Quantum metrology with unitary parametrization processes

Quantum Fisher information is a central quantity in quantum metrology. We discuss an alternative representation of quantum Fisher information for unitary parametrization processes. In this representation, all information of parametrization transformation, i.e., the entire dynamical information, is totally involved in a Hermitian operator . Utilizing this representation, quantum Fisher information is only determined by and the initial state. Furthermore, can be expressed in an expanded form. The highlights of this form is that it can bring great convenience during the calculation for the Hamiltonians owning recursive commutations with their partial derivative. We apply this representation in a collective spin system and show the specific expression of . For a simple case, a spin-half system, the quantum Fisher information is given and the optimal states to access maximum quantum Fisher information are found. Moreover, for an exponential form initial state, an analytical expression of quantum Fisher information by operator is provided. The multiparameter quantum metrology is also considered and discussed utilizing this representation.

How to precisely measure the values of physical quantities, such as the phases of light in interferometers, magnetic strength, gravity and so on, is always an important topic in physics.Obtaining high-precision values of these quantities will not only bring an obvious advantage in applied sciences, including the atomic clocks, physical geography, civil navigation and even military industry, but also accelerate the development of fundamental theories.One vivid example is the search for gravitational waves.Quantum metrology is such a field attempting to find optimal methods to offer highest precision of a parameter that under estimation.In recently decades, many protocols and strategies have been proposed and realized to improve the precisions of various parameters [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Some of them can even approach to the Heisenberg 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 of the parameter's variance according to Cramé-Rao inequality [21,22].The quantum Fisher information for parameter θ is defined as F = Tr(ρL 2 ), where ρ is a density matrix dependent on θ and L is the symmetric logarithmic derivative (SLD) operator and determined by the equation ∂ θ ρ = (ρL + Lρ)/2.For a multiparameter system, the counterpart of 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 [23] 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 the selection of initial states.Utilizing this * Electronic address: xgwang@zimp.zju.edu.cnrepresentation, the quantum Fisher information is only determined by H and the initial state.
In this report, we give a general expression of quantum Fisher information and quantum Fisher information matrix utilizing H operator.For a unitary parametrization process, H can be expressed in an expanded form.We calculate the specific expression of H in a collective spin system, and provide an analytical expression of quantum Fisher information in a spin-half system for any initial state.Based on this expression, all optimal states to access maximum quantum Fisher information are found in this system.Furthermore, considering this spin-half system as a multiparameter system, the quantum Fisher information matrix, as well as the optimal states, can be easily obtained by the known form of H in single parameter estimations.On the other hand, inspired by a recent work [24], for an exponential form initial state, we provide an analytical expression of quantum Fisher information using H operator.A demonstration with a spin thermal initial state is given in this scenario.The maximum quantum Fisher information and the optimal condition are also discussed.

Results
Quantum Fisher information with H operator.For a general unitary parametrization transformation, the parametrized state ρ(θ) is expressed by ρ(θ) = U (θ)ρ 0 U † (θ), where ρ 0 is a state independent of θ.In this paper, since the parameter θ is only brought by U (θ), not the initial state ρ 0 , we use U instead of U (θ) for short.Denote the spectral decomposition of ρ 0 as ρ 0 = M i=1 p i |ψ i ψ i |, where p i and |ψ i are the ith eigenvalue and eigenstate of ρ 0 and M is the dimension of the support of ρ 0 .It is easy to see that p i and U |ψ i are the corresponding eigenvalue and eigenstate of ρ(θ), respectively.The quantum Fisher information for ρ(θ) can then be expressed by [25,26] where is a Hermitian operator since the equality is the variance of H on the ith eigenstate of ρ 0 .When ∂ θ U commutes with U , H can be explained as the generator of the parametrization transformation [23].The expression (1) of quantum Fisher information is not just a formalized representation.The operator H is only determined by the parametrization process, that is the dynamics of the system or the device.For a known dynamical process of a parameter, i.e., known system's Hamiltonian, H is a settled operator and can be obtained in principle.
In this representation, the calculation of quantum Fisher information is separated into two parts: the diagonalization of initial state and calculation of H.For a general 2-dimensional state, the quantum Fisher information reduces to The subscript of the variance can be chosen as 1 or 2 as any Hermitian operator's variances on two orthonormal states are equivalent in 2-dimentional Hilbert space.For a pure state, the quantum Fisher information can be easily obtain from Eq. ( 4) with taking det ρ 0 = 0 and the variance on that pure state, i.e., For a well applied form of parametrization transformation U = exp(−itH θ ) [23], being aware of the equation H can then be expressed by Defining a superoperator H can be written in an expanded form where the coefficient In many real problems, the recursive commutations in Eq. ( 8) can either repeat or terminate [24], indicating an analytical expression of H. Thus, this representation of quantum Fisher information would be very useful in these problems.For the simplest case that H θ = θH, all terms vanish but the first one, then In the following we give an example to exhibit Eq. ( 8).Consider the interaction Hamiltonian of a collective spin system in a magnetic field where J n0 = n 0 • J with n 0 = (cos θ, 0, sin θ) T and J = (J x , J y , J z ) T .B is the amplitude of the external magnetic field. Here the Pauli matrix for kth spin.Taking θ as the parameter under estimation, H can be expressed by where J n1 = n 1 • J with the vector where µ = sgn(sin(Bt/2)) is the sign function and n 1 is normalized.Utilizing Eq. ( 12), one can immediately obtain the form of H for a spin-half system with σ = (σ x , σ y , σ z ) T , which was also discussed in the Hamiltonian eigenbasis in Ref. [23].For any 2dimensional state, based on Eq. ( 4), the quantum Fisher information can be expressed by where r ρ0 = ( σ x , σ y , σ z ) is the Bloch vector of the initial state ρ 0 and r e is the Bloch vector of any eigenstate of ρ 0 .For pure states, there is r e = r ρ0 and |r ρ0 | = 1.
Since the Bloch vector of a 2-dimensional state satisfies |r ρ0 | ≤ 1, it can be found that the maximum value of Eq. ( 14) is which can be saturated when |r ρ0 | = 1 and n 1 • r ρ0 = 0, namely, the optimal state to access maximum quantum Fisher information here is a pure state perpendicular to n 1 , as shown in Fig. 1.In this figure, the yellow sphere represents the Bloch sphere and the blue arrow represents the vector n 1 .It can be found that all states on the joint ring of the green plane and surface of Bloch sphere can access the maximum quantum Fisher information, i.e., all states on this ring are optimal states.One simple example is r opt = n 0 , and another one is the superposition state of two eigenstates of H [20,23].
Alternatively, B could be the parameter that under estimation.In the spin-half case, with respect to B, H B = −tn 0 • σ/2, then the quantum Fisher information can be expressed by The optimal states to access the maximum value F max B = t 2 are the pure states vertical to n 0 .
Exponential form initial state.For an exponential form initial state ρ 0 = exp(G 0 ), the parametrized state reads Recently, Jiang [24] studied the quantum Fisher information for exponential states and gave a general form of SLD operator.In his theory, the SLD operator can be expanded as where the coefficient 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 0 U † .
Through some straightforward calculation, the derivative of G on θ reads Based on this equation, the nth order term in Eq. ( 18) is where H is given by Eq. ( 8).It is known that the quantum Fisher information reads where the effective SLD operator L eff = U † LU .Substituting Eq. ( 21) into Eq.( 18), the effective SLD operator can be expanded as 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 eigenbasis of G 0 .We denote the ith eigenvalue and eigenstate of G 0 as a i and |φ i , and in the eigenbasis of G 0 , the element of G ×n 0 H satisfies the recursion relation where [•] ij := φ i | • |φ j .Utilizing this recursive equation, a general formula of nth order term can be obtained, Substituting above equation into the expression of L eff and being aware of the equality the element of effective SLD operator in Eq. ( 23) can be written as Based on the equation F = Tr(e G0 L 2 eff ), the quantum Fisher information in the eigenbasis of G 0 can finally be expressed by This is one of the main results in this paper.In some real problems, the eigenspace of G 0 could be find easily.For instance, the eigenspace of a bosonic thermal state is the Fock space.Thus, as long as the formula of H in Fock space is established, the quantum Fisher information can be obtained from Eq. ( 28).Now we exhibit Eq. ( 28) with a spin-half thermal state.The initial state is taken as Immediately, the quantum Fisher information can be obtained from Eq. ( 28) as The maximum value of above expression is obtained at Bt = (4k + 1)π for k = 0, 1, ... and From this equation, one can see that the value of maximum quantum Fisher information is only affected by the temperature.With the increase of temperature, the maximum value reduces.In the other hand, quantum Fisher information in Eq. ( 31) 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, 2π] 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 2, indicating that the quantum Fisher information can be still very robust and near its maximum value even when Bt is hard to set exactly at π.
Multiparameter processes.For a multiparameter system, the element of quantum Fisher information matrix in Ref. [26] can also be written with H operator, where U is dependent on a series of parameters α, β and so on, and with the index m = α, β, ....The covariance matrix on the ith eigenstate of initial state is defined as with {•, •} the anti-commutation.For a single qubit system, Eq. (33) reduces to Similarly with the single-parameter scenario, the subscript in Eq. ( 35) can be chosen as 1 or 2 since the covariance for two 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 namely, for pure states, the element of quantum Fisher information matrix is actually the covariance between two H operators.For the diagonal elements, they are exactly the quantum Fisher information for the corresponding parameters.
We still consider the spin-half system with Hamiltonian H = Bn 0 • σ/2.Take both B and θ as the parameters under estimations.First, based on aforementioned calculation, the H function for B and θ read Based on the property of Pauli matrices {n 0 •σ, n 1 •σ} = 2n 0 • n 1 , the anti-commutation in the covirance reads For a pure initial state, the off-diagonal element of the quantum Fisher information matrix is expressed by where r ρ0 is the Bloch vector of the initial pure state and the equality n 0 • n 1 = 0 is used.When the initial pure state is vertical to n 0 or n 1 , this off-diagonal element vanishes and the quantum Fisher information matrix is achievable.The optimal state r opt in this multiparameter system is a pure state vertical to both n 0 and n 1 simultaneously.There are two states in the Bloch sphere are qualified to this condition, as shown in Fig. 3. Thus, these two optimal states can access maximum quantum Fisher information for both B and θ simultaneously.

Discussion
We have discussed the quantum Fisher information with unitary parametrization utilizing an alternative representation.The total information of the parametrization process is involved in a H operator in this representation.This operator is totally determined by the parameter and parametrization transformation U .As long as the parameter and transformation are taken, H is a settled operator and independent of the initial state.Utilizing this representation, we give a general analytical expression of quantum Fisher information for an exponential form initial state.We also study and discuss the corresponding representation in multiparameter processes.
As a demonstration, we apply this representation in a collective spin system and show the expression of H. Furthermore, we provide an analytical expression of quantum Fisher information in a spin-half system.If we consider this system as a multiparameter system, the corresponding quantum Fisher information matrix can also be straightforwardly obtained by this representation.From these expressions, one can find the optimal states to access the maximum quantum Fisher information.For the parameter B, the optimal state is a pure state vertical to n 0 , and for the parameter θ, the optimal one is also a pure state, but vertical to n 1 .By analyzing the off-diagonal element of quantum Fisher information matrix, the optimal states for both B and θ to access maximum quantum Fisher information simultaneously are found.Two pure states on Bloch sphere can fulfill this task, which are vertical to n 0 and n 1 simultaneously.

Methods
Collective spin system in a magnetic field.For the Hamiltonian (11), its derivative on parameter θ is ∂ θ H θ = n ′ 0 • J = J n ′ 0 = −iH × θ J y with the vector n ′ 0 = dn 0 /dθ = (− sin θ, 0, cos θ) T .Based on Eq. ( 8), H can be written as It is worth to notice that Being aware of the commutation relations one can straightforwardly obtain the nth order term as below J ×n n0 J y = J y , for even n; iJ n ′ 0 .for odd n. ( With this equation, H can be expressed by 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) − 1 and r z = − sin(Bt) cos θ.After the normalization process, H is rewritten into the form of Eq. ( 12).For a spin-half system, the quantum Fisher information can be expressed by where r ρ0 is the Bloch vector of ρ 0 and can be obtained through the equation with 1 1 the identity matrix.σ i = ( σ x i , σ y i , σ z i ) T is the vector of expected values on the ith (i = 1, 2) eigenstate of ρ 0 .It can also be treated as the Bloch vector of the eigenstates.In previous sections, we denote r e := σ i .

n 1 Figure 1 :
Figure 1: Optimal states to access maximum quantum Fisher information in a spin-half system.The blue arrow represents the vector n1 and all vectors in the green plane are vertical to n1.All the states in the joint ring of green plane and Bloch sphere's surface can access maximum quantum Fisher information.

where β = 1 / 2 n 1 •
(k b T ) with k b the Boltzmann constant and T the temperature.In Planck unit, k b = 1.The partition function reads Z = Tr[exp(−βσ z )] = 2 cosh β.In this case, G 0 = −βσ z − ln z.Denoting the eigenstates of σ z as |0 and |1 , i.e., σ z = |0 0| − |1 1|, the eigenvalues of G 0 read a 1 = −βσ z − ln z and a 2 = βσ z − ln z.The parametrization process is still taken as H θ = Bn 0 • σ/2 with θ the parameter under estimation, indicating that H = sin Bt σ, then the squared norm of the offdiagonal element of H in the eigenbasis of σ z reads

Figure 2 :
Figure 2: Quantum Fisher information as a function of Bt and θ.The initial state is a spin-half thermal state and the temperature is set as T = 1 here.

Figure 3 :
Figure 3: Optimal states to access maximum quantum Fisher information for B and θ simultaneously.Two states in the Bloch sphere, represented as ropt in solid and dashed lines, are vertical to n0 and n1 simultaneously.These two states are capable to access the maximum quantum Fisher information for both B and θ simultaneously.
with C a constant matrix or proportional to H θ , only the first and second terms remain.In this case, H reduces to −t(∂ θ H θ + itC/2).A more interesting case is that [H θ , ∂ θ H θ ] = c∂ θ H θ , with c a nonzero constant number, then H can be written in the form