As a particularly sensitive carrier of information, light represents an exceptional platform for precision measurements, with applications including spectroscopy, interferometry, positioning, timing, and imaging. The properties of quantum light are determined, on one hand, by its quantum state, which may be coherent, thermal, or a Fock state, for instance. On the other hand, a full description of the electromagnetic field further requires knowledge of the modes on which this quantum state is defined [1]. These modes are normalized solutions of Maxwell’s equations that determine the spatial intensity distribution, frequency spectrum, and polarization of the light field. We refer to the parameters that determine these modal properties as mode parameters. In contrast, the number of photons, purity, and temperature are parameters that define the state on these modes. Other parameters, such as phase shifts, can be equivalently considered as parameters of the mode or the state.

The ultimate precision limit on the measurement of any parameter can be determined within the framework of quantum metrology [2–8]. A lower bound on the variance of any unbiased estimator for the parameter of interest is given by the quantum Cramér–Rao bound. This bound can be achieved asymptotically when a large number of measurement results are available, for instance, by applying maximum likelihood estimation to the data obtained from an optimal observable. This approach has led to the improvement of the error scaling in several measurements [9–11], e.g., in gravitational wave detectors [12,13] or atomic clocks and interferometers [10].

Methods to construct the optimal observable and to determine the quantum Cramér–Rao bound are, in principle, available for arbitrary states and parameters. Explicit expressions can be obtained, e.g., from the spectral decomposition of the state [7,8,10,14], from integral representations [7], or using matrix vectorization techniques [15]. Furthermore, convenient decompositions in terms of the covariance matrix are available for Gaussian states [16–20]. However, all of these techniques require the explicit description of the evolution of the quantum state under variations of the parameter. This is particularly simple when the parameter is imprinted by a unitary transformation and the evolution is given in terms of a parameter-independent Hamiltonian via Schrödinger’s equation. But when the parameter of interest describes a property of the mode, the resulting evolution of the state is less evident.

So far, mode parameter estimation has been approached on a case-by-case basis. Known results that explicitly discuss the precision limits for a mode parameter are either limited to specific observables and estimators [21–23] or to specific states, such as a combination of a coherent state with a pure Gaussian state [24]. A notable recent breakthrough was the identification of sub-Rayleigh imaging strategies that outperform traditional direct imaging [11,25] through a systematic optimization over all measurement observables. Formally, the imaging problem addresses the estimation of the separation of the two sources' transverse spatial modes. The quantum limit was first derived in the limit of very faint sources [25], and has since been generalized to arbitrary number-diagonal states [26], which include thermal states [27].

One of the most interesting prospects of quantum metrology is the improvement of the error scaling beyond the standard quantum limit (SQL). Quantum-enhanced measurement strategies exploit squeezed or entangled states [4–6,10,28,29] to reduce the relevant quantum fluctuations beyond those of the vacuum, which define the SQL. Several results in the existing literature suggest that such enhancements are impossible for mode parameter estimation problems: known precision bounds, e.g., in superresolution imaging [26], and the estimation of beam displacement with a single mode field [21] depend only on the average number of photons $N = \langle {\hat N}\rangle$. Quantum strategies can suppress measurement fluctuations beyond the SQL only if the precision bound actually depends on these fluctuations, e.g., when terms such as $\langle {\hat N^2}\rangle$ are present. In such cases, which include interferometric measurements, quantum metrology offers a wealth of well-known strategies ranging from squeezed to NOON states that lead to improved precision [8,10]. The maximal theoretically possible improvement is known as the Heisenberg limit and corresponds to a factor of $N$ over the SQL. For estimation of beam displacement, it was shown that by populating a second, carefully designed mode, quantum enhancements are possible [21,30].

In this paper, we derive a quantum theory that identifies the quantum precision limit on the estimation of any mode parameter by optimizing over all possible quantum measurements without making assumptions about the state or the modes on which it is defined. We find that any mode parameter can be estimated with quantum-enhanced precision if suitable modes are populated with nonclassical states. Our results reveal mode design strategies that enable a scaling improvement, which consist of either (1) choosing initial modes that have nonvanishing overlap with their own derivative modes or (2) populating suitably chosen auxiliary modes. Both approaches open the way to reduce measurement noise below the SQL using standard techniques based on squeezed or other nonclassical states. We illustrate the general applicability of our framework with various examples, including the estimation of beam displacements and superresolution imaging.

Quantum theory of mode parameter estimation. The quantum limit on the estimation of a parameter $\theta$ is given by the quantum Cramér–Rao bound ${(\Delta {\theta _{{\rm est}}})^2} \ge 1/{F_Q}[\hat \rho (\theta)]$, where quantum Fisher information (QFI) ${F_Q}[\hat \rho (\theta)]$ [2,7,10,14] describes the sensitivity of the state $\hat \rho (\theta)$ under variations of $\theta$. The central quantity of interest, the QFI, can be determined explicitly with a variety of methods [7,8,14–16], which all require explicit knowledge of $\frac{\partial}{{\partial \theta}}\hat \rho (\theta)$. The evolution $\frac{\partial}{{\partial \theta}}\hat \rho (\theta)$ describes the variation of the quantum state under changes of the parameter of interest. Note that it is generally not necessary to vary parameter $\theta$ in an experiment. Typically, we detect small deviations of $\theta$ around a fixed value that we can use to define $\theta = 0$.

A particularly simple situation arises when the parameter is imprinted via a unitary transformation with a parameter-independent Hamiltonian $H$. In this case, $\frac{\partial}{{\partial \theta}}\hat \rho (\theta) = - i[\hat H,\hat \rho (\theta)]$ is governed by the von Neumann equation, and explicit expressions for the QFI ${F_Q}[\hat \rho ,\hat H]$ are available as a function of the initial state $\hat \rho = \hat \rho (0)$ and the Hamiltonian $\hat H$. The QFI ${F_Q}[\hat \rho ,\hat H]$ expresses the quantum fluctuations of the state $\hat \rho$ and, indeed, depends quadratically on $\hat H$. In fact, for pure states $\hat \psi$, the QFI for unitary evolutions reduces to the variance ${F_Q}[\hat \psi ,\hat H] = 4(\Delta \hat H)_{\hat \psi}^2 = 4({\langle {\hat H^2}\rangle _{\hat \psi}} - \langle \hat H\rangle _{\hat \psi}^2)$ [14]. The nonlinear term ${\langle {\hat H^2}\rangle _{\hat \psi}}$ can be exploited to reduce the quantum noise beyond the SQL with the help of nonclassical states, such as metrologically useful entangled states or squeezed states whose quantum fluctuations are smaller than those of coherent and vacuum states [4–6,10,28,29].

Let us now address the variation of a quantum state $\frac{\partial}{{\partial \theta}}\hat \rho (\theta)$ that is caused by changes of a parameter $\theta$ that determines the modes on which the state is defined. We focus on the common situation where a parameter-independent quantum state is prepared in one or several modes whose shape depends on a single parameter of interest. Consider a mode basis ${\{{f_k}\} _k}$ with an inner product $({f_k}|{f_l}) = \int {\rm d}xf_k^*(x){f_l}(x) = {\delta _{\textit{kl}}}$. Here, $x$ denotes an abstract set of arguments of the modes, which may be, e.g., spatial, temporal, frequency, or other coordinates. A perturbation of these modes gives rise to the shifted mode basis ${\{{f_k}[\theta]\} _k}$, parametrized by $\theta$, such that at $\theta = 0$, we recover the original modes, i.e., ${f_k}[0] = {f_k}$ for all $k$. As our first result, we show that changes of an arbitrary state $\hat \rho$ due to a variation of mode parameter $\theta$ around $\theta = 0$ can be described by a unitary beam splitter evolution (see Supplement 1) as $\frac{\partial}{{\partial \theta}}\hat \rho = - i[\hat H,\hat \rho]$, with the effective Hamiltonian

Here, $\hat a_k^\dagger$ and $d_k^\dagger = \frac{i}{{{w_k}}}\sum\nolimits_j ({f_j}|f_k^\prime)\hat a_j^\dagger$ create a photon in the modes ${f_k}$ and $\frac{{if_k^\prime}}{{{w_k}}} = \frac{i}{{{w_k}}}\frac{\partial}{{\partial \theta}}{f_k}[\theta {]|_{\theta = 0}}$, respectively, where ${w_k} = \sqrt {(f_k^\prime |f_k^\prime)}$. The resulting mode-mixing evolution thus coherently redistributes populations from the original modes ${f_k}$ into the derivative modes $\frac{{if_k^\prime}}{{{w_k}}}$. The coupling coefficients of the effective Hamiltonian depend on the shape and normalization ${w_k}$ of the modes via the overlap integral $i({f_j}|f_k^\prime)$. Note that orthonormality of the modes implies that $(f_k^\prime |{f_j}) = - ({f_k}|f_j^\prime)$, and thus $\hat H$ is Hermitian.

The effective Hamiltonian Eq. (1) provides a remarkably simple description of a quantum state’s dependence on a mode parameter and translates an evolution of the modes into an evolution of the state. Most importantly in our context, this result allows us to determine the quantum limits on any mode parameter estimation using well-known expressions for the QFI ${F_Q}[\hat \rho ,\hat H]$ for unitary evolutions, which apply to arbitrary quantum states.

In practical situations, typically only a finite number of modes ${f_k}$ will be occupied by the initial state $\hat \rho$. By explicitly distinguishing populated modes from vacuum modes, we gain insight into the strategies that allow us to optimize the quantum limits on mode parameter estimation. To this end, let us introduce

In this expression, the sensitivity of the state is described exclusively in terms of modes that are initially populated. The first term in Eq. (3) is the QFI for a unitary evolution generated by ${\hat H_I}$ and thus contains relevant quantum fluctuations of the state $\rho$. As we will detail below, nonclassical states with quantum fluctuations below the SQL are able to generate quantum enhancements by increasing this term beyond classical limits. In contrast, the second term always scales linearly with the average number of photons. It is therefore independent of the state’s quantum fluctuations and consequently cannot beat the scaling of the SQL. This less favorable scaling ensues from the quantum noise of vacuum modes that exchange relevant information about the parameter through the effective beam splitter. Whenever the derivative modes $\frac{{if_k^\prime}}{{{w_k}}}$ have nonvanishing overlap with some initially unpopulated modes ${f_k}$, part of the information about the parameter will end up in vacuum modes whose fluctuations limit measurement precision. To see this explicitly, note that we may rewrite Eq. (3) as ${F_Q}[\hat \rho (\theta {)]|_{\theta = 0}} = {F_Q}[\hat \rho ,{\hat H_I}] + {\langle \hat O\rangle _{\hat \rho}}$, where

Mode design. We are in the position to make a simple but important observation about the origin of quantum sensitivity scaling enhancements in mode parameter estimations: any improvement of measurement precision must have its origin in the unitary QFI ${F_Q}[\hat \rho ,{\hat H_I}]$, which is the only term that actually depends on the quantum fluctuations of the state. However, this term vanishes when the Hamiltonian Eq. (2) is zero, and in this case, the SQL cannot be overcome. This can be avoided if there exist $j,k \in I$ such that $i({f_j}|f_k^\prime) \ne 0$.

In other words, a necessary condition for quantum-enhanced mode parameter estimation can be formulated as follows: for at least one initially populated mode ${f_k}$, a mode with nonzero overlap with the derivative mode $\frac{{if_k^\prime}}{{{w_k}}}$ must also be populated. There are generally two ways to achieve this: (1) the mode ${f_k}$ may already be nonorthogonal to its own derivative mode $\frac{{if_k^\prime}}{{{w_k}}}$, or (2) one may populate additional auxiliary modes that are proportional to $\frac{{if_k^\prime}}{{{w_k}}}$.

In the situation of Eq. (1), quantum enhancements are possible even if ${f_k}$ is the only populated mode. The most general single-mode scenario is discussed in detail in Supplement 1. We demonstrate that single-mode approaches are sufficient to achieve quantum-enhanced estimation of a mode parameter only if this parameter is encoded in the phase of the mode, which applies to the estimation of frequency and time, as well as to orbital angular momentum of Laguerre–Gauss modes [31].

In practical situations, it may not always be possible to manipulate the shape of the modes of interest as they are usually determined by the problem at hand. Nevertheless, even when ${f_k}$ is orthogonal to its derivative mode $\frac{{if_k^\prime}}{{{w_k}}}$, we may achieve quantum enhancements by following approach (2). To this end, we employ a multimode setting by incorporating suitable auxiliary modes with nonvanishing overlap with $\frac{{if_k^\prime}}{{{w_k}}}$. The unitary QFI ${F_Q}[\hat \rho ,{\hat H_I}]$ increases as the overlap between the populated modes ${f_k}$ and their derivatives $\frac{{if_k^\prime}}{{{w_k}}}$ grows. An extreme situation is found when all of the derivative modes $\frac{{if_k^\prime}}{{{w_k}}}$ can be expanded using only the initially populated modes ${f_k}$. In this case, of which a Mach–Zehnder interferometer is an important instance (see Supplement 1), the second line in Eq. (3) vanishes, and all the information about the precision limit is contained in the unitary QFI.

In summary, the population of suitable modes allows us to establish the necessary condition for achieving quantum-enhanced measurement precision. However, this condition is not sufficient: these modes must also be populated with suitable nonclassical states to overcome the fluctuations of the vacuum.

State design. Quantum states that lead to sensitivity improvement beyond the SQL can be identified for any nonzero effective Hamiltonian using standard methods from quantum metrology by maximization of QFI over a set of quantum states under appropriate constraints [8,10]. The choice of suitable nonclassical states depends on the limitations of the experimental setup at hand. Maximal quantum enhancements that achieve the Heisenberg limit typically require large and fragile superposition states that are hard to prepare, such as NOON states. Nevertheless, other classes of more accessible states are also able to achieve useful and scalable quantum enhancements under realistic conditions. For instance, we demonstrate in Supplement 1 that a strongly populated coherent state in a mode that is orthogonal to its own derivative can always be complemented by a squeezed state in a suitably designed auxiliary mode to improve the measurement precision of any mode parameter with simple homodyne measurements.

Applications. Given any mode parameter estimation task, a suitable measurement strategy is identified in two steps. First, a study of the effective Hamiltonian identifies a set of modes whose nonvanishing population establishes the necessary condition to achieve quantum-enhanced precision. Second, the precision limit for any quantum state, pure or mixed, Gaussian or non-Gaussian, prepared in those modes can be determined by virtue of the QFI. In the following, we apply our formalism to transverse spatial modes and superresolution imaging, focusing on the design of suitable modes. Additional examples are given in Supplement 1.

Transverse spatial modes. Whenever the mode of interest is orthogonal to its own derivative, quantum-enhanced precision can be achieved only by populating an additional auxiliary mode. This is the case for the measurement of transverse displacements of a beam described by Hermite–Gauss modes ${f_n}[\theta](x,y) = {{\rm HG}_{\textit{nm}}}(x + \theta ,y)$ with $m$ fixed (see Supplement 1 for details), which is a fundamental task in optics known as beam positioning. Populating only the single mode ${{\rm HG}_{\textit{nm}}}$ thus leads to the precision ${F_Q}[\hat \rho (\theta {)]|_{\theta = 0}} = 4\frac{{(2n + 1)}}{{{w^2}}}{\langle \hat N\rangle _{\hat \rho}}$, which scales linearly with $N$ and depends on the beam waist $w$. A classical enhancement of sensitivity is offered by modes of higher order $n$. By complementing with suitable auxiliary modes, the quadratic scaling and the potential for quantum enhancements can be recovered. These auxiliary modes correspond to the derivatives $w\frac{\partial}{{\partial x}}{{\rm HG}_{\textit{nm}}} = \sqrt n {{\rm HG}_{n - 1,m}} - \sqrt {n + 1} {{\rm HG}_{n + 1,m}}$, and can again be expressed in terms of Hermite–Gauss modes. For arbitrary multimode quantum states that occupy a basis of Hermite–Gauss modes, we obtain the effective Hamiltonian Eq. (1):

Superresolution imaging. The resolution of two point sources with a diffraction-limited imaging system is a mode parameter estimation problem of fundamental relevance for astronomy and microscopy [11,25]. The ultimate quantum limit was derived for thermal sources [26,27], but a general upper sensitivity bound reveals that even strongly nonclassical states cannot yield quantum scaling enhancement [26]. Following Refs. [25,26], we describe the problem in an orthogonal mode basis containing (anti-)symmetric combinations ${f_ \pm}$ of the local source modes. This orthogonalization procedure leads to parameter-dependent populations that are not described by the mode transformation due to losses [26]. We thus amend our general expression in Eq. (3) by adding the classical Fisher information of the populations: ${F_c} = \sum\nolimits_n {(p_n^\prime)^2}/{p_n}$ [7,14].

Whenever the phase of the point-spread function (PSF) of the imaging system [32] is independent of the transverse coordinate, we have $(f_ \mp ^\prime |f_ \pm ^\prime) = ({f_ \mp}|f_ \pm ^\prime) = ({f_ \pm}|f_ \pm ^\prime) = 0$. Since the derivative modes are orthogonal to the populated modes, the vanishing of the Hamiltonian Eq. (1) implies SQL scaling even if the source modes could be prepared in arbitrary nonclassical states [26]. The remaining terms in Eq. (3) read ${F_Q}{[\hat \rho (\theta)]_{\theta = 0}} = {F_c} + 4(f_ + ^\prime |f_ + ^\prime)\langle {\hat N_ +}\rangle + 4(f_ - ^\prime |f_ - ^\prime)\langle {\hat N_ -}\rangle$ and produce exactly the expression that was derived in Ref. [26] for number-diagonal states, demonstrating its validity for arbitrary states whose eigenstates are independent of the source separation (see Supplement 1).

Our general theory for quantum mode parameter estimation allows us to discuss possibilities for achieving beyond-SQL quantum enhancements in superresolution imaging, assuming that some control is available over the state of the sources as is the case, e.g., in certain microscopy settings or in the time–frequency domain. First, if the phase of the PSF is nonconstant, suitably constructed (anti-)symmetric modes will no longer be orthogonal to their derivatives. While for superresolution of spatial modes in the paraxial regime the assumption of a constant phase is well justified [26,32], nonconstant phases emerge naturally in the time–frequency domain, which has been studied experimentally [33]. For example, we show in Supplement 1 that a linear phase $\psi (x) = u(x){e^{- ikx}}$ adds to the sensitivity the term ${F_Q}[\hat \rho ,\hat H]$ with the Hamiltonian Eq. (1) $\hat H = k({\hat N_ +} + {\hat N_ -})/2$. Because ${F_Q}[\hat \rho ,\hat H] = 0$ whenever $\hat \rho$ and $\hat H$ commute, the sensitivity of number-diagonal states, in particular thermal states, remains unaffected by this additional term, reflecting their inability to overcome the SQL. However, suitable nonclassical emitters are able to exploit this term to achieve nonlinear sensitivity scalings with the number of photons $N$. Second, as we have seen in our general discussion as well as in previous examples, another possibility to achieve quantum scaling enhancements even if the mode shape cannot be modified consists of populating the derivative modes. These strategies open up interesting avenues for quantum-enhanced superresolution.

Conclusions. Any mode parameter estimation problem can be modeled by a suitable effective bilinear Hamiltonian that contains information about the shape of the modes. This reveals the general quantum limits for high-precision measurements of mode parameters, without requiring assumptions about specific states, modes, measurement observables, or estimators. Our general result predicts precisely how the shape of modes influences the quantum limits and ultimately determines whether or not quantum enhancements beyond the SQL are possible. We find that, generally, such quantum enhancements can be achieved by a suitable design of the modes on which the probe state is being prepared. These results reveal strategies to optimize precision measurements of mode properties in quantum optical settings with light and atoms, including in spectroscopy and imaging.