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, quantum-enhanced 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: \(\delta \hat x \ge \frac{1}{{\sqrt {I\left( x \right)} }},\) where \(\delta \hat x\) is the standard deviation of the estimation of x, and I(x) is the Fisher information of the measurement results, \(I\left( x \right) = \mathop {\sum}\nolimits_y {p\left( {y\left| x \right.} \right){{\left( {\frac{{\partial lnp\left( {y\left| x \right.} \right)}}{{\partial x}}} \right)}^2}} \).29 The Fisher information can be further optimized over all POVMs, which gives

$$\delta \hat x \ge \frac{1}{{\sqrt {{{\max }_{{E_y}}}I\left( x \right)} }} = \frac{1}{{\sqrt {J\left( {{\rho _x}} \right)} }},$$
(1)

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 \(\delta \hat x \ge \frac{1}{{\sqrt {nJ({\rho _x})} }}.\)

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, \({\rho _0}\mathop{\longrightarrow}\limits^{{{\phi _x}}}{\rho _x}\). 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.

Results

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

$$d_{Bures}^2\left( {{\rho _x},{\rho _{x + dx}}} \right) = \frac{1}{4}J\left( {{\rho _x}} \right)d{x^2},$$
(2)

where \({d_{Bures}}\left( {{\rho _1},{\rho _2}} \right) = \sqrt {2 - 2F\left( {{\rho _1},{\rho _2}} \right)} \); here \(F\left( {{\rho _1},{\rho _2}} \right) = Tr\sqrt {\rho _1^{1/2}{\rho _2}\rho _1^{1/2}} \) 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 \(\mathop {{\min }}\nolimits_\rho F\left[ {{\phi _x}\left( \rho \right),{\phi _{x + dx}}\left( \rho \right)} \right]\). 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 \(\Theta \left( {{\phi _x},{\phi _{x + dx}}} \right) = \mathop {{\max }}\nolimits_\rho \mathop {{\cos }}\nolimits^{ - 1} \left[ {F\left( {{\phi _x}\left( \rho \right),{\phi _{x + dx}}\left( \rho \right)} \right)} \right]\). This generalizes the Bures angle on quantum states33 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 \(\mathop {{\max }}\nolimits_\rho d_{Bures}^2\left[ {{\phi _x}\left( \rho \right),{\phi _{x + dx}}\left( \rho \right)} \right] = 2 - 2\cos \Theta \left( {{\phi _x},{\phi _{x + dx}}} \right)\), thus from Eq. (2) we have

$$\mathop {{\max }}\limits_\rho \,J\left[ {{\phi _x}\left( \rho \right)} \right] = \mathop {{\lim }}\limits_{dx \to 0} \frac{{8\left[ {1 - \cos \Theta \left( {{\phi _x},{\phi _{x + dx}}} \right)} \right]}}{{d{x^2}}}.$$
(3)

The ultimate precision limit under the evolution ϕ x is thus determined by the Bures angle between ϕ x and the neighboring channels

$$\delta \hat x \ge \frac{1}{{{{\lim }_{dx \to 0}}\frac{{\sqrt {8\left[ {1 - \cos \Theta \left( {{\phi _x},{\phi _{x + dx}}} \right)} \right]} }}{{\left| {dx} \right|}}\sqrt n }},$$
(4)

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, Θ(ϕ x , ϕ x + dx )→Θ(ϕ x , ϕ x ) = 0, in this case

$$\begin{array}{ccccc}\\ \mathop {{\max }}\limits_\rho J\left[ {{\phi _x}\left( \rho \right)} \right] = \mathop {{\lim }}\limits_{dx \to 0} \frac{{8\left[ {1 - \cos \Theta \left( {{\phi _x},{\phi _{x + dx}}} \right)} \right]}}{{d{x^2}}}\\ \\ \qquad\qquad= \mathop {{\lim }}\limits_{dx \to 0} \frac{{16\mathop {{\sin }}\nolimits^2 \frac{{\Theta \left( {{\phi _x},{\phi _{x + dx}}} \right)}}{2}}}{{d{x^2}}}\\ \\ \qquad\qquad= \mathop {{\lim }}\limits_{dx \to 0} \frac{{4{\Theta ^2}\left( {{\phi _x},{\phi _{x + dx}}} \right)}}{{d{x^2}}},\\ \end{array}$$
(5)

the ultimate precision limit is then given by

$$\delta \hat x \ge \frac{1}{{{{\lim }_{dx \to 0}}2\frac{{\Theta \left( {{\phi _x},{\phi _{x + dx}}} \right)}}{{\left| {dx} \right|}}\sqrt n }}.$$
(6)

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 \(F\left( {{U_1}\rho U_1^\dag ,{U_2}\rho U_2^\dag } \right) = F\left( {\rho ,U_1^\dag {U_2}\rho U_2^\dag {U_1}} \right)\), we have \(\Theta \left( {{U_1},{U_2}} \right) = \Theta \left( {I,U_1^\dag {U_2}} \right)\), 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 \({e^{ - i{\theta _j}}}\) 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 \(\Theta \left( {I,U} \right) = \frac{{{\theta _{\max }} - {\theta _{\min }}}}{2}\) when θ max − θ min ≤ π,34,35,36,37,38,39 specifically if U = e −iHt, then \(\Theta \left( {I,U} \right) = \frac{{\left( {{\lambda _{\max }} - {\lambda _{\min }}} \right)t}}{2}\) if \(\left( {{\lambda _{\max }} - {\lambda _{\min }}} \right)t \le \pi \), 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

$$\begin{array}{ccccc}\\ \Theta \left[ {U\left( x \right),U\left( {x + dx} \right)} \right] = \Theta \left[ {I,{U^\dag }\left( x \right)U\left( {x + dx} \right)} \right]\\ \\ \qquad\qquad\quad\qquad= \Theta \left[ {I,{{\left( {{e^{ - iHtdx}}} \right)}^{ \otimes N}}} \right].\\ \end{array}$$
(7)

It is easy to see that the difference between the maximal eigen-angle and the minimal eigen-angle of (e −iHtdx)⊗N is \({\theta _{\max }} - {\theta _{\min }} = N\left( {{\lambda _{\max }}\left| {dx} \right|t - {\lambda _{\min }}\left| {dx} \right|t} \right)\). Thus \(\Theta \left( {I,{{\left( {{e^{ - iHtdx}}} \right)}^{ \otimes N}}} \right) = \frac{{{\theta _{\max }} - {\theta _{\min }}}}{2} = \frac{{\left( {N{\lambda _{\max }}\left| {dx} \right| - N{\lambda _{\min }}\left| {dx} \right|} \right)t}}{2},\) Eq. (6) then recovers the Heisenberg limit

$$\delta \hat x \ge \frac{1}{{\sqrt n \left( {{\lambda _{\max }} - {\lambda _{\min }}} \right)t}}\frac{1}{N}.$$
(8)

This also has close connection to the quantum speed limit,40,41,42 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 2-dimensional Hilbert space, the evolution can be represented by a Kraus operation \(K\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_j}{\rho _S}F_j^\dag } \); here the Kraus operators F j , 1 ≤ j ≤ d are of the size m 2 × m 1, \(\mathop {\sum}\nolimits_{j = 1}^d {F_j^\dag {F_j}} = {I_{{m_1}}}\). The channel can be equivalently represented as follows:

$$K\left( {{\rho _S}} \right) = T{r_E}\left( {{U_{ES}}\left( {\left| {{0_E}} \right\rangle \left\langle {{0_E}} \right| \otimes {\rho _S}} \right)U_{ES}^\dag } \right),$$
(9)

where \(\left| {{0_E}} \right\rangle \) 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:

$${U_{ES}} = ({W_E} \otimes {I_{{m_2}}})\underbrace {\left[ {\begin{array}{*{20}{l}}{{F_1}} \hfill & * \hfill & * \hfill & \cdots \hfill & * \hfill \\ {{F_2}} \hfill & * \hfill & * \hfill & \cdots \hfill & * \hfill \\ \vdots \hfill & {} \hfill & \vdots \hfill & {} \hfill & \vdots \hfill \\ {{F_d}} \hfill & * \hfill & * \hfill & \cdots \hfill & * \hfill \\ 0 \hfill & * \hfill & * \hfill & \cdots \hfill & * \hfill \\ \vdots \hfill & {} \hfill & \vdots \hfill & {} \hfill & \vdots \hfill \\ 0 \hfill & * \hfill & * \hfill & \cdots \hfill & * \hfill \\ \end{array}} \right]}_U,$$
(10)

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

$$\left( {K \otimes {I_A}} \right)\left( {{\rho _{SA}}} \right) = \mathop {\sum}\limits_j {\left( {{F_j} \otimes {I_A}} \right){\rho _{SA}}{{\left( {{F_j} \otimes {I_A}} \right)}^\dag }} ,$$

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 have43

$$\begin{array}{ccccc}\\ \Theta \left( {{K_1} \otimes {I_A},{K_2} \otimes {I_A}} \right) = \mathop {{\min }}\limits_{{U_{ES1}},{U_{ES2}}} \Theta \left( {{U_{ES1}},{U_{ES2}}} \right)\\ = \mathop {{\min }}\limits_{{U_{ES1}}} \Theta \left( {{U_{ES1}},{U_{ES2}}} \right)\\ = \mathop {{\min }}\limits_{{U_{ES2}}} \Theta \left( {{U_{ES1}},{U_{ES2}}} \right).\\ \end{array}$$
(11)

This extends Uhlmann’s purication theorem on mixed states44 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 \({K_1}\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_{1j}}{\rho _S}F_{1j}^\dag } \), \({K_2}\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_{2j}}{\rho _S}F_{2j}^\dag } \), then \(\cos \Theta \left( {{K_1} \otimes {I_A},{K_2} \otimes {I_A}} \right) = \mathop {{\max }}\nolimits_{\left\| W \right\| \le 1} \frac{1}{2}{\lambda _{\min }}\left( {{K_W} + K_W^\dag } \right)\); here \({\lambda _{\min }}\left( {{K_W} + K_W^\dag } \right)\) denotes the minimum eigenvalue of \({K_W} + K_W^\dag \), where \({K_W} = \mathop {\sum}\nolimits_{ij} {{w_{ij}}F_{1i}^\dag {F_{2j}}} \), with w ij as the ij-th entry of a d × d matrix W, which satisfies \(\left\| W \right\| \le 1\) (\(\left\| \cdot \right\|\) denotes the operator norm, which is equal to the maximum singular value). If we substitute K 1 = K x and K 2 = K x + dx , where \({K_x}\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_j}\left( x \right){\rho _S}F_j^\dag \left( x \right)} \) and \({K_{x + dx}}\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_j}\left( {x + dx} \right){\rho _S}F_j^\dag \left( {x + dx} \right)} \) with x being the interested parameter, then

$$\begin{array}{ccccc} \cos \Theta \left( {{K_x} \otimes {I_A},{K_{x + dx}} \otimes {I_A}} \right)\\ = \mathop {{\max }}\limits_{\left\| W \right\| \le 1} \frac{1}{2}{\lambda _{\min }}\left( {{K_W} + K_W^\dag } \right),\\ \end{array}$$
(12)

where \({K_W} = \mathop {\sum}\nolimits_{ij} {{w_{ij}}F_i^\dag \left( x \right){F_j}\left( {x + dx} \right)} \).

By substituting ϕ x  = K x ⊗I A and ϕ x + dx  = K x + dx ⊗I A in Eq. (3), we then get the maximal quantum Fisher information for the extended channel K x ⊗I A ,

$$\max J = \mathop {{\lim }}\limits_{dx \to 0} \frac{{8\left[ {1 - {{\max }_{\left\| W \right\| \le 1}}\frac{1}{2}{\lambda _{\min }}\left( {{K_W} + K_W^\dag } \right)} \right]}}{{d{x^2}}}.$$
(13)

The maximization in Eq. (13) can be formulated as semi-definite programming: \(\mathop {{\max }}\nolimits_{\left\| W \right\| \le 1} \frac{1}{2}{\lambda _{\min }}\left( {{K_W} + K_W^\dag } \right) = \)

$$\begin{array}{ccccc}\\ maximize\quad \frac{1}{2}t\\ \\ s.t.\quad \quad \left( {\begin{array}{*{20}{c}}\\ I & {{W^\dag }} \\ \\ W & I \\ \end{array}} \right)\ge 0,\\ \\ {K_W} + K_W^\dag - tI\ge 0.\\ \end{array}$$
(14)

For example, consider two qubits with independent dephasing noises, which can be represented by four Kraus operators: F 1(x)⊗F 1(x), F 1(x)⊗F 2(x), F 2(x)⊗F 1(x), F 2(x)⊗F 1(x) with \({F_1}\left( x \right) = \sqrt {\frac{{1 + \eta }}{2}} U\left( x \right)\), \({F_2}\left( x \right) = \sqrt {\frac{{1 - \eta }}{2}} {\sigma _3}U\left( x \right)\); here \(U\left( x \right) = \exp \left( { - i\frac{{{\sigma _3}}}{2}x} \right).\) Figure 1 shows the maximal quantum Fisher information and the quantum Fisher information for the separable input state \(\left| { + + } \right\rangle \), where \(\left| + \right\rangle = \frac{{\left| 0 \right\rangle + \left| 1 \right\rangle }}{{\sqrt 2 }}\). 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.

Fig. 1
figure 1

Quantum Fisher information for the optimal input state and separable input state \(\left| { + + } \right\rangle \) for two qubits with independent dephasing noises

Full size image

In Fig. 2 the quantum Fisher information for the optimal state, GHZ state, and the separable state are plotted.

Fig. 2
figure 2

Quantum Fisher information for optimal probe states, GHZ state, and separable state for five qubits under independent dephasing noises

Full size image

Parallel scheme

Previous results on the SQL (standard quantum limit)-like scaling for certain independent noise processes12, 13, 15, 18, 45 can also be recaptured in this framework. In ref. 43 we showed that given any two channels \({K_1}\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_{1j}}{\rho _S}F_{1j}^\dag } \), \({K_2}\left( {{\rho _S}} \right) = \mathop {\sum}\nolimits_{j = 1}^d {{F_{2j}}{\rho _S}F_{2j}^\dag } \), we have

$$\begin{array}{ccccc}\\ & 2 - 2\cos \Theta \left( {K_1^{ \otimes N} \otimes {I_A},K_2^{ \otimes N} \otimes {I_A}} \right)\\ \\ & \le N\left\| {2I - {K_W} - K_W^\dag } \right\| + N\left( {N - 1} \right){\left\| {I - {K_W}} \right\|^2},\\ \end{array}$$
(15)

where K ⊗N denote N channels in parallel as in Fig. 3, \({K_W} = \mathop {\sum}\nolimits_{ij} {{w_{ij}}F_{1i}^\dag {F_{2j}}} \), with w ij as the ij-th entry of a d × d matrix W which satisfies \(\left\| W \right\| \le 1\). This inequality is valid for any W with \(\left\| W \right\| \le 1\), the smaller the right side of the inequality, the tighter the bound is. In the asymptotical limit, \(N\left( {N - 1} \right){\left\| {I - {K_W}} \right\|^2}\) is the dominating term, in that case we would like to choose a W minimizing \(\left\| {I - {K_W}} \right\|\) for a tighter bound. This can be formulated as semi-definite programming with

$$\begin{array}{ccccc}\mathop {{\min }}\nolimits_{\left\| W \right\| \le 1} \left\| {I - {K_W}} \right\| = \\ &s.t.\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}} {\min } & t \\ \end{array}\,} \\ {\left( {\begin{array}{*{20}{c}} I & {{W^\dag }} \\ W & I \end{array}} \right)} \end{array}\ge 0,\\ & \left( {\begin{array}{*{20}{c}}{tI} & {{{\left( {I - {K_W}} \right)}^\dag }} \\ {I - {K_W}} & {tI} \\ \end{array}} \right)\ge 0.\\ \end{array}$$
(16)
Fig. 3
figure 3

N probes with independent noisy processes

Full size image

For quantum parameter estimation with the noisy channel \({K_x}\left( \rho \right) = \mathop {\sum}\nolimits_{i = 1}^d {{F_i}\left( x \right)\rho F_i^\dag \left( x \right)} \), we can substitute K 1 = K x and K 2 = K x + dx into Eq. (15). If there exists a d × d matrix W with \(\left\| W \right\| \le 1\) such that \(\left\| {I - {K_W}} \right\| \le Dd{x^2}\), where \({K_W} = \mathop {\sum}\nolimits_{ij} {{w_{ij}}F_i^\dag \left( x \right){F_j}\left( {x + dx} \right)} \), then the precision limit of \(K_x^{ \otimes N}\) will scale at most \(\frac{1}{{\sqrt N }}\). As by substituting K 1 = K x and K 2 = K x + dx into Eq. (15),

$$\begin{array}{ccccc} 2 - 2\cos \Theta \left( {K_x^{ \otimes N} \otimes {I_A},K_{x + dx}^{ \otimes N} \otimes {I_A}} \right)\\ \le N\left\| {2I - {K_W} - K_W^\dag } \right\| + N\left( {N - 1} \right){\left\| {I - {K_W}} \right\|^2}\\ \le N\left( {\left\| {I - {K_W}} \right\| + \left\| {I - K_W^\dag } \right\|} \right) + N\left( {N - 1} \right){D^2}d{x^4}\\ \le 2DNd{x^2} + N\left( {N - 1} \right){D^2}d{x^4}.\\ \end{array}$$
(17)

The quantum Fisher information is then bounded by

$$\begin{array}{ccccc}\\ \max J = \mathop {{\lim }}\limits_{dx \to 0} 8\frac{{1 - \cos \left[ {\Theta \left( {K_x^{ \otimes N} \otimes {I_A},K_{x + dx}^{ \otimes N} \otimes {I_A}} \right)} \right]}}{{d{x^2}}}\\ \le 8DN,\\ \end{array}$$

thus the precision limit has SQL scaling

$$\delta x \ge \frac{1}{{\sqrt {nJ} }} \ge \frac{1}{{\sqrt {8nDN} }}.$$

For example, consider the dephasing channel with

$${K_x}\left( {{\rho _0}} \right) = U\left( x \right)\left( {\frac{{1 + \eta }}{2}{\rho _0} + \frac{{1 - \eta }}{2}{\sigma _3}{\rho _0}{\sigma _3}} \right){U^\dag }\left( x \right),$$

where \(U\left( x \right) = \exp \left( { - i\frac{{{\sigma _3}}}{2}x} \right)\), \({\sigma _1} = \left( {\begin{array}{*{20}{l}}0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill \\ \end{array}} \right)\), \({\sigma _2} = \left( {\begin{array}{*{20}{c}}0 & { - i} \\ i & 0 \\ \end{array}} \right)\) and \({\sigma _3} = \left( {\begin{array}{*{20}{l}}1 \hfill & 0 \hfill \\ 0 \hfill & { - 1} \hfill \\ \end{array}} \right)\), η∈[0, 1]. In this case \({F_1}\left( x \right) = \sqrt {\frac{{1 + \eta }}{2}} U\left( x \right)\), \({F_2}\left( x \right) = \sqrt {\frac{{1 - \eta }}{2}} {\sigma _3}U\left( x \right).\) We choose \(W = \left[ {\begin{array}{*{20}{c}}{\cos (\xi dx)} & {i\sin (\xi dx)} \\ {i\sin (\xi dx)} & {\cos (\xi dx)} \\ \end{array}} \right]\) and vary ξ to minimize \(\left\| {I - {K_W}} \right\|\). In this case

$$\begin{array}{ccccc}{K_W} & \\ = & \cos \left( {\xi dx} \right)F_1^\dag \left( x \right){F_1}\left( {x + dx} \right) + i\sin \left( {\xi dx} \right)F_1^\dag \left( x \right){F_2}\left( {x + dx} \right) \hfill + i\sin \left( {\xi dx} \right)F_2^\dag \left( x \right){F_1}\left( {x + dx} \right) + \cos \left( {\xi dx} \right)F_2^\dag \left( x \right){F_2}\left( {x + dx} \right) \hfill \\ = &\left( {\begin{array}{*{20}{l}}\\ {\left[ {\cos \left( {\xi dx} \right) + i\sqrt {1 - {\eta ^2}} \sin \left( {\xi dx} \right)} \right]{e^{i\frac{{dx}}{2}}}} \hfill & 0 \hfill \\ \\ 0 \hfill & {\left[ {\cos \left( {\xi dx} \right) - i\sqrt {1 - {\eta ^2}} \sin \left( {\xi dx} \right)} \right]{e^{ - i\frac{{dx}}{2}}}} \hfill \\ \end{array}} \right),\\ \end{array}$$
(18)

thus

$$I - {K_W} = \left( {\begin{array}{*{20}{l}}{R - iI} \hfill & 0 \hfill \\ 0 \hfill & {R + iI} \hfill \\ \end{array}} \right),$$
(19)

where \(R = 1 - \cos \left( {\xi dx} \right)\cos \frac{{dx}}{2} + \sqrt {1 - {\eta ^2}} \sin \left( {\xi dx} \right)\sin \frac{{dx}}{2}\) and \(I = \cos \left( {\xi dx} \right)\sin \frac{{dx}}{2} + \sqrt {1 - {\eta ^2}} \sin \left( {\xi dx} \right)\cos \frac{{dx}}{2}\), then \(\left\| {I - {K_W}} \right\| = \sqrt {{R^2} + {I^2}} \). Expanding R and I to the second order of dx, we can get \(R = \frac{{1 + 4{\xi ^2} + 4\xi \sqrt {1 - {\eta ^2}} }}{8}d{x^2} + O\left( {d{x^3}} \right)\) and \(I = \frac{{dx}}{2} + \sqrt {1 - {\eta ^2}} \xi dx + O(d{x^3})\). To minimize \(\left\| {I - {K_W}} \right\|\) we should choose \(\xi = - \frac{1}{{2\sqrt {1 - {\eta ^2}} }}\) 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

$$\begin{array}{ccccc}\\ & \left\| {I - {K_W}} \right\| = \left| R \right|\\ \\ = & \frac{{{\eta ^2}}}{{8\left( {1 - {\eta ^2}} \right)}}d{x^2} + O\left( {d{x^3}} \right),\\ \end{array}$$
(20)

thus \(\max \,J = \mathop {{\lim }}\nolimits_{dx \to 0} 8\frac{{1 - \cos \left[ {{\Theta _{QC}}\left( {K_x^{ \otimes N} \otimes {I_A},K_{x + dx}^{ \otimes N} \otimes {I_A}} \right)} \right]}}{{d{x^2}}} \le \frac{{{\eta ^2}}}{{1 - {\eta ^2}}}N\), and the precision limit \(\delta x \ge \frac{1}{{\sqrt {nJ} }} \ge \frac{{\sqrt {1 - {\eta ^2}} }}{{\eta \sqrt {nN} }}\), which scales as \(\frac{1}{{\sqrt N }}\) for any η ≠ 1. This is consistent with previous studies12, 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 studies12, 13 show that for an extended channel K x ⊗I A the maximal quantum Fisher information is given by

$$\max J = 4\mathop {{\min }}\limits_{\left\{ {{{\hat F}_j}\left( x \right)} \right\}} \left\| {\mathop {\sum}\limits_{j = 1}^d {\dot{\hat F}_j^\dag \left( x \right){{\dot{\hat F}}_j}\left( x \right)} } \right\|$$
(21)

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

$$\begin{array}{cccc}\\ \kern -4.9pc\max J = 4\mathop {{\min }}\limits_{\left\{ {{{\hat F}_j}\left( x \right)} \right\}} \left\| {\mathop {\sum}\limits_{j = 1}^d {\dot{\hat F}_j^\dag \left( x \right){{\dot{\hat F}}_j}\left( x \right)} } \right\|\\ \\ = 4\mathop {{\min }}\limits_{\left\{ {{{\hat F}_j}\left( x \right)} \right\}} \left\| {\mathop {\sum}\limits_{j = 1}^d {\mathop {{\lim }}\limits_{dx \to 0} \frac{{\left( {\hat F_j^\dag \left( {x + dx} \right) - \hat F_j^\dag \left( x \right)} \right)}}{{dx}}\frac{{\left( {{{\hat F}_j}\left( {x + dx} \right) - {{\hat F}_j}\left( x \right)} \right)}}{{dx}}} } \right\|\\ \\ \kern -2.6pc = 4\mathop {{\min }}\limits_{\left\{ {{{\hat F}_j}\left( x \right)} \right\}} \left\| {\frac{{2I - \mathop {\sum}\nolimits_{j = 1}^d {\left( {\hat F_j^\dag \left( x \right){{\hat F}_j}\left( {x + dx} \right) + \hat F_j^\dag \left( {x + dx} \right){{\hat F}_j}\left( x \right)} \right)} }}{{d{x^2}}}} \right\|\\ \\ \kern -0.6pc= 4\frac{{2 - {{\max }_{\left\{ {{{\hat F}_j}\left( x \right)} \right\}}}{\lambda _{\min }}\left[ {\mathop {\sum}\nolimits_{j = 1}^d {\left( {\hat F_j^\dag \left( x \right){{\hat F}_j}\left( {x + dx} \right) + \hat F_j^\dag \left( {x + dx} \right){{\hat F}_j}\left( x \right)} \right)} } \right]}}{{d{x^2}}},\\ \end{array}$$
(22)

where the optimization is over all smooth representations of equivalent Kraus operators. In previous studies the equivalent Kraus operators are represented by \({\hat F_j}\left( x \right) = \mathop {\sum}\nolimits_{i = 1}^d {{\omega _{ji}}\left( x \right){F_i}\left( x \right)} \) and \({\hat F_j}\left( {x + dx} \right) = \mathop {\sum}\nolimits_{i = 1}^d {{\omega _{ji}}\left( {x + dx} \right){F_i}\left( {x + dx} \right)} \), where ω ji (x) is ji-entry of W E (x)∈U(d), and W E (x) is required to be smooth with respect to x. It is easy to see that in this case Eq. (22) is a special case of Eq. (13) with W restricted to the form \(W_E^\dag \left( x \right){W_E}\left( {x + dx} \right)\). This restriction arises as the operator W E in Eq. (10), which originally can be arbitrary chosen, was assumed to depend on x smoothly in previous studies.12, 13 Such restriction is not intrinsic to the dynamics. Furthermore without the restriction the set \(\left\{ {\left. W \right|\left\| W \right\| \le 1} \right\}\) is a convex set, which allows a direct formulation as the semi-definite programming. While with the restriction \(W = W_E^\dag \left( x \right){W_E}\left( {x + dx} \right)\) needs to be unitary which does not form a convex set, to circumvent this difficulty previous study needs to resort to the Lie algebra of the unitaries and formulated the semi-definite 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 N-qubit system, m 1 = m 2 = 2N, 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 \(\left\{ {\left. W \right|\left\| W \right\| \le 1} \right\}\) also 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 semi-definite 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.