Standard Quantum Limit and Heisenberg Limit in Function Estimation

TL;DR

This paper establishes fundamental quantum error bounds for function estimation, revealing the limits imposed by quantum mechanics and the role of entanglement, with implications for quantum metrology and sampling theory.

Contribution

It derives optimal error bounds for quantum function estimation, connecting quantum limits with classical sampling theorems, and compares entanglement's impact on estimation accuracy.

Findings

Error bounds correspond to standard quantum limit and Heisenberg limit.

Bounds are achievable with position- or wavenumber-localized states.

Quantum metrology on functions adheres to Nyquist-Shannon sampling theorem.

Abstract

Unlike well-established parameter estimation, function estimation faces conceptual and mathematical difficulties despite its enormous potential utility. We establish the fundamental error bounds on function estimation in quantum metrology for a spatially varying phase operator, where various degrees of smooth functions are considered. The error bounds are identified in both cases of absence and presence of interparticle entanglement, which correspond to the standard quantum limit and the Heisenberg limit, respectively. Notably, these error bounds can be reached by either position-localized states or wavenumber-localized ones. In fact, we show that these error bounds are theoretically optimal for any type of probe states, indicating that quantum metrology on functions is also subject to the Nyquist-Shannon sampling theorem, even if classical detection is replaced by quantum measurement.

Equations

$$\delta^{2}=\mathbb{E}\biggl{[}\int_{0}^{L}\frac{dx}{L}[\tilde{\varphi}(x)-\varphi(x)]_{2\pi}^{2}\biggr{]}.$$

$$\int_{0}^{L}\frac{dx}{L}\big{\lvert}{\varphi^{\prime}(x)}\big{\rvert}^{2}\leq\frac{M^{2}}{L^{2}}$$

$$\sup_{0<\epsilon<a}\int_{0}^{L}\frac{dx}{L}\bigg{\lvert}{\frac{\varphi(x+\epsilon)-\varphi(x)}{\epsilon^{q}}}\bigg{\rvert}^{2}\leq\frac{M^{2}}{L^{2q}},$$

$$\tilde{\phi }(x)=\sum _{j=1}^{n_1}{\tilde{\phi }}_{j}f(x-x_j),$$

$$\delta\geq O(n_{1}^{-q}M)+O(n_{2}^{-1/2})\geq O\bigl{(}(M^{1/q}N^{-1})^{q/(2q+1)}\bigr{)}$$

$$\delta\geq O(n_{1}^{-q}M)+O(n_{2}^{-1})\geq O\bigl{(}(M^{1/q}N^{-1})^{q/(q+1)}\bigr{)}$$

$${\delta }_{\mathrm{PS}}=O(K^{-q}M),{\delta }_{\mathrm{QT}}=O(K^{1/2}{N}^{-1/2}).$$

$${\delta }_{\mathrm{PS}}=O(n_{\mathrm{c}}^{-q}M),{\delta }_{\mathrm{QT}}=O(n_{\mathrm{p}}^{-1}).$$

$${\phi }_k={\int }_0^L\frac{dx}{L}{e}^{-2\pi ikx/L}\phi (x).$$

$$\sum _{k=1}^{\infty }{k}^{2q}|{\phi }_k{|}^2\le \frac{M^2}{2c_0^2},$$

$${\phi }_{\mathbf{u}}(x)=\sum _{k=1}^K\sqrt{2}{u}_k\sin (2\pi ikx/L).$$

$$\delta_{\mathbf{u}}\geq\delta_{\mathrm{UUB}}:=K\bigl{(}\tr[J(\mathbf{u})]\bigr{)}^{-1/2},$$

$$[J(\mathbf{u})]_{jk}=4\mathop{\mathrm{Re}}\nolimits\big{\langle}{\tfrac{\partial}{\partial u_{j}}\psi_{\mathbf{u}}}\big{\rvert}\bigl{[}1-\lvert{\psi_{\mathbf{u}}}\rangle\langle{\psi_{\mathbf{u}}}\rvert\bigr{]}\big{\lvert}{\tfrac{\partial}{\partial u_{k}}\psi_{\mathbf{u}}}\big{\rangle}.$$

$${\delta }_{\mathbf{u}}\ge {\delta }_{\mathrm{WBB}},{\delta }_{\mathrm{WBB}}^{-1}:={\delta }_{\mathrm{UUB}}^{-1}+{\rho }^{-1}.$$

$$\delta \ge c_1(M^{1/q}{N}^{-1}{)}^{q/(2q+1)},$$

$$n_{\mathrm{p}}\le [(\pi c_1^{-1}{)}^{(2q+1)/2}{M}^{-1}{N}^q{]}^{1/(q+1)}.$$

$$\delta \ge c_2(M^{1/q}{N}^{-1}{)}^{q/(q+1)},$$

$$\sup_{0<\epsilon<a}\int_{0}^{L}\frac{dx}{L}\bigg{\lvert}{\frac{\varphi^{(m)}(x+\epsilon)-\varphi^{(m)}(x)}{\epsilon^{\sigma}}}\bigg{\rvert}^{2}\leq\frac{M^{2}}{L^{2q}}.$$

Full text

Accurate estimation of signals with a limited amount of resource is a fundamental problem in physics. Quantum metrology has made a profound contribution to this problem by demonstrating a classically unattainable scaling of the estimation error Holland and Burnett (1993); Giovannetti et al. (2004, 2006); Higgins et al. (2007); Giovannetti et al. (2011). This non-classical accuracy, called the Heisenberg limit, can be reached by various forms of quantum features including entanglement Giovannetti et al. (2004), quantum circuit Higgins et al. (2007), bosonic Anisimov et al. (2010); Gagatsos et al. (2016) and spin Kitagawa and Ueda (1993); Wineland et al. (1992) squeezing, and quantum statistics Datta and Shaji (2012). Applications of the Heisenberg-limited measurement range from detection of fundamental signals such as atomic clocks de Burgh and Bartlett (2005); Kessler et al. (2014), gravitational waves Schnabel et al. (2010); The LIGO Scientific Collaboration (2011), a scalable cat state Gao et al. (2010) and polariton condensates Deng et al. (2007); Huang et al. (2012).

The Heisenberg limit is significantly better than the standard quantum limit (SQL) $\delta=O(N^{-1/2})$, which sets the accuracy bound arising from uncorrelated noises, where $N$ is the size of resource. While the Heisenberg limit $\delta=O(N^{-1})$ and the SQL $\delta=O(N^{-1/2})$ apply to both scalar estimation and vector estimation Macchiavello (2003); Ballester (2004); Humphreys et al. (2013); Yao et al. (2014); Szczykulska et al. (2016); Tsang (2017), it is generally considered that the continuity of the signal may alter the scaling law. Such a problem can be categorized as function estimation, which has attracted growing attention. For example, atomic clocks Galleani and Tavella (2008); Fraas (2016) and graviational waves The LIGO Scientific Collaboration (2011); Kolkowitz et al. (2016) involves time-varying singals, which offer richer information when treated as functions. More generally, exploration for new phenomena involves observing structures in a continuous in space and/or time, which can be represented as functions. This indicates that functional structures play crucial roles in generic continuous systems including measurements on magnetometry Budker and Romalis (2007); Vengalattore et al. (2007), nanostructured materials Johnson et al. (1998); Tsang (2009); Brida et al. (2010); Samantaray et al. (2017), live cells Stephens (2003); Michalet (2005), and event horizons The EHT Collaboration (2019a, b). Hence, it is not only a fundamental question but also relevant to a wide range of applications to ask how quantum metrology can contribute to the detection of functions with ultimate accuracy.

The quantum version of function estimation has been investigated in terms of the signal detection theory in Refs. Berry and Wiseman (2002, 2006); Tsang et al. (2011); Berry et al. (2015); Fraas (2016); Dinani and Berry (2017). In fact, weaker scaling laws are implied when the target parameter can change continuously in time, such as a Gaussian signal. The demonstration of such unconventional limits has recently become within the experimental reach due to the realization of, e.g., high-N00N states Afek et al. (2010) and optical phase tracking Yonezawa et al. (2012). Although the detection theory is applicable to stochastic noises, it does not support the case where the relevant parameter is not inherently stochastic, which is often the case with quantum imaging and quantum signal processing Mamaev et al. (2003); Galleani and Tavella (2008); Brida et al. (2010); Lupo and Pirandola (2016).

In this Letter, we present a fundamental framework of quantum metrology on functions. Unlike parametric estimation, function estimation involves infinite degrees of freedom and inevitably requires further assumptions on the target function. Assuming only the smoothness of the function, we find the SQL of $O(N^{-q/(2q+1)})$ and the Heisenberg limit of $O(N^{-q/(q+1)})$, where $q$ indicates the degree of smoothness of the function. Our framework allows analysis of estimation errors of data series under given smoothness, such as a bound on the amplitude of derivatives. This includes the previous results on Gaussian processes through computation of their smoothness Belayev (1961), as demonstrated later. The data series requires neither to have a prior distribution nor even to be continuous, allowing, for example, a sample with a finite number of discontinuous points Brida et al. (2010); Samantaray et al. (2017). Moreover, we have found that the error limit can equally be saturated by states which are localized in position or wavenumber. This result implies the equivalence between space discretization and momentum cutoff in quantum information processing, reminiscent of the Nyquist–Shannon sampling theorem in classical statistics.

Setup.

We consider the estimation of an unknown function $\varphi(x)$ defined over the interval $0\leq x\leq L$ by using the position-dependent phase-shift gate $\hat{U}_{\varphi}$ shown in Fig. 1. For simplicity, we also assume the periodic boundary condition.

To estimate $\varphi$, we prepare an $N$-particle state as a probe, which evolves according to the unitary operator $\hat{U}_{\varphi}$ and then is measured. These particles are distributed in the interval $[0,L]$ and have two internal states: one state $\lvert{+}\rangle$ interacts with the phase-shift gate $\hat{U}_{\varphi}$, and the other state $\lvert{-}\rangle$ does not.

We work in the first-quantization formalism and denote by $\lvert{x;\pm}\rangle$, which is the position eigenstate at $x$ with internal state $\pm$. Let the phase-shift gate act as $\hat{U}_{\varphi}\lvert{x;+}\rangle=e^{i\varphi(x)}\lvert{x;+}\rangle$ and $\hat{U}_{\varphi}\lvert{x;-}\rangle=\lvert{x;-}\rangle.$ The unknown function $\varphi$ can be estimated from measurement on this output. When the probe is composed of $N$ separable particles, the error of the function estimation is bounded by the SQL. A probe with appropriately entangled $N$ particles, on the other hand, leads to the Heisenberg limit.

We recall that the estimation error of a scalar parameter $\theta$ is computed as $\delta^{2}=\mathbb{E}[\lvert{\tilde{\theta}-\theta}\rvert^{2}]$, where $\tilde{\theta}$ is the estimator depending on the stochastic nature of measurement outcome. Similarly, we consider a stochastic estimator $\tilde{\varphi}$ for the function, and compute the mean-square periodic error (MSPE)  Routtenberg and Tabrikian (2013) as

[TABLE]

In other words, the estimation error is averaged over $x$ and the modulus is replaced by [\tilde{\varphi}(x)-\varphi(x)]_{2\pi}=\min_{n\in\mathbb{Z}}\big{\lvert}{\tilde{\varphi}(x)-\varphi(x)+2\pi n}\big{\rvert}, i.e. the minimal modulus regarding the $2\pi$ periodicity of a phase.

The main difficulty in function estimation lies in the fact that the problem involves an the infinite degrees of freedom. In particular, the lower bound on $\delta^{2}$ cannot be established for an arbitrary function, since we cannot exclude any rapidly fluctuating functions from a finite number of measurements. Hence, we impose the following constraint on the target function $\varphi$:

[TABLE]

for some positive number $M>0$. With this constraint, we can establish a suitable lower bound on sufficiently smooth and slowly varying functions $\varphi$.

The condition \eqrefDiffReg can be applied only when the target function is differentiable. In our framework, we consider more general functions without differentiability: the Hölder continuity $\lvert{\varphi(x+\epsilon)-\varphi(x)}\rvert=O(\epsilon^{q})$ for fixed $0<q\leq 1$ Lunardi (2012). More rigorously, we impose a general constraint:

[TABLE]

where $a>0$ is a constant that does not affect the estimation error in the limit of large $N$. The special case with $q=1$ reduces to Eq. \eqrefDiffReg.

Estimation methods.

Given the target function $\varphi$ under the constraint \eqrefHolderReg, there exist estimation methods that ensure a finite estimation error $\delta$ defined in Eq. \eqrefdef-MSPE. We here compare the following two different methods:

Position-state (PS) method – We estimate the individual phases $\varphi(x_{j})$ at several positions $x_{j}$, and then computationally reconstruct the entire function.

Wavenumber-state (WS) method – We prepare a sufficiently large number of wavefunctions $\psi(x)\propto e^{i\varphi(x)}$, and estimate the function $\varphi$ by reconstructing the quantum state $\lvert{\psi}\rangle$ by the quantum tomography.

We find that the numbers of particles $N$ required for these two methods are the same up to a constant factor.

Position-state method.

The PS method can be used when the target function is relatively small, say, $\lvert{\varphi(x)}\rvert\leq\pi/3$ for all $x$. In this case, we can circumvent the phase wrapping problem and employ a method analogous to the kernel density estimation Stone (1980).

In the first step, we sample $n_{1}$ positions $x_{1},x_{2},\dotsc,x_{n_{1}}$ in the interval $0\leq x\leq L$ with equal spacing. Then, the phase $\varphi_{j}=\varphi(x_{j})$ at each position $x_{j}$ is measured by using $n_{2}$ particles localized at $x_{j}$. The estimation error $\delta_{\mathrm{ind}}$ of the individual phase $\varphi_{j}$ is known Kitaev (1996); Higgins et al. (2007) as it is the quantum metrology on a scalar parameter; the SQL $\delta_{\mathrm{ind}}=O(n_{2}^{-1/2})$ is established by the probe $\frac{1}{\sqrt{2}}(\lvert{x_{j};-}\rangle+\lvert{x_{j};+}\rangle)^{\otimes n_{2}}$ and the Heisenberg limit $\delta_{\mathrm{ind}}=O(n_{2}^{-1})$ by $\frac{1}{\sqrt{2}}(\lvert{x_{j};-}\rangle^{\otimes n_{2}}+\lvert{x_{j};+}\rangle^{\otimes n_{2}})$. Finally, the function estimator $\tilde{\varphi}(x)$ is computed from the individual estimators $\tilde{\varphi}_{j}$ by local linear smoothing Wand and Jones (1994):

[TABLE]

where $f$ is a smoothing function. In the present case, we may just set $f(x)=1$ for $x\leq\frac{L}{2n_{1}}$ and $f(x)=0$ otherwise. This corresponds to the approximation by the value at the nearest site, i.e., we set $\tilde{\varphi}(x)=\tilde{\varphi}_{j}$ where $x_{j}$ is the point nearest to $x$.

The estimation error can be decomposed into two parts: the statistical error $\delta_{\mathrm{stat}}$ caused by the measurement, and the deterministic error $\delta_{\mathrm{det}}$ due to smoothing. The balance between these errors can be tuned by the width $l$ of smoothing. The estimated value $\tilde{\varphi}_{j}(x)$ is of the same order as $\delta_{\mathrm{ind}}$: $\delta_{\mathrm{stat}}=O(n_{2}^{-1/2})$ for the SQL and $\delta_{\mathrm{stat}}=O(n_{2}^{-1})$ for the Heisenberg limit. On the other hand, the deterministic error $\delta_{\mathrm{det}}$ is the variation of $\varphi(x)$ within the width $\frac{L}{2n_{1}}$, which turns out to be $\delta_{\mathrm{det}}=O(n_{1}^{-q}M)$ by virtue of the constraint \eqrefHolderReg.

For a given number of particles $N=n_{1}n_{2}$, the optimal accuracy is determined by the trade-off between $\delta_{\mathrm{stat}}$ and $\delta_{\mathrm{det}}$. As a consequence of Young’s inequality, we obtain

[TABLE]

for the SQL and

[TABLE]

for the Heisenberg limit. Therefore, the overall estimation error $\delta$ is significantly larger than the traditional quantum limit, which is an expected feature of the function estimation. We note that entanglement of particles in different positions is not necessary to achieve the Heisenberg limit; such intersite entanglement does not enhance the estimation of linear parameters, as suggested in studies of quantum network sensors Proctor et al. (2018); Ge et al. (2018).

Wavenumber-state method.

In the WS method, we begin with the wavenumber eigenstate with zero eigenvalue: $\int_{0}^{L}\frac{dx}{\sqrt{2L}}[\lvert{x;-}\rangle^{\otimes n_{\mathrm{p}}}+\lvert{x;+}\rangle^{\otimes n_{\mathrm{p}}}].$ We use the one-particle state ($n_{\mathrm{p}}=1$) for the SQL and a multipartite EPR state ($n_{\mathrm{p}}>1$) for the Heisenberg limit.

By the phase-shift gate $\hat{U}_{\varphi}$, one obtains the output probe state {gather} \lvertS_φ⟩ = ∫_0^L dx2L [\lvertx;-⟩^⊗n_p + e^in_pφ(x)\lvertx;+⟩^⊗n_p].

The estimation is conducted by reconstructing $\lvert{S_{\varphi}}\rangle$ as accurate as possible by measuring $n_{\mathrm{c}}$ copies of the probe state. For this purpose, we consider the projection $P_{K}$ onto the subspace of wavenumbers $k$ such that $\lvert{k}\rvert\leq 2\pi K/L$. Since the postselected state $\lvert{S^{*}_{\varphi}}\rangle\propto P_{K}\lvert{S_{\varphi}}\rangle$ belongs to a $(2K+1)$-dimensional Hilbert space, it can be identified by the quantum tomography.

The error of the state reconstruction can be quantified by the infidelity $1-\lvert{\langle{S_{\varphi}}|{S_{\tilde{\varphi}}}\rangle}\rvert,$ where $\lvert{S_{\tilde{\varphi}}}\rangle$ denotes the reconstructed state. In fact, we show in the Supplemental Material 111See Supplemental Material for the rigorous derivation of the biased Cramér-Rao bound, details of the explicit estimation procedures, and an application of Kitaev’s method on function estimation. that the MSPE has can be bounded by the expected infidelity as {align} δ^2 &≤π2np2 E[ 1 - \lvert⟨ S_φ — S_~φ ⟩\rvert ] \notag

≤π2np2(1 - \lvert⟨ S_φ — S^_φ* ⟩\rvert^2) + π2np2 E[ 1 - \lvert⟨ S_φ^* — S_~φ ⟩\rvert^2 ] \notag

= δ^2_PS + δ^2_QT.

Here, the error is divided into the postselection part $\delta^{2}_{\mathrm{PS}}$ and the quantum-tomography part $\delta^{2}_{\mathrm{QT}}$. The postselection error can be bounded by the constraint \eqrefHolderReg as $\delta_{\mathrm{PS}}\leq O(K^{-q}M)$ Quade (1937); Prossdorf (1975); Note (1), while the results of the finite-dimensional tomography imply $n_{\mathrm{p}}\delta_{\mathrm{QT}}\leq O(K^{1/2}n_{\mathrm{c}}^{-1/2})$ Fujiwara and Nagaoka (1995).

When $n_{\mathrm{p}}=1$ and $N=n_{\mathrm{c}}$, we have the trade-off relation between $\delta_{\mathrm{PS}}$ and $\delta_{\mathrm{QT}}$ for the SQL:

[TABLE]

By setting $n_{1}=K$ and $n_{2}=N/K$, the errors $\delta_{\mathrm{PS}}$ and $\delta_{\mathrm{QT}}$ can be mapped to the errors $\delta_{\mathrm{det}}$ and $\delta_{\mathrm{stat}}$ in the PS method, respectively. Therefore, the SQL in the WS method reduces to that in Eq. \eqrefPS-SQL obtained by the PS method.

The error bound can be lowered for $n_{\mathrm{p}}>1$ and $N=n_{\mathrm{p}}n_{\mathrm{c}}$, while $K\leq O(n_{\mathrm{c}})$ must be maintained in order to robustly conduct the quantum tomography. Therefore, the optimal trade-off relation for the Heisenberg limit is

[TABLE]

By setting $n_{1}=n_{\mathrm{c}}$ and $n_{2}=n_{\mathrm{p}}$, this tradeoff relation corresponds exactly to the PS method, and we obtain the same Heisenberg limit as Eq. \eqrefPS-HL. However, there is a caveat that, in the output state \eqrefWS-output the phase ambiguity of $\varphi(x)$ modulo $2\pi/n_{\mathrm{p}}$ needs to be removed. We show in the Supplemental Material Note (1) that this removal can be handled by analogy with the Kitaev’s method Kitaev (1996).

Optimality of the SQL.

We have preposed the SQL \eqrefPS-SQL and the Heisenberg limits \eqrefPS-HL that can be achieved by both the PS and the WS methods. We show that these limits are in fact optimal; any theoretical method is subject to the same bounds on the estimation error.

We first derive the theoretical lower bound on the SQL. We consider the Fourier transform of the function $\varphi(x)$:

[TABLE]

On the wavenumber basis, the constraint \eqrefHolderReg corresponds to the suppression of high-wavenumber components: $\varphi_{k}=o(k^{-q})$. In particular, the special constraint \eqrefDiffReg is equivalent to $\sum_{k=1}^{\infty}k^{2}\lvert{\varphi_{k}}\rvert^{2}\leq\frac{M^{2}}{8\pi^{2}},$ which can be seen from Perseval’s equality. A generalization of this argument leads to a sufficient condition on the constraint \eqrefHolderReg:

[TABLE]

where $c_{0}=2\pi^{q}\max_{0\leq x\leq\pi}[x^{-q}\sin x]$ is a $q$-dependent constant.

To utilize the known results in the discrete parameter estimation Humphreys et al. (2013); Baumgratz and Datta (2016), we focus on the functions with only some low-wavenumber components. Using a $K$-dimensional vector $\mathbf{u}=(u_{1},\dotsc,u_{K})$, we parametrize the function $\varphi$ as

[TABLE]

Such function $\varphi_{\mathbf{u}}$ meets the constraint if $\lVert{\mathbf{u}}\rVert\leq\rho$ is satisfied for $\rho=c_{0}^{-1}MK^{-q}$. Since $\rho=o(1)$ and $\sqrt{2}\sin(2\pi ikx/L)$ forms an orthonormal basis, the MSPE $\delta^{2}$ of the function $\varphi$ can be bounded by MSE $\delta_{\mathbf{u}}^{2}$ of the vector $\mathbf{u}$.

Hence, instead of the function estimation, we may consider the vector estimation in which case the error can be evaluated by the quantum Cramér-Rao bound (QCRB) Holevo (1973); Helstrom (1969). For an unbiased estimation, the QCRB is given as

[TABLE]

where $[J(\mathbf{u})]$ is the Fisher information matrix defined for the output probe state $\lvert{\psi_{\mathbf{u}}}\rangle$ as

[TABLE]

Since the Fisher information is bounded from above by the SQL Pang and Brun (2014): $[J(\mathbf{u})]_{jj}\leq 8N$ for each $1\leq j\leq K$, we obtain a uniform, unbiased bound $\delta_{\mathrm{UUB}}=(K/8N)^{1/2}.$

Although such uniform bound is not applicable to the biased estimation, there exists the worst-case biased bound $\delta_{\mathrm{WBB}}$ Note (1). Within the region $\lVert{\mathbf{u}}\rVert\leq\rho$, one can find a vector $\mathbf{u}$ satisfying

[TABLE]

Since this biased version of QCRB holds for any integer $K\geq 1$, we choose $K$ that gives the maximal bound $\delta_{\mathrm{WBB}}$. This is satisfied when $\rho$ and $\delta_{\mathrm{UUB}}$ are comparable to each other — this case holds when we set K=O\bigl{(}(M^{2}N)^{1/(2q+1)}\bigr{)}. Hence the SQL is given as

[TABLE]

where the constant factor $c_{1}$ is lower-bounded by $\frac{q}{2(2q+1)^{2}}[c_{0}^{-1}q]^{2q/(2q+1)}$.

Optimality of the Heisenberg limit.

We consider the case in which entanglement between at most ${n_{\mathrm{p}}}$ ($\geq 1$) particles is allowed. It is known that the quantum information of a probe state is maximal when their wavefunction is completely symmetric Imai and Fujiwara (2007). With completely symmetric probe states, the problem becomes equivalent to the estimation of an effective phase ${n_{\mathrm{p}}}\varphi(x)$ with $n_{\mathrm{c}}=n_{\mathrm{p}}^{-1}N$ separate particles; the probe states in \eqrefWS-output serve as an example for the WS method.

Since the function of interest $\varphi$ is replaced by its effective one ${n_{\mathrm{p}}}\varphi$, the MSPE $\delta^{2}$ and the normalization constant $M$ are replaced by $n_{\mathrm{p}}^{2}\delta^{2}$ and ${n_{\mathrm{p}}}M$, respectively. This argument leads to a generalized limit: {align} n_pδ&≥c_1 [(n_p M)^1/q(n_p^-1N)^-1]^q/(2q+1) \notag

= c_1 (M n_p^q+1 N^-q)^1/(2q+1).

To restore the original function $\varphi(x)$ from the estimate of ${n_{\mathrm{p}}}\varphi(x)$, we need to resolve the phase ambiguity by $2\pi/{n_{\mathrm{p}}}$. For this purpose, the left-hand side of \eqrefReduced-SQL should not exceed $\pi$, giving

[TABLE]

With the maximal ${n_{\mathrm{p}}}$ substituted in \eqrefReduced-SQL, we obtain the Heisenberg limit

[TABLE]

where the constant $c_{2}$ is at least $(\pi^{-1}c_{1})^{(2q+1)/(q+1)}$.

Extension to smoother functions.

The degree of smoothness can further be extended into $q>1$, where the target function is known to be more than just differentiable. For an integer $m$ and $0<\sigma\leq 1$ satisfying $q=m+\sigma$, the constraint for smoother functions is given as

[TABLE]

Our results in the PS method and the optimality are also valid for $q>1$, thus leaving the quantum limits \eqrefPS-SQL and \eqrefPS-HL unchanged. On the other hand, the straightforward extension of the WS method into $q>1$ does not work. Therefore, the asymptotic equivalence between the PS method and the WS method can be obtained only for $0<q\leq 1$. See the Supplemental Material for more details Note (1).

Comparison with Gaussian signal estimation.

The error bounds we have obtained here are related to that of the Gaussian signal estimation Berry et al. (2013, 2015); Dinani and Berry (2017), in which the time-dependent phase $\varphi_{t}$ is subject to a Gaussian process with the power spectrum $I(\omega)\sim\lvert{\omega}\rvert^{-p}$. The estimation error of an instantaneous phase $\varphi_{t=0}$ is $O(\mathcal{N}^{-(p-1)/2p})$ for a coherent state and $O(\mathcal{N}^{-(p-1)/(p+1)})$ for a squeezed state, where $\mathcal{N}$ is the photon flux Dinani and Berry (2017). This can exactly be mapped into the SQL $O(N^{-q/(2q+1)})$ and the Heisenberg limit $O(N^{-q/(q+1)})$ in our study by setting $p=2q+1$. In fact, almost all sample functions of the Gaussian process $\varphi_{t}$ satisfy Eq. \eqrefGenHolderReg by taking the large time span Belayev (1961); Kono (1970). Therefore, the estimation error of the Gaussian process is subject to the quantum metrology of function estimation. We suspect that this fact is related to the minmax theorem Berger (2013); Tanaka (2012), which explains the consistency between the Bayesian and non-Bayesian estimation methods, though no clear connection is established yet.

Conclusion and outlook.

In this Letter, we have established the fundamental limits on function estimation subject to a bounded $q$th-order differentiability. The estimation error is bounded from below by $O(N^{-q/(2q+1)})$ in the standard quantum limit and $O(N^{-q/(q+1)})$ in the Heisenberg limit. These results can be reduced to the previous studies on the signal estimation in quantum optics Berry et al. (2015); Dinani and Berry (2017) when the target function is an infinitely extended stochastic process. We have presented two theoretical methods of the functional quantum metrology, both of which saturate the fundamental limits for $0<q\leq 1$.

This is a fundamental result for the effecient detection of functional structures — continuous signals and images, for example — which is a common target of estimation today. In fact, our results set theoretical bounds on various types of analysis relying on the function structure, such as model prediction or feature extraction Moon and Stirling (2000); Prochazka et al. (2013). On one hand, these bounds indicate the turning point where quantum methods outperform classical methods on functional data, with the scaling laws different from those obtained from parameter estimation. On the other hand, our result shows the optimal strategies for the quantum estimation of functions, such as an appropriate choice of temporal/spatial resolution or the size of entanglement. We note that choice of resolution is crucial in the real application Tsang et al. (2016); Lupo and Pirandola (2016); Tsang (2017); Lu et al. (2018), and what is more in the quantum case, we have seen that larger entanglement does not necessarily mean better accuracy.

The framework presented here also enables further quantum information-theoretic analysis on functions, such as a the quantum version of the Nyquist–Shannon sampling theorem which concerns the exact equivalence between the position- and wavenumber-states in the signal detection, including the $O(1)$ prefactor that has remained undetermined.

Acknowledgements.

We gratefully acknowledge D. W. Berry for helpful discussions on the quantum signal estimation. Special thanks are due to Y. Ashida and R. Hamazaki for critical advice. This work was supported by a Grand-in-Aid for Scientific Research on innovative Areas “Topological Materials Science” (KAKENHI Grant No. JP15H05855) from MEXT of Japan. N. K. was supported by the Leading Graduate Schools “ALPS.”