Introduction

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 parameters1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22. 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 Cramér-Rao inequality23,24. The quantum Fisher information for parameter α is defined as F = Tr(ρL2), where ρ is a density matrix dependent on α and L is the symmetric logarithmic derivative (SLD) operator and determined by the equation ∂αρ = (ρL + Lρ)/2. For a multiparameter system, the counterpart of quantum Fisher information is called quantum Fisher information matrix , of which the element is defined as , where Lα, Lβ are the SLD operators for parameters α and β, respectively.

Recently, it has been found27 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 . This operator characterizes the dynamical property of the parametrization process and totally independent of the selection of initial states. Utilizing this representation, the quantum Fisher information is only determined by and the initial state.

In this report, we give a general expression of quantum Fisher information and quantum Fisher information matrix utilizing operator. For a unitary parametrization process, can be expressed in an expanded form. This form is particularly useful when the Hamiltonian owns a recursive commutation relation with its derivative on parameter estimation. We calculate the specific expression of 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, can be easily obtained by the known form of in single parameter estimations. On the other hand, inspired by a recent work28, for an exponential form initial state, we provide an analytical expression of quantum Fisher information using 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 operator

For a general unitary parametrization transformation, the parametrized state ρ(α) is expressed by ρ(α) = U(α)ρ0U†(α), where ρ0 is a state independent of α. In this paper, since the parameter α is only brought by U(α), not the initial state ρ0, we use U instead of U(α) for short. Denote the spectral decomposition of ρ0 as , where pi and |ψi〉 are the ith eigenvalue and eigenstate of ρ0 and M is the dimension of the support of ρ0. It is easy to see that pi and U|ψi〉 are the corresponding eigenvalue and eigenstate of ρ(α), respectively. The quantum Fisher information for ρ(α) can then be expressed by29,30

where25,26

is a Hermitian operator since the equality (∂αU†)U = −U†(∂αU). Meanwhile,

is the variance of on the ith eigenstate of ρ0. When ∂αU commutes with U, can be explained as the generator of the parametrization transformation27. The expression (1) of quantum Fisher information is not just a formalized representation. The operator 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, 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 . 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 the purity Trρ2 = 1 and the variance on that pure state, i.e., ref. 27

Namely, the quantum Fisher information is proportional to the variance of on the initial state. In this scenario, denote the initial state ρ0 = |ψ0〉〈ψ0|, the quantum Fisher information can be rewritten into , with the effective SLD operator

For a well applied form of parametrization transformation U = exp(−itHα)27, where has been set as 1 in Planck unit and being aware of the equation

can then be expressed by

Defining a superoperator A× as A×(·) : = [A, ·], can be written in an expanded form

where the coefficient

In many real problems, the recursive commutations in Eq. (9) can either repeat or terminate28, indicating an analytical expression of . Thus, this representation of quantum Fisher information would be very useful in these problems. For the simplest case that Hα = αH, all terms vanish but the first one, then . When [Hα, ∂αHα] = C, with C a constant matrix or proportional to Hα, only the first and second terms remain. In this case, reduces to −t(∂αHα + itC/2). A more interesting case is that [Hα, ∂αHα] = c∂αHα, with c a nonzero constant number, then can be written in the form

 In the following we give an example to exhibit Eq. (9). Consider the interaction Hamiltonian of a collective spin system in a magnetic field

where with n0 = (cos θ, 0, sin θ)T and J = (Jx, Jy, Jz)T. B is the amplitude of the external magnetic field and θ is the angle between the field and the collective spin. Here for i = x, y, z with the Pauli matrix for kth spin. Taking θ as the parameter under estimation, can be expressed by

where with the vector

where µ = sgn(sin(Bt/2)) is the sign function and n1 is normalized.

The operator for Hamiltonian (12) may be also available to be solved using the procedure in Ref. 27, in the (2j + 1)-dimensional eigenspace of Hθ (j is the total spin). In principle, the eigenstates of Hθ can be found by rotating the Dicke state into the same direction of Hθ. However, even one can analytically obtain all the eigenvalues and eigenvectors, it still requires a large amount of calculations to obtain , especially when the spin numbers are tremendous. 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 .

Utilizing Eq. (13), one can immediately obtain the form of for a spin-half system

with σ = (σx, σy, σz)T, which was also discussed in the Hamiltonian eigenbasis in Ref. 27. For any 2-dimensional state, based on Eq. (4), the quantum Fisher information can be expressed by

where rin = (〈σx〉, 〈σy〉, 〈σz〉)T is the Bloch vector of the initial state ρ0 and re is the Bloch vector of any eigenstate of ρ0. For pure states, there is re = rin and |rin| = 1. Since the Bloch vector of a 2-dimensional state satisfies |rin| ≤ 1, it can be found that the maximum value of Eq. (15) is

which can be saturated when |rin| = 1 and n1 · rin = 0, namely, the optimal state to access maximum quantum Fisher information here is a pure state perpendicular to n1, as shown in Fig. 1. In this figure, the yellow sphere represents the Bloch sphere and the blue arrow represents the vector n1. 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 ropt = n0 and another one is the superposition state of two eigenstates of 22,27.

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.

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Alternatively, B could be the parameter that under estimation. In the spin-half case, with respect to B, , then the quantum Fisher information can be expressed by

The optimal states to access the maximum value are the pure states vertical to n0.

Exponential form initial state

For an exponential form initial state ρ0 = exp(G0), the parametrized state reads

Recently, Jiang28 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 gn vanishes for odd n. Here is the (n + 2)th Bernoulli number and in our case, G = UG0U†. Through some straightforward calculation, the derivative of G on α reads

Based on this equation, the nth order term in Eq. (19) is

where is given by Eq. (9). Generally, it is known that the quantum Fisher information reads

where the effective SLD operator Leff = 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

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 G0. We denote the ith eigenvalue and eigenstate of G0 as ai and |ϕi〉 and in the eigenbasis of G0, the element of satisfies the recursion relation

where [·]ij : = 〈ϕi| · |ϕj〉. Utilizing this recursive equation, a general formula of nth order term can be obtained,

Substituting above equation into the expression of Leff and being aware of the equality

the element of effective SLD operator in Eq. (24) can be written as

Based on the equation , the quantum Fisher information in the eigenbasis of G0 can finally be expressed by

This is one of the main results in this paper. In some real problems, the eigenspace of G0 could be find easily. For instance, the eigenspace of a bosonic thermal state is the Fock space. Thus, as long as the formula of in Fock space is 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

where β = 1/(kbT) with kb the Boltzmann constant and T the temperature. In Planck unit, kb = 1. The partition function reads Z = Tr[exp(−βσz)] = 2 cosh β. In this case, G0 = −βσz − ln z. Denoting the eigenstates of σz as |0〉 and |1〉, i.e., σz = |0〉〈0| − |1〉〈1|, the eigenvalues of G0 read a1 = −βσz − ln z and a2 = βσz − lnz. The parametrization process is still taken as Hθ = Bn0 · σ/2 with θ the parameter under estimation, indicating that , then the squared norm of the off-diagonal element of in the eigenbasis of σz reads

Immediately, the quantum Fisher information can be obtained from Eq. (29) 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. (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, 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.

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.

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Multiparameter processes

For a multiparameter system, the element of quantum Fisher information matrix in Ref. 30 can also be written with 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. (34) reduces to

Similarly with the single-parameter scenario, the subscript in Eq. (36) 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 operators on the initial state. When the total Hamiltonian can be written as and [Hi, Hj] = 0 for any i, j, above equation can reduce to the covariance between Hi and Hj31. For the diagonal elements, they are exactly the quantum 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〈ψout|LαLβ|ψout〉 = 0, ∀α, β32,33. Here |ψout〉 is dependent on the parameter under estimation. In the unitary parametrization, |ψout〉 = U|ψ0〉 and this condition can be rewritten into , ∀α, β. Here is the effective SLD operator for parameter α(β). Utilizing Eq. (6), this condition can be expressed in the form of operator,

In other word, needs to be a real number for any α and β. When commutes with 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 be expressed by . From the definition equation of SLD, one can see that satisfies the equation ∂θρ = U{ρ0, Leff}U†/2. The quantum Fisher information matrix has more than one definitions. One alternative candidate is using the so-called Right Logarithmic Derivative (RLD)24,34,35, which is defined as ∂αρ = ρRα, with Rα the RLD. The element of RLD quantum Fisher information matrix can be written as

where the effective RLD reads . For a unitary parametrization process, assuming the initial state has nonzero determinant, can be expressed by and the initial state ρ0, i.e.,

With this equation, the element of RLD quantum Fisher information matrix can be expressed by

When the parametrization process is displacement, this equation can reduces to the corresponding form in Ref. 35. For pure states, the element reads . Recently, Genoni et al.35 proposed a most informative Cramér-Rao bound for the total variance of all parameters under estimation. From the relation between and , one can see that is always larger than , namely, the SLD Cramér-Rao bound is always more informative than the RLD counterpart in this scenario.

We still consider the spin-half system with the Hamiltonian H = Bn0 · σ/2. Take both B and θ as the parameters under estimations. First, based on aforementioned calculation, the operator for B and θ read

Based on the property of Pauli matrices {n0 · σ, n1 · σ} = 2n0 · n1, the anti-commutation in the covariance reads

For a pure initial state, the off-diagonal element of the quantum Fisher information matrix is expressed by

where rin is the Bloch vector of the initial pure state and the equality n0 · n1 = 0 has been used. When the initial pure state is vertical to n0 or n1, this off-diagonal element vanishes. Compared with the optimal condition for maximum quantum Fisher information for B and θ individually, the Bloch vector n2 = n0 × n1 can optimize both the diagonal elements of quantum Fisher information matrix and vanish the off-diagonal elements. However, all above is only necessary conditions for the achievement of Cramér-Rao bound. To find out if the bound can be really achieved, the condition (38) needs to be checked. In this case,

With this equation, condition (38) reduces to n2 · rin = 0, i.e., to make the Cramér-Rao bound achievable, the Bloch vector of the initial state needs to in the plane of n0 and n1. Unfortunately, n2 is not in this plane. Thus, B and θ cannot be optimally joint measured simultaneously.

In the plane constructed by n0 and n1, any Bloch vector of pure state can be written as rin = n0 cos ϕ + n1 sin ϕ, then we have , and . From these expressions, one can see that the determinant of quantum Fisher information matrix is zero, i.e., det . This fact indicates that, utilizing any pure state in this plane, the variances of B and θ cannot be estimated simultaneously through the Cramér-Rao theory.

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 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, is a settled operator and independent of the initial state. More interestingly, can be expressed in an expanded form. For the Hamiltonians owning recursive commutations with their partial derivative on the parameter under estimation, this expanded form shows a huge advantage. Utilizing this representation, we give a general analytical expression of quantum Fisher information for an exponential form initial state. Moreover, we have also studied the representation in multiparameter processes. The condition of Cramér-Rao bound to be achievable for pure states are also presented in the form of operator. In addition, we give the representation of Right 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 . 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 straight-forwardly 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 n0 and for the parameter θ, the optimal one is also a pure state, but vertical to n1. By analyzing the off-diagonal element of quantum Fisher information matrix, the states to optimize the diagonal elements and make the off-diagonal elements vanish are found. However, these states fail to satisfy the condition of achievement. Thus, B and θ cannot be optimally jointed measured.

Methods

Collective spin system in a magnetic field

For the Hamiltonian (12), its derivative on parameter θ is with the vector . Based on Eq. (9), can be written as

It is worth to notice that , then is

Being aware of the commutation relations

one can straightforwardly obtain the nth order term as below

With this equation, can be expressed by

equivalently, it can be written in a inner product form: , where the elements of r read rx = sin(Bt) sin θ, ry = cos(Bt) − 1 and rz = − sin(Bt) cos θ. After the normalization process, is rewritten into the form of Eq. (13).

For a spin-half system, the quantum Fisher information can be expressed by

where rin is the Bloch vector of ρ0 and can be obtained through the equation

with the identity matrix. 〈σ〉i = (〈σx〉i, 〈σy〉i, 〈σz〉i)T is the vector of expected values on the ith (i = 1, 2) eigenstate of ρ0. It can also be treated as the Bloch vector of the eigenstates. In previous sections, we denote re : = 〈σ〉i.