Quantum metrology for the Ising Hamiltonian with transverse magnetic field

We consider quantum metrology for unitary evolutions generated by parameter-dependent Hamiltonians. We focus on the unitary evolutions generated by the Ising Hamiltonian that describes the dynamics of a one-dimensional chain of spins with nearest-neighbour interactions and in the pres- ence of a global, transverse, magnetic field. We analytically solve the problem and show that the precision with which one can estimate the magnetic field (interaction strength) given one knows the interaction strength (magnetic field) scales at the Heisenberg limit, and can be achieved by a linear superposition of the vacuum and N free fermion states. In addition, we show that GHZ-type states exhibit Heisenberg scaling in precision throughout the entire regime of parameters. Moreover, we numerically observe that the optimal precision using a product input state scales at the standard quantum limit.


I. INTRODUCTION
Quantum metrology is one of the archetypical applications where quantum mechanics demonstrably exhibits a vast improvement over the best known classical strategies. The use of entangled input states of N qubits, such as GHZ states [1], is known to allow for a precise determination of an unknown parameter such as the relative phase in a Mach-Zender interferometer [2][3][4], or the frequency of an atomic transition [5,6] with a precision that scales inversely proportional to N , the Heisenberg limit. In comparison the best classical estimation strategy, which employs separable states, gives a precision scaling inversely proportional to √ N , the standard quantum limit [7,8]. This observation has proven very useful in the development of ultra-precise atomic clocks [9][10][11], high resolution imaging [12], as well as the detection of gravitational waves [13,14].
In the absence of any noise or decoherence effects the parameter of interest, λ, in a metrological scenario is imprinted onto the state of N probes via the unitary operator U = e iH(λ)t , where H(λ) is the Hamiltonian describing the dynamics of the N probes. For the case of phase and frequency estimation-where the parameter to be estimated is λt and λ respectively-the parameter is a multicplicative factor of the Hamiltonian, H(λ) = λH, and the Hamiltonian is local, i.e., H = N i=1 h (i) where h (i) is the Hamiltonian describing the evolution of the i th probe. Indeed, quantum metrology using local Hamiltonians where the parameter of interest enters only as a multiplicative factor have been studied extensively both in the absence as well as in the presence of noise [15][16][17][18][19][20][21][22][23].
In stark contrast quantum metrology with more general Hamiltonians is only now beginning to attract attention. Some instances of quantum metrology with parameter dependent Hamiltonians concern the estimation of time-varying signals [24][25][26], or the estimation of magnetic-field gradients along a spin chain [27,28]. Quantum metrology using parameter dependent local Hamiltonians was also considered in [29].
In this work we focus on noiseless quantum metrology for the Ising Hamiltonian where we are interested in determining the precision with which one can estimate either the strength of the transverse magnetic field, B, or the coupling interaction, J provided the remaining quantity is known. The Hamiltonian of Eq. (1) is known to exhibit a phase transition [30], which have been discussed previously in the context of quantum metrology and were shown to be resourceful [31][32][33][34]. Moreover, as the Ising Hamiltonian is entanglement generating [35], it has found applications in ion-trap quantum computing architectures [36,37], where either J or B can be controlled at will by modifying either the separation of the ions, or the global magnetic field strength.
The main results in this work can be listed as follows: 1. The ultimate precision with which one can estimate either J or B, having complete knowledge of B or J, using N probe systems scales at the Heisenberg limit, and we provide an analytic expression for this achievable precision.
2. We provide numerical evidence that the ultimate achievable precision strictly outperforms the optimal classical strategy, which deploys the N initial probes in a pure product state. For up to N = 11 our numerical study shows that the optimal product input state yields a precision that scales at the standard quantum limit.
3. For any Hamiltonian that is the sum of two noncommuting terms, where we wish to estimate the strength of the first term while having complete control over the second, the best strategy is always to set the the second term to zero.
This paper is organised as follows. In Sec. II we review the basics of quantum metrology as well as some important mathematical results regarding unitary operators generated by parameter dependent Hamiltonians.
In Sec. III we use the Jordan-Wigner transformation to determine the maximal possible precision with which one can estimate either J and B using N systems as probes. We compare this to the best possible precision that can be achieved by a separable state of N probes which we determine numerically for up to N = 11 qubits. In Sec. IV we tackle the general question of whether full control of part of the Hamiltonian evolution can help in estimation of relevant parameters not under our control. Finally, Sec. V contains the conclusions of our investigation as well as some open questions for future work.

II. BASICS OF QUANTUM METROLOGY
In this section we review the main results in quantum metrology and derive some important facts pertaining to parameter-dependent Hamiltonians in general and to the Ising Hamiltonian (Eq. (1)) in particular. Specifically, we concentrate on noiseless quantum metrology and the quantum Fisher information (QFI); the central quantity of interest in quantum metrology. After introducing the QFI we derive a formula for calculating it for the case of general parameter-dependent Hamiltonians and then to the specific case of the Ising Hamiltonian.
A standard protocol in noiseless quantum metrology can be formulated as follows: N probes are prepared in a suitable state and undergo an evolution for some time, t, described by the unitary operator U (λ, t) = e −itH(λ) , where H(λ) is the Hamiltonian describing the dynamics of the N probes and explicitly depends on the vector of parameters λ ≡ (λ 1 , . . . , λ M ). Finally the N probes are measured and an estimate ofλ is obtained from the measurement statistics of ν repetitions of the above procedure. In what follows we shall assume that all other parameters except λ i are known, and shall be concerned with estimating λ i as precisely as possible. A lower bound on the error, δλ i = λ i −λ i , for any unbiased estimatorλ i is given by the quantum Cramér-Rao bound [38] where F(ρ λ,t ) is the quantum Fisher information (QFI) of the state ρ λ,t describing the N probes after the unitary dynamics have acted. In the most general case the QFI can be computed as [38] with the symmetric logarithmic derivative, where α (β) are the eigenvalues of ρ λ,t , |ψ α , (|ψ β ) the corresponding eigenvectors, and the sum in Eq. (4) is over all α, β satisfying α + β = 0. Here and in what follows ∂ x ≡ ∂ ∂x . We now review the case where the parameter of interest enters as a multiplicative factor of the Hamiltonian

A. Parameter independent Hamiltonians
In the case of noiseless metrology, an easier expression for computing the QFI exists if one initializes the N probes in a pure state ρ = |ψ ψ|. In this case the QFI can be shown to be ) ⊗N , where α min (max) are the eigenstates corresponding to the minimum and maximum eigenvalues of h, then F (|ψ λ,t ) = t 2 N (α max − α min ) 2 and give the standard quantum limit in estimation precision [8]. On the other hand if the probes are prepared in the Greenberger-Horne-Zeilinger (GHZ) state the Heisenberg scaling in estimation precision [8]. Hence, in the case where the parameter to be estimated is a multiplicative factor of a local Hamiltonian, the use of highly entangled states leads to a quadratic improvement in scaling precision. Note that besides the GHZ states there exists a large class of pretty good states that scale at the Heisenberg limit up to a multiplicative factor [23].

B. Parameter dependent Hamiltonians
We now use Eq. (5) to compute the QFI for Hamiltonians of the form where [H 1 , H 2 ] = 0. We make no assumptions on the structure of H 1 , H 2 ; in particular we do not assume that they are local Hamiltonians. Notice that the Ising Hamiltonian of Eq. (1) is a special case of Eq. (7) with Eq. (5) reads where and the variance of O i is computed with respect to the evolved state |ψ λ,t . Thus, the QFI is maximised by the states that are linear superpositions of the eigenstates corresponding to the minimum and maximum eigenvalues of O i (λ, t).
In the next section we use the expressions in Eq. (9) to determine the optimal QFI for either the magnetic field B or interaction strength J of the Ising Hamiltonian using separable and entangled states respectively.

III. ESTIMATION OF MAGNETIC FIELD AND INTERACTION STRENGTH
We are interested in determining the optimal precision in estimating either the magnetic field strength, B, or interaction strength, J, of the Ising Hamiltonian (Eq. (1)). In particular, we will show that the optimal precision in estimating either B or J scales at the Heisenberg limit, up to a constant factor which depends only on the ratio of J and B, and is achievable by states that are linear superpositions of the vacuum and fully occupied states of free fermions of a suitable type. Furthermore, we numerically determine the best achievable precision using a separable state for up to 11 qubits and show that the optimal precision scales, to within best fit errors, linearly with N . Hence, our results provide strong evidence that the entanglement generated by the Ising Hamiltonian when acting on an initially pure separable state is not enough to boost the precision in estimation from the SQL to the Heisenberg limit.
We begin by first determining the optimal precision in estimation of either B, or J, and the corresponding optimal states. To do so we note that via the use of the Jordan-Wigner transformation [39] the Ising Hamiltonian of Eq. (1) can be expressed as a quadratic Hamiltonian in fermionic creation and annihilation operators which can be suitably diagonalized. Specifically the mapping and its inverse where {σ x , σ y , σ z } are the Pauli matrices [40] with σ y , and a j , a † j are the annihilation and creation operators for mode j respectively and satisfy the anti- where α k = J cos 2πk of the mode operators a k . After substituting Eq. (15) into Eq. (13) the Hamiltonian can be written in matrix form as and φ j = π 2 , ∀j. We now determine the optimal achievable precision in estimating J given that B is known.

A. Estimating the Interaction strength
We now proceed to estimate the interaction strength J for the Ising Hamiltonian. The QFI is given by Eq. (9) with where . Writing the latter in terms of the fermionic operators b k one obtains In order to calculate the operator O J (J, B, t) of Eq. (18) we need to determine the action of H(J, B) on the fermionic operators b k , i.e., we need to determine b k (s) = U (J, B, s)b k (0)U † (J, B, s). This is simply the Heisenberg equation of motion for the mode operators b k . Due to the block-diagonal structure of H(J, B), when written in terms of fermionic operators b k , the solution to the Heisenberg equation of motion can be seen to be Henceforth, we drop the explicit time dependence of the mode operators b k for convenience.
Substituting Eq. (20) into Eq. (18) yields Performing the integration over s gives where with ω (±) k = α 2 k ± β 2 k . As Eq. (22) is of the same form as Eq. (16) it can be brought to the diagonal form by a suitable Bogoliubov transformation (see Eq. (17)). Note that the free fermions corresponding to d † k , d k are different from those of Eq. (14), and we may choose without loss of generality the positive square square root of Ω 2 k + |∆ k | 2 , which corresponds to choosing the vacuum state for free fermions to be zero.
The optimal precision in estimating the interaction strength J is now easy to determine. As the latter is inversely proportional to the square root of the variance of O J (J, B, t), we simply need to determine the maximum achievable variance for the operator in Eq. (24). This is achieved by preparing the equally weighted superposition of the vacuum state and the state where all N modes are occupied by fermions. The variance with respect to this state is simply given by As the sum includes N summands, the variance of O J (J, B, t) scales as N 2 up to some constant factor that depends solely on the ratio between J and B as we now explain. The coefficients Ω k and ∆ k given by Eq. (23) can be separated into two parts; a part that depends linearly on t, and a second part that oscillates with t (see also [29]). For long interaction times, i.e. t → ∞ the linear part completely dominates and both Ω k and ∆ k can be greatly simplified. Moreover, for N large the sum in Eq. (25) can be replaced, to a good approximation, by an integral resulting in the following simple expression for the variance Thus, the variance scales as N 2 t 2 , i.e., at the Heisenberg limit, up to an overall constant factor that only depends on the ratio J/B. The function G is plotted in Fig. 1. Notice that G(J/B) exhibits a phase transition at J = B.
We now proceed to determine the optimal precision with which one can estimate J, knowing B, if we restrict the input state of N systems to be a product state. As it is not immediately evident what separable states look like in the basis that diagonalizes the operator O J (J, B, t), we work directly with Eq. (10). However, due to the form of Eq. (10) it is highly non-trivial to perform an analytical optimization over all possible separable states of N qubits. To that end we perform a brute-force optimization of the variance of Eq. (25) over all possible input product states for up to N = 11 qubits. The results are shown in Fig. 2 where one can already see a difference in scaling between the optimal product state strategy and the optimal quantum strategy even for small system sizes. Moreover, within a high margin of certainty the scaling of the QFI using the optimal product input state is at most linear in N .

B. Estimating the field strength
We now proceed to estimate the magnetic field B, given we know J exactly. The procedure is identical to , i.e., when B in Eq. (1) is set to zero, using the optimal product state |ψ = |01 ⊗N/2 .

that of Sec. III A. Writing
z in terms of the fermionic operators b k we obtain Substituting Eqs. (20,27) into Eq. (10) and performing the integration over s yields where As Eq. (28) is of the same form as Eq. (16) it can be brought to the diagonal form by a suitable Bogoliubov transformation (see Eq. (17)). The optimal achievable precision for estimating the field strength B is given by the optimal variance of O B (J, B, t) in Eq. (30) which can be easily computed to be and is again achieved by the equally weighted superposition of the vacuum state and the state where all N fermionic modes are occupied. We note that because the Bogoliubov transformation diagonalizing Eq. (28) explicitly depends on the coefficients of Eq. (29) the fermions described by modes f k here are different than those described by modes d k and c k in Eqs. (24,14) respectively. As the sum in Eq. (31) includes N summands, the maximum variance of O B (J, B, t) scales as N 2 up to some factor. As in the case of estimating the interaction strength, the coefficients A k and B k in Eq. (29) can be separated into two parts: one that depends linearly on t, and a second part that oscillates with t. For t → ∞ and N very large the expression for the maximal variance can be given explicitly as where G is the function given in Eq. (26) and Fig. 1 with g = B/J. This is a rather remarkable result; namely that the constant factor multiplying the ultimate scaling in precision for estimating either J or B is exactly the same. The optimal precision for estimating B, given we known J, using a separable strategy is again numerically calculated for up to N = 11 qubits with the results shown in Fig. 3. Just as in the case of estimating J, one can already observe a difference in scaling of the QFI between the optimal product state strategy and the corresponding optimal quantum strategy. Moreover, within a high margin of certainty, the scaling of the QFI using the optimal product state is at most linear in N .

IV. LOCAL ESTIMATION WITH AUXILIARY HAMILTONIANS
Hitherto, we considered estimating either λ 1 or λ 2 , assuming that λ 2 or λ 1 was known and fixed. We now consider the case of estimating λ 1 (λ 2 ) where we have complete control of λ 2 (λ 1 ) and ask whether this additional power can aid in the precision of estimation. Assuming that no noise or decoherence process is present we know (see Sec. (II)) that the QFI is proportional to the variance of the operator given in Eq. (10). We will now show that the optimal strategy for estimating λ 1 (λ 2 ) given full control of λ 2 H 2 (λ 1 H 1 ) is to set the latter equal to zero.
The proof is as follows. Assume without loss of gener- ality that we are interested in estimating λ 1 so that For any operator A and any unitary U , the spectrum of U AU † is the same as the spectrum of A. Now as (∆ 2 H 1 ) max = (α max − α min ) 2 , where α max(min)which are functions of N -are the corresponding maximum and minimum eigenvalues of H 1 , it follows that ∆ 2 O 1 (λ, t) ≤ t 2 (∆ 2 H 1 ) max as either the unitary transformation U (λ, s) leaves the eigenspaces corresponding to the maximum and minimum eigenvalues of H 1 invariant, or it couples them with the eigenspaces of a lower (higher) eigenvalue respectively. Hence, given full control over λ 2 H 2 the best strategy for estimating λ 1 is to set λ 2 H 2 equal to zero. In this case the dynamics on the N probes are given by U (λ 1 , t) = e iλ1tH1 , i.e., the problem reduces to that of estimating a multiplicative factor of the Hamiltonian, for which the optimal strategy is simply to prepare the GHZ state 1 √ 2 (|α max + |α min ), where α max(min) are the eigenstates of H 1 corresponding to the maximum (minimum) eigenvalue. This observation is quite general; given any Hamiltonian of the form of Eq. (7) where, we have complete control over part of the dynamics and have access to the optimal state for estimating the parameter of the remaining part, the best choice is to set the dynamics under our control to zero. Note however, that if one does not have access to the optimal state for estimating λ 1 , switching λ 2 H 2 on may actually help increase precision. Whereas we have shown that this is not the case for the Ising Hamiltonian if we are restricted to product input states (compare red and green lines in Figs. 2 and 3), it may still be the case that states which yield suboptimal precision in the estimation of J (B) in the absence of B (J), can offer improved precision when B(J) are switched on.

V. CONCLUSION
In this work we investigated precision limits for noiseless quantum metrology in the presence of parameter dependent Hamiltonians, and in particular the Ising Hamiltonian. We showed that the ultimate limit in estimating the interaction or magnetic field strength scales quadratically with the number of probe systems N , i.e., at the Heisenberg limit, and that the states that achieve this precision are linear superposition of the vacuum and fully occupied states of N free fermions. Moreover, whereas the Ising Hamiltonian generates entanglement this entanglement does not help in boosting the precision scaling with respect to N that can be achieved with product states. Finally, we observed the general property that given access to the optimal state for estimating part of the dynamics generated by a Hamiltonian and full control over the reaming dynamical evolution then the optimal strategy is to set the latter part of the dynamics to zero.
Whereas we have shown that the entanglement generating properties of the Ising Hamiltonian do not boost the precision in estimation for product states, it may still be the case that we can exploit this property of the Ising Hamiltonian to reduce the amount of entanglement required in the initial input state of the N probes. This would be of high interest for practical realizations of quantum metrology where the creation of highly entangled states of N systems remains a challenge.
In addition, our analysis deals with optimal states and bounds in the absence of noise. It would be interesting to investigate the achievable precision bounds in the presence of several physical noise models, such as uncorrelated dephasing or depolarizing noise, as well as spatial and temporal correlated noise. Furthermore, it would be interesting to determine which of these types of noise can we readily combat via the use of error-correcting techniques, or by dynamical decoupling [41][42][43] VI. ACKNOWLEDGEMENTS This work was supported by the Austrian Science Fund (FWF): P24273-N16 and the Swiss National Science Foundation grant P2GEP2 151964.