Quantum parameter estimation with general dynamics

One of the main quests in quantum metrology, and quantum parameter estimation in general, is to find out the highest achievable precision with given resources and design schemes to attain it. In this article we present a general framework for quantum parameter estimation and provide systematic methods for computing the ultimate precision limit, which is more general and efficient than conventional methods. Measuring the parameters of interest with high precision is essential for science and technology, where the main quest is to find the ultimate precision limit and the optimal schemes to attain it. For a general noisy dynamics, the identification of the ultimate precision limit usually requires certain assumptions. In this work, Haidong Yuan from the Chinese University of Hong Kong and Chi-Hang Fred Fung from Munich Research Center, Huawei Technologies provided a systematic method to obtain the ultimate precision limit for general dynamics which does not make any assumptions and is more efficient than existing methods. Their method is based on an extension of a popular tool, the Bures angle, from quantum states to quantum channels, which is expected to have wide applications not only in quantum metrology but in many other fields of quantum information science.


INTRODUCTION
A pivotal task in science and technology is to identify the highest achievable precision in measurement and estimation and design schemes to reach it. Quantum metrology, which exploits quantum mechanical effects such as entanglement, can achieve better precision than classical schemes and has found wide applications in quantum sensing, gravitational wave detection, quantumenhanced reading of digital memory, quantum imaging, atomic clock synchronization, etc.; [1][2][3][4][5][6][7][8][9][10][11] this has gained increasing attention in recent years. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] A typical situation in quantum parameter estimation is to estimate the value of a continuous parameter x encoded in some quantum state ρ x of the system. To estimate the value, one needs to first perform measurements on the system, which, in the general form, are described by Positive Operator Valued Measurements (POVM), {E y }, which provides a distribution for the measurement results p(y|x) = Tr(E y ρ x ). According to the Cramér-Rao bound in statistical theory, 2, 3, 27, 28 the standard deviation for any unbiased estimator of x, based on the measurement results y, is bounded below by the Fisher information: δx ! 1 ffiffiffiffiffi ffi I x ð Þ p ; where δx is the standard deviation of the estimation of x, and I(x) is the Fisher information of the measurement results, I x ð Þ ¼ P y p y x j ð Þ ∂lnp y x j ð Þ ∂x 2 . 29 The Fisher information can be further optimized over all POVMs, which gives where the optimized value J(ρ x ) is called quantum Fisher information. 2,3,30,31 If the above process is repeated n times, then the standard deviation of the estimator is bounded by To achieve the highest precision, we can further optimize the encoding procedures x→ρ x so that J(ρ x ) is maximized. Typically the encoding is achieved by preparing the probe in some initial state ρ 0 , then let it evolve under a dynamics that contains the interested parameter, ρ 0 À! ϕ x ρ x . Usually ϕ x is determined by a given physical dynamics which is then fixed, while the initial state is up to our choice and can be optimized. A pivotal task in quantum metrology is to find out the optimal initial state ρ 0 and the corresponding maximum quantum Fisher information under any given evolution ϕ x . When ϕ x is unitary the GHZ-type of states are known to be optimal, which leads to the Heisenberg limit. However when ϕ x is noisy, such states are in general no longer optimal. Finding the optimal probe states and the corresponding highest precision limit under general dynamics has been the main quest of the field. Recently using the purification approach much progress has been made on developing systematical methods of calculating the highest precision limit. 12,13,15,18,19 These methods, however, require smooth representations of the Kraus operators, which is not intrinsic to the dynamics. In this article, we provide an alternative purification approach that does not require smooth representations of the Kraus operators. This framework provides systematic methods for computing the ultimate precision limit, which can be formulated as semi-definite programming and solved more efficiently than conventional methods. We also extend the Bures angle on quantum states to quantum channels, which is expected to find wide application in various fields of quantum information science.

Ultimate precision limit
The precision limit of measuring x from a set of quantum states ρ x is determined by the distinguishability between ρ x and its neighboring states ρ x + dx . 30,32 This is best seen if we expand the Bures distance between the neighboring states ρ x and ρ x + dx up to the second order of dx: 30 q is the fidelity between two states. Thus maximizing the quantum Fisher information is equivalent as maximizing the Bures distance, which is equivalent as minimizing the fidelity between ρ x and ρ x + dx . If the evolution is given by ϕ x , ρ x = ϕ x (ρ) and ρ x + dx = ϕ x + dx (ρ), the problem is then equivalent to finding out min ρ F ϕ x ρ ð Þ; ϕ xþdx ρ ð Þ Â Ã . We now develop tools to solve this problem for both unitary and open quantum dynamics.
Given two evolution ϕ x and ϕ x + dx , we define the Bures angle between them as Θ ϕ This generalizes the Bures angle on quantum states 33 to quantum channels. Θ(ϕ x , ϕ x + dx ) can be seen as an induced measure on quantum channel from the Bures angle on quantum states, it thus also defines a metric on quantum channels. From the definition of the Bures distance it is easy to see max ρ d 2 Bures The ultimate precision limit under the evolution ϕ x is thus determined by the Bures angle between ϕ x and the neighboring channels where n is the number of times that the procedure is repeated. If ϕ x is continuous with respect to x, then when dx→0, the ultimate precision limit is then given by The problem is thus reduced to determine the Bures angle between quantum channels. We will first show how to compute the Bures angle between unitary channels, then generalize to noisy quantum channels.
Ultimate precision limit for unitary channels. Given two unitaries U 1 and U 2 of the same dimension, since i.e., the Bures angle between two unitaries can be reduced to the Bures angle between the identity and a unitary. For a m × m unitary matrix U, let e Àiθj be the eigenvalues of U, where θ j ∈(−π, π], 1 ≤ j ≤ m, which we will call the eigen-angles of U. If θ max = θ 1 ≥ θ 2 ≥⋯≥θ m = θ min are arranged in decreasing order, then Θ I; U ð Þ ¼ θmaxÀθmin where λ max(min) is the maximal (minimal) eigenvalue of H. This provides ways to compute Bures angles on unitary channels. For example, suppose the evolution takes the form U(x) = (e −ixHt ) ⊗N (tensor product of e −ixHt for N times, which means the same unitary evolution e −ixHt acts on all N probes). Then It is easy to see that the difference between the maximal eigen-angle and the minimal eigen-angle of (e −iHtdx ) ⊗N is ; Eq. (6) then recovers the Heisenberg limit This also has close connection to the quantum speed limit, 40-42 essentially the optimal probe state in this case, which is the equal superposition of the eigenvectors corresponding to λ max and λ min , is also the state that has the fastest speed of evolution.
Ultimate precision limit for noisy quantum channels. For a general quantum channel that maps from a m 1 -dimensional to m 2dimensional Hilbert space, the evolution can be represented by a The channel can be equivalently represented as follows: where 0 E j i denotes some standard state of the environment, and U ES is a unitary operator acting on both system and environment, which we will call as the unitary extension of K. A general U ES can be written as follows: where only the first m 1 columns of U are fixed and W E ∈U(p)(p × p unitaries) only acts on the environment and can be chosen arbitrarily; here p ≥ d as p − d zero Kraus operators can be added. Given a channel an ancillary system can be used to improve the precision limit, this can be described as the extended channel where ρ SA represents a state of the original and ancillary systems. Without loss of generality, the ancillary system can be assumed to have the same dimension as the original system. Given two quantum channels K 1 and K 2 of the same dimension, let U ES1 and U ES2 as unitary extensions of K 1 and K 2 , respectively, we have 43 This extends Uhlmann's purication theorem on mixed states 44 to noisy quantum channels. Furthermore, Θ(K 1 ⊗I A , K 2 ⊗I A ) can be explicitly computed from the Kraus operators of K 1 and K 2 (please see supplemental material for detail): if with w ij as the ij-th entry of a d × d matrix W, which satisfies W k k 1 ( Á k k denotes the operator norm, which is equal to the maximum singular value). If we substitute K 1 = K x and Þ with x being the interested parameter, then where (3), we then get the maximal quantum Fisher information for the extended channel K x ⊗I A , The maximization in Eq. (13) can be formulated as semi-definite programming: max W k k 1 1 For example, consider two qubits with independent dephasing noises, which can be represented by four Kraus operators: À Á : Figure 1 shows the maximal quantum Fisher information and the quantum Fisher information for the separable input state þþ j i, where þ j i ¼ 0 p . It can be seen that the gain of entanglement is only obvious in the region of high η, i.e., low noises. It is also found that there exists a threshold for η, above the threshold the GHZ state is the optimal state that achieves the maximal quantum Fisher information, but with the decreasing of η the optimal state gradually changes from GHZ state to separable state, and this threshold increases with the number of qubits. In Fig. 2 the quantum Fisher information for the optimal state, GHZ state, and the separable state are plotted.
Parallel scheme Previous results on the SQL (standard quantum limit)-like scaling for certain independent noise processes 12,13,15,18,45 can also be recaptured in this framework. In ref. 43 we showed that given any two channels where K ⊗N denote N channels in parallel as in Fig. 3, This inequality is valid for any W with W k k 1, the smaller the right side of the inequality, the tighter the bound is. In the asymptotical limit, N N À 1 ð Þ I À K W k k 2 is the dominating term, in that case we would like to choose a W minimizing I À K W k kfor a tighter bound. This can be formulated For quantum parameter estimation with the noisy channel Þ, then the precision limit of K N x will scale at most 1 ffiffi ffi N p . As by substituting K 1 = K x and K 2 = K x + dx into Eq. (15), The quantum Fisher information is then bounded by thus the precision limit has SQL scaling For example, consider the dephasing channel with where . Expanding R and I to the second order of dx, we can p when η ≠ 1(η = 1 corresponds to the case of no dephasing error) so the first-order term in I cancels. In this case up to the second order , which scales as 1 ffiffi ffi N p for any η ≠ 1. This is consistent with previous studies 12,13,18,19 but here with a clear procedure to obtain the value for ξ.

DISCUSSION
We discuss how our results are related to previous studies. Previous studies 12,13 show that for an extended channel K x ⊗I A the maximal quantum Fisher information is given by where the minimization is over all smooth representations of equivalent Kraus operators of the channel K x . Note that this can be equivalently written as where the optimization is over all smooth representations of equivalent Kraus operators. In previous studies the equivalent Kraus operators are represented byF j x ð Þ ¼ Quantum parameter estimation H Yuan and C-H F Fung convex set, to circumvent this difficulty previous study needs to resort to the Lie algebra of the unitaries and formulated the semidefinite programming on the tangent space instead. 15 That, however, comes with a cost on the computational complexity. The complexity of semi-definite programming is determined by the number of variables (A) and the size of the constraining matrices (B) as O(A 2 B 2 ), 46 while the number of variables in the semi-definite programming here is in the same order as previous studies (both in the order of d 2 ), the size of the constraining matrices differ: the constraining matrices here have the total size of 2d + m 1 , while previous formulation needs a size of m 1 + dm 2 . 15 The difference can be significant when the system gets large (note that for generic channels d is in the order of m 1 m 2 ). For example, for Nqubit system, m 1 = m 2 = 2 N , the difference quickly becomes large with the increase of N. Also since any choice of allowed W leads to a lower bound on the precision limit, expanding the set of allowed W from the unitaries to Wj W k k 1 f galso provides more room for obtaining useful lower bounds.

CONCLUSION
In conclusion, we presented a general framework for quantum metrology that provides systematical ways to obtain the ultimate precision limit. This framework relates the ultimate precision limit directly to the geometrical properties of the underlying dynamics, which eases the analysis on utilizing quantum control methods to alter the underlying dynamics for better precision limit. 47,48 The tools developed here, such as the generalized Bures angle on quantum channels that can be efficiently computed using semidefinite programming, are expected to find wide applications in various fields of quantum information science.

METHODS
For more details on the derivation of the formulas for the ultimate precision limit, please see the Supplemental Information.