ = (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) Bt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n · σ . (43)Based on the property of Pauli matrices { n · σ , n · σ } = 2 n · n , the anti-commutation in the covariance reads {H B , H θ } = − t (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) Bt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n · n . (44)For a pure initial state, the off-diagonal element of the quantum Fisher information matrix is expressed by F Bθ = 2 t (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) Bt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( n · r in ) ( n · r in ) , (45)where r in is the Bloch vector of the initial pure state and the equality n · n = 0 has been used. When the initial purestate is vertical to n or n , this off-diagonal element vanishes. Compared with the optimal condition for maximumquantum Fisher information for B and θ individually, the Bloch vector n = n × n can optimize both the diagonalelements of quantum Fisher information matrix and vanish the off-diagonal elements. However, all above is onlynecessary conditions for the achievement of Cramér-Rao bound. To find out if the bound can be really achieved, thecondition (38) needs to be checked. In this case, [ H B , H θ ] = − it (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) Bt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n · σ . (46)With this equation, condition (38) reduces to n · r in = 0 , i.e., to make the Cramér-Rao bound achievable, the Blochvector of the initial state needs to in the plane of n and n . Unfortunately, n is not in this plane. Thus, B and θ cannot be optimally joint measured simultaneously.In the plane constructed by n and n , any Bloch vector of pure state can be written as r in = n cos φ + n sin φ ,then we have F BB = t sin φ , F θθ = 4 sin ( Bt/
2) cos φ , and F Bθ = 2 t | sin( Bt/ | cos φ sin φ . From these expressions,one can see that the determinant of quantum Fisher information matrix is zero, i.e., det F = 0 . This fact indicatesthat, utilizing any pure state in this plane, the variances of B and θ cannot be estimated simultaneously through theCramér-Rao theory. Discussion
We have discussed the quantum Fisher information with unitary parametrization utilizing an alternative representa-tion. The total information of the parametrization process is involved in a H operator in this representation. Thisoperator is totally determined by the parameter and parametrization transformation U . As long as the parameterand transformation are taken, H is a settled operator and independent of the initial state. More interestingly, H canbe expressed in an expanded form. For the Hamiltonians owning recursive commutations with their partial derivativeon the parameter under estimation, this expanded form shows a huge advantage. Utilizing this representation, wegive a general analytical expression of quantum Fisher information for an exponential form initial state. Moreover,we have also studied the H representation in multiparameter processes. The condition of Cramér-Rao bound to beachievable for pure states are also presented in the form of H operator. In addition, we give the H representation ofRight Logarithmic Derivative and the corresponding quantum Fisher information matrix.As a demonstration, we apply this representation in a collective spin system and show the expression of H . Fur-thermore, we provide an analytical expression of quantum Fisher information in a spin-half system. If we considerthis system as a multiparameter system, the corresponding quantum Fisher information matrix can also be straight-forwardly obtained by this representation. From these expressions, one can find the optimal states to access themaximum quantum Fisher information. For the parameter B , the optimal state is a pure state vertical to n , andfor the parameter θ , the optimal one is also a pure state, but vertical to n . By analyzing the off-diagonal elementof quantum Fisher information matrix, the states to optimize the diagonal elements and make the off-diagonal ele-ments vanish are found. However, these states fail to satisfy the condition of achievement. Thus, B and θ cannot beoptimally jointed measured. MethodsCollective spin system in a magnetic field.
For the Hamiltonian (12), its derivative on parameter θ is ∂ θ H θ = n ′ · J = J n ′ = − iH × θ J y with the vector n ′ = d n /dθ = ( − sin θ, , cos θ ) T . Based on Eq. (9), H can be written as H = (cid:2) exp (cid:0) itH × θ (cid:1) − (cid:3) J y . (47)It is worth to notice that H × θ = BJ × n , then H is H = (cid:2) exp (cid:0) iBtJ × n (cid:1) − (cid:3) J y . (48)Being aware of the commutation relations [ J n , J y ] = iJ n ′ , (49) [ J n , J n ′ ] = − iJ y , (50)one can straightforwardly obtain the n th order term as below J × n n J y = ( J y , for even n ; iJ n ′ . for odd n. (51)With this equation, H can be expressed by H = [cos ( Bt ) − J y − sin ( Bt ) J n ′ , (52)equivalently, it can be written in a inner product form: H = r · J , where the elements of r read r x = sin( Bt ) sin θ , r y = cos( Bt ) − and r z = − sin( Bt ) cos θ . After the normalization process, H is rewritten into the form of Eq. (13).For a spin-half system, the quantum Fisher information can be expressed by F = 4 sin (cid:18) Bt (cid:19) | r in | (cid:2) − ( n · h σ i ) (cid:3) , (53)where r in is the Bloch vector of ρ and can be obtained through the equation ρ = 12 + 12 X i = x,y,z r in ,i σ i , (54)with h σ i i = ( h σ x i i , h σ y i i , h σ z i i ) T is the vector of expected values on the i th ( i = 1 , ) eigenstateof ρ . It can also be treated as the Bloch vector of the eigenstates. In previous sections, we denote r e := h σ i i . [1] Napolitano, M. et al. Interaction-based quantum metrology showing scaling beyond the Heisenberg limit. Nature ,486–489 (2011); DOI: 10.1038/nature09778.[2] Riedel, M.F. et al. Atom-chip-based generation of entanglement for quantum metrology.
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The authors thank Dr. X.-M. Lu for helpful discussion. This work was supported by the NFRPC through Grant No.2012CB921602 and the NSFC through Grants No. 11475146.
Author contributions
X.W. and J.L. contributed the idea. J.L. performed the calculations and prepared the figures. X.J. checked thecalculations. J.L. wrote the main manuscript and X.W. made an improvement. All authors contributed to discussionand reviewed the manuscript.
Additional information
Competing financial interests: The authors declare no competing financial interests.0 Bt θ FIG. 2:
Quantum Fisher information as a function of Bt and θ . The initial state is a spin-half thermal state and thetemperature is set as T = 1= 1