## Abstract

The quantum Fisher information (FI), when applied to the estimation of the separation of two point sources, has been shown to be non-zero in cases where the coherence between the sources is known. Although it has been claimed that ignorance of the coherence causes the quantum FI to vanish (a resurgence of Rayleigh’s curse), a more complete analysis including both the magnitude and phase of the coherence parameter is given here. Partial ignorance of the coherence is shown to potentially break Rayleigh’s curse, whereas complete ignorance guarantees its resurgence.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

It is intuitive that in the problem of estimating the separation of two point sources, the closer the sources become, the more difficult the task [1]. This limitation is intimately related to the study of resolution, whose definition has been the topic of discussion in recent investigations. Among the earliest definitions was the notion of the minimum separation between two point sources for which they can be distinguished with an imaging system. In this classical context, resolution was directly related to the maximum spatial frequency content that is adequately passed by the imaging system (which in turn is related to the system’s aperture) [2,3]. The term *superresolution* in this context, which is an ongoing area of research, refers to concepts that allow for the production of images where two point sources are resolved to be closer than is allowed by the classical resolution limit; various techniques include, but are not limited to, stochastic optical reconstruction microscopy (STORM) [4], stimulated emission depletion microscopy (STED) [4], and using superoscillations in conjunction with confocal microscopes [5–8]. However, in the context of image processing, resolution takes the definition of the minimum separation between point sources that can be estimated with a desired precision [3]. It is this latter definition of resolution that is the focus of this paper, which uses the language of the Fisher information (FI) [9]. The *classical* FI quantifies the upperbound of information that one can obtain regarding the separation of the two point sources, for a specified measurement scheme [10]. For the simplest schemes (such as the direct imaging of two point sources through an optical system), the classical FI vanishes as the separation vanishes; this phenomenon has been termed Rayleigh’s curse.

However, it has recently been shown that it is possible to circumvent Rayleigh’s curse through the implementation of new measurement schemes. Such developments were based on the *quantum* FI [9], which provides an experiment-free information lowerbound on the two-point separation problem. Quantum FI calculations showed that it was theoretically possible to obtain a non-vanishing information lowerbound even as the source separation approaches zero [11]. The proposal in [11] for incoherent point sources was verified in a series of experiments [12–15]. These results were followed by other experiments and discussions that commented on the true attainability of the quantum FI (the reciprocal of the variance upperbound of a parameter estimation) [16,17]. In particular, the coherence between the two point sources was identified as a relevant parameter to explore.

When the coherence, along with the separation, between two point sources is unknown, one must account for the need to jointly estimate both the coherence and the separation in a multi-parameter estimation framework. Previous work claimed, through quantum FI calculations, that the ignorance of coherence gives rise to the return of Rayleigh’s curse [16]. However, the theoretical framework of the analysis was valid only for real coherence parameters. Because of this, the phase of the coherence was implicitly assumed to be known. In the language of multi-parameter estimation, the coherence between two point sources is complex and therefore comprises two real parameters. Previous works therefore implicitly assumed that one of these parameters (the phase) is known, an assumption that we relax in our work.

A complete analysis of the effects of coherence in estimating the separation of two point sources is provided in this work. By allowing the coherence to be expressed as two real parameters (either or both of which may be unknown), novel phenomena, regarding the behavior of the quantum FI when the separation of the two point sources vanishes, are revealed. The discussion regarding whether Rayleigh’s curse persists when coherence is unknown is shown to be more complicated than initially expected. In addition to clarifying the effects of coherence on two-point separation estimation, the results presented here also serve to append the known results in the literature regarding the quantum FI of separation estimation for arbitrary coherence. So far, the cases where the sources are incoherent, real and partially coherent, and fully coherent have been studied [11,16,18]. This work provides the framework necessary to fill out the remainder of the complex coherence disk, as shown in Fig. 1(b). In doing so, as this work’s results demonstrate, we obtain both a deeper understanding of the connections of previously studied cases and novel phenomena evident only in a full treatment of the complex coherence. For example, we find a singular/multivalued behavior in the quantum FI for some scenarios of multi-parameter estimation.

It is important to point out that the quantum FI framework has been used extensively to study other generalizations of the incoherent two-point separation problem [19,20]. These include extending the incoherent treatment to more than two point sources and finding information bounds for separation estimation in three-dimensional space. Other parameters that have been analyzed are the estimation of longitudinal separation [21], simultaneous estimation of the separation, centroid, and relative intensities between two incoherent point sources [22].

## 2. THEORY

Our treatment begins in a fashion analogous to the approaches in [11,16,18] by stipulating an image-plane density matrix in the nonorthogonal (as indicated by the subscript) basis $\{|{\psi _ +}\rangle ,|{\psi _ -}\rangle \}$:

Equation (1) indicates that the there are four parameters to be treated in the framework of parameter estimation: ${\cal P} = \{A,r,\phi ,s\}$. The goal then is to obtain a $4 \times 4$ quantum FI matrix (QFIM), from which the quantum FI is calculated. This process begins with finding an orthonormal basis that spans $\hat \rho$ and its parametric derivatives ${\partial _j}\hat \rho$, where $j \in {\cal P}$. Once this basis is obtained, symmetric logarithmic derivative (SLD) matrices for each parameter can be calculated. Finally, the QFIM is readily obtained from the SLD matrices [23].

To begin, $\hat \rho$ is re-expressed in terms of an orthogonal basis $\{|{\psi _ +},|{\chi _ -}\rangle \}$, where $|{\chi _ -}\rangle$ is defined as

With ${\cal E}$, it is now possible write down the SLD matrix ${\hat {\cal L}^j}$ for each parameter $j \in {\cal P}$:

Although the SLD matrices are presented here as an intermediate step in determining the QFIM, they importantly serve the role of determining whether a single measurement of multiple parameters can be optimal with regard to the quantum Cramer–Rao bound determined by the QFIM. Namely, optimal measurements for $i,j \in {\cal P}$ can be simultaneously obtained if

where $[{\hat {\cal L}^i},{\hat {\cal L}^j}]$ is a commutator.Finally, upon obtaining the QFIM, it is possible to calculate the quantum FI regarding a measurement done on the parameters in ${\cal P}$ under a variety of conditions. First, one must identify the parameters ${{\cal P}_{\rm u}} \buildrel \Delta \over = \{{j_1}, \ldots ,{j_m}\} \subseteq {\cal P}$, where $1 \le m \le |{\cal P}|$, that are *unknown* (to be estimated through measurement). Once this is done, one then considers the submatrix $\hat {\cal Q}({{\cal P}_{\rm u}})$ that contains the rows and columns that correspond to the parameters in ${{\cal P}_{\rm u}}$. Note that $\hat {\cal Q}({{\cal P}_{\rm u}})$ will be a $m \times m$ matrix. The quantum FI ${H_{{j_i}}}$, for the parameter ${j_i}$, with $i = 1, \ldots ,m$, is then obtained as

## 3. RESULTS

Since our analysis involves the potential of calculating multi-parameter quantum FI, it is important to understand when this FI is simultaneously achievable. As noted before, this occurs for parameters $i,j \in {{\cal P}_{\rm u}}$ when Eq. (8) is satisfied. Of particular interest is the case of $i = s$, i.e., when one of the parameters is the separation between the two point sources. The conditions for which Eq. (8) is satisfied for various $j \in {{\cal P}_{\rm u}}$ are summarized in Table 1.

For simplicity, the remainder of our discussion will assume that the intensity PSF takes the form of a Gaussian:

#### A. Estimating Only the Separation Between Two Point Sources

The simplest ${H_s}$ is derived for the case when ${{\cal P}_{\rm u}} = \{s\}$. That is, all the parameters are assumed known aside from $s$. Here, Eq. (9) reduces to ${H_s} = {\hat {\cal Q}_{\textit{ss}}}$. This quantum FI is plotted in Fig. 2 as a function of the complex coherence parameter $\Gamma$ (on transverse disks) and the normalized separation $s/\sigma$ (longitudinally).

The three special cases of Tsang, Larson, and Hradil (which are the incoherent, purely real, and purely coherent cases) are encapsulated in the transverse disks as the origin, the region of ${\rm Im}(\Gamma) = 0$, and the circumference of the $\Gamma$ disk, respectively. The inset shows several special cases of ${H_s}$ that were previously mentioned by Tsang and Larson. Furthermore, note that Fig. 2 shows that for ${\rm Re}(\Gamma) = 0$, ${H_s}$ is a constant for all $s/\sigma$ with value $H = 1/4$ (olive green color in Fig. 2). This is a generalization of Tsang’s result regarding the incoherent case. For all other values of $\Gamma$, ${H_s}$ asymptotically approaches $H$ as $s/\sigma$ increases. Note that this asymptotic behavior is also true for ${H_s}$ in the cases, discussed in later sections, where ${{\cal P}_{\rm u}}$ contains more unknown parameters than just $s$.

The transverse disk $s = 0$ depicts the well-known anomalous behavior of ${H_s}$ as $\Gamma \to - 1$, which is the case where the two point sources become perfectly anti-correlated (the limit is naturally defined along the ${\rm Re}(\Gamma)$ axis since ${H_s}$ is independent of ${\rm Im}(\Gamma)$). This behavior, shown further in Fig. 3, describes the divergence of ${H_s}$ as $\Gamma \to - 1$ over an infinitesimally small region of $s/\sigma$. The cause of this anomalous behavior may be traced to the fact that the image of two perfectly anti-correlated point sources is dark when their separation vanishes. Indeed, this is reinforced by Fig. 3, which shows that the anomalous behavior does not exist for other values of $A$. That is, for $A \ne 1$, the value of ${H_s}$ at $s = 0$ and $\Gamma = - \sqrt A$ is finite (and relatively large). Therefore, the theory described in Section 2 indicates that if the intensities of the two sources are known to be even slightly unequal, the value of ${H_s}$ at $s = 0$, for $\Gamma = - \sqrt A$, is finite and large.

#### B. Estimating Both the Separation and Another Parameter

The case of ${{\cal P}_{\rm u}} = \{j,s\}$, where $j \ne s$, is now considered. The quantum FI is given here by

The case of ${{\cal P}_{\rm u}} = \{r,s\}$ is analyzed first, which includes Larson’s analysis in which it was claimed that the lack of knowledge of both the coherence and the separation causes Rayleigh’s curse to return (${H_s} = 0$ as $s \to 0$). However, their study was limited to the case of real $\Gamma$, which inherently assumes knowledge regarding $\phi$, the *phase* of the coherence. Therefore, their case actually corresponds to when only $r$ (and not $\phi$) is unknown in addition to $s$. This is precisely the case to be discussed now. Figure 4 shows ${H_s}$; note that ${H_s} \ne 0$ over the transverse disk of $s = 0$, which indicates that it is possible to avoid Rayleigh’s curse even when ${{\cal P}_{\rm u}} = \{r,s\}$. It should be noted that in Fig. 4 through the particular cross-section of ${\rm Im}(\Gamma) = 0$, ${H_s} = 0$ at $s = 0$ (see inset). In other words, the case of ${\rm Im}(\Gamma) = 0$, where Rayleigh’s curse returns, is the only case for which it does. Other values of $\Gamma$ allow for a non-zero ${H_s}$ at $s = 0$.

One can also consider the case ${{\cal P}_{\rm u}} = \{\phi ,s\}$. Unlike the preceding case, it is now assumed that $\phi$, the phase of the coherence parameter, is unknown (and $r$ is known). The corresponding ${H_s}$ is shown in Fig. 5, which indicates, as in the case of ${{\cal P}_{\rm u}} = \{r,s\}$, that it is also possible to avoid Rayleigh’s curse since ${H_s}$ does not necessarily vanish over the disk $s = 0$. In fact, it vanishes only when ${\rm Re}(\Gamma) = 0$. Note that, in contrast to the case of ${{\cal P}_{\rm u}} = \{r,s\}$, this condition involves the zero set of ${\rm Re}(\Gamma)$ rather than that of ${\rm Im}(\Gamma)$.

Note that ${H_s}$ is singular at $\Gamma = 0$ for ${{\cal P}_{\rm u}} = \{j,s\}$, where $j \in \{r,\phi \}$ (indicated by black dots in Figs. 4 and 5). This behavior is shown in an alternative manner in Fig. 6, where the singular nature is represented by the multi-valued nature of $r = 0$. That is, depending on the trajectory one takes (which $\phi$, for instance) in the limit of $\Gamma \to 0$, ${H_s}$ approaches a different value. Although Fig. 6 shows this behavior only for $s = 0$, this phenomenon persists as $s$ increases. However, as indicated by Figs. 4 and 5, the range of multi-values that ${H_s}$ can take for $r = 0$ collapses to $H = 1/4$ as $s$ increases. Finally, we note that this singular behavior is evident when the analysis includes both $r$ and $\phi$. It is possible to miss this phenomenon if knowledge regarding either $r$ or $\phi$ is assumed to be known in Eq. (1). For instance, if one were to assume ${\rm Im}(\Gamma) = 0$ (as was done in Larson’s work), then the limit as $\Gamma \to 0$ is automatically restricted to the vertical trajectories in Fig. 6(a) that correspond to $\phi \in \{0,\pi \}$. Since ${H_s} = 0$ for both of these trajectories, the singular behavior is consequently missed. The appearance of this singular behavior is perhaps indicative of some ambiguity regarding ${H_s}$ when $\Gamma$ is partially known. Presently, it is unclear as to what may determine (theoretically or experimentally) which of the multiple values ${H_s}$ may take at $\Gamma = 0$.

Only the plots of ${H_s}$ are shown in Figs. 4 and 5, since $s$ is arguably the more important parameter to estimate under the framework of this analysis. However, ${H_j}$, the quantum FI for the other unknown parameter $j$, can also be calculated. The two quantities ${H_s}$ and ${H_j}$ then represent the quantum FI for the two unknown parameters $j,s$, keeping in mind that these information bounds can be simultaneously reached (measured) only if the conditions in Table 1 are satisfied.

Although it is possible to consider the case of ${{\cal P}_{\rm u}} = \{A,s\}$, this case is not relevant to the analysis of how partial ignorance of $\Gamma$ affects ${H_s}$. Therefore, for brevity, this case is not further discussed despite the fact that the theory in Section 2 fully encapsulates this route of inquiry as well.

#### C. Estimating the Separation and the Complex Coherence

The case of ${{\cal P}_{\rm u}} = \{r,\phi ,s\}$ is now considered. This corresponds to the situation of, in addition to $s$, both the magnitude and phase of $\Gamma$. With three unknown parameters, ${H_s}$ is given by the more complicated expression of

*complete*ignorance of the coherence parameter $\Gamma$ does indeed lead to a vanishing quantum FI for $s$. However, despite the fact that ${H_s}$ vanishes here at $s = 0$, the FI is still larger than the classical FI of direct intensity measurements in a comparable scenario for $s \gt 0$. Therefore, the quantum FI calculations here indicate a possible advantage over conventional imaging methods even when $\Gamma$ is completely unknown. Further discussion is found in Supplement 1.

Although it is possible to look at other combinations of three unknown parameters (namely, those that include $A$ as an unknown), those results are not shown here since the main purpose of this work is to analyze the relationship between the quantum FI for $s$ and how it is affected by the ignorance (partial or full) of $\Gamma$. Finally, it is also possible to consider ${{\cal P}_{\rm u}} = {\cal P}$, i.e., the case where all four parameters are considered unknown. However, it is not possible for ${H_s}$ to increase from that shown in Fig. 7 since having additional unknowns will only serve to lower ${H_s}$. Therefore, Rayleigh’s curse cannot be avoided when all four parameters in ${\cal P}$ are unknown.

## 4. CONCLUDING REMARKS

The question of whether the ignorance of coherence, $\Gamma$, causes the resurgence of Rayleigh’s curse is shown to be more complicated when its magnitude and phase ($r$ and $\phi$, respectively) are considered to be distinct parameters to be estimated. To properly address this question, a theoretical framework for the multi-parameter estimation of two point sources is introduced, where the set of possibly unknown parameters, ${\cal P} = \{A,r,\phi ,s\}$, consists of the intensity ratio, magnitude of coherence, phase of coherence, and separation between the two sources.

The simplest case, where only the separation is unknown, corroborated the previously known results regarding the violation of Rayleigh’s curse. Namely, it is shown that as long as ${\rm Re}(\Gamma) \ne 1$ (for $A = 1$), then ${H_s}$, the quantum FI for source separation, is non-zero. Unsurprisingly, the anomalous behavior for when the sources are fully anti-correlated persists in the present framework. However, this anomaly disappears when the intensities of the two sources are unequal.

The main results of this work concern the cases where $\Gamma = r{e^{{\rm i}\phi}}$ is partially or completely unknown. When $\phi$ is known and $r$ is unknown, ${H_s}$ is shown to vanish over the $s = 0$ disk for the region ${\rm Im}(\Gamma) = 0$. This result agrees with a previous work’s assertion that, for real $\Gamma$, ${H_s}$ vanishes with the ignorance of $r$. However, we have shown that the region defined by ${\rm Im}(\Gamma) = 0$ is the only one where ${H_s}$ vanishes. For all other (complex) values of $\Gamma$, it is evidently possible to break Rayleigh’s curse. Similar results are true for when $r$ is known and $\phi$ is unknown. Hence, when $\Gamma$ is partially known within the context of multi-parameter estimation, it is possible to break Rayleigh’s curse. However, when $\Gamma$ is completely unknown, ${H_s}$ vanishes over the $s = 0$ disk. Therefore, complete ignorance of $\Gamma$, unlike partial ignorance (where either $r$ or $\phi$ is known), necessarily causes Rayleigh’s curse to reappear. Additional findings include the singular behavior at $\Gamma = 0$ for the cases when $\Gamma$ is only partially known.

The theoretical framework developed in this work allows for a wider scope of analysis than the results presented. This includes the effects of considering unequal intensities ($A \ne 1$), allowing $A$ to be an unknown parameter, and analyzing the quantum FI, ${H_j}$, for parameters $j \ne s$ that would need to be jointly estimated with the separation $s$. Additional discussions are possible regarding the attainability of optimal joint measurements and possible experimental verification of the results regarding the partial ignorance of $\Gamma$. However, these aforementioned topics, although interesting in their own right, fall outside the scope of this paper, which serves primarily to clarify the effects of coherence in two-point separation estimation. Nevertheless, an overview of an experimental design, used in [17], that can explore the theoretical results is provided as follows: the measured intensity arising from two partially coherent point sources can be generated by summing the intensities from its two coherent modes via the coherent mode decomposition. These modes can be generated from Gaussian laser light passing through a mode converter, and their intensities measured from a parity sorter to perform a binary spatial-mode demultiplexing (BSPADE) measurement. By adjusting the weights of the intensity summation between the two modes and performing the subsequent maximum likelihood estimation on the measured intensities, it is possible to explore the FI from different portions of the complex coherence disk.

## Funding

Defense Advanced Research Projects Agency (D19AP00042).

## Acknowledgment

The authors thank Andrew N. Jordan for useful discussions.

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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