The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.
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Resolution continues to be a central issue in optical imaging. Earlier notions of resolution were concerned with the minimum separation necessary for two incoherent point sources of equal intensities to be discerned from measurement of the intensity of their image through a diffraction-limited imaging system. Different definitions and criteria were based on various measures of discernibility [1–4]. Mathematical methods based on analytic continuation were later shown to formally solve the inverse problem associated with diffraction-limited imaging, i.e., restore details finer than the resolution limit [5–8]. Although such solutions were deemed to offer superresolution, it was noted that the precision of such solutions deteriorates rapidly in the presence of measurement noise or uncertainty since the inverse problem is ill-posed. More recently, superresolution was demonstrated under different imaging conditions, namely, scanning systems with single points or subwavelength areas emitting one at a time [9–11].
With the theoretical development of statistical signal and image processing tools, image restoration was cast as a formal parameter estimation problem [12,13]. Two-point resolution was defined as the minimum separation between two point sources that can be estimated from measurement of the optical intensity in the imaging plane with a predefined precision, under the assumption of a given probability model for the measurement noise. The Fisher information (and corresponding Cramér–Rao bound)  was used for this purpose , and it was shown that the precision of optimal estimators diminishes as the separation is reduced, a problem that has become known as Rayleigh’s curse. The realization that reducing uncertainty is a requirement for enhancing precision has also led to investigations of the use of nonclassical states of light, such as squeezed, entangled, and sub-Poisson quantum states [16–19].
The recent strong interest in quantum information science and the development of new versatile quantum tools have revived interest in the venerable subject of two-point resolution. Also, the earlier formulation of a general quantum estimation theory [20,21] has provided a theoretical foundation based on the concept of quantum Fisher information for establishing rigorous fundamental bounds on the precision of estimates of parameters of an optical source achievable by employing optimal quantum measurements on the optical field in the image plane. By applying these principles to the problem of estimation of the separation between two incoherent point sources, it was recently shown that the quantum Fisher information bound offers precision significantly greater than that afforded by direct intensity measurement. Moreover, the quantum bound remains constant as the separation diminishes, making it possible in principle to resolve infinitesimally small separations [22–25]. Consequently, it was implied that the optimal resolution is limitless, and hence it was declared that Rayleigh’s curse is not a fundamental obstacle to imaging.
With insights from quantum metrology, it was further shown that, together with appropriate measurements in the image plane, a specific simple processing scheme based on linear projections of the optical field onto an even and an odd spatial mode offers estimates of the two-point separation with precision approaching the ultimate quantum bound [22–29]. Experimental verification of these relatively simple schemes has confirmed that Rayleigh’s curse can be practically overcome.
Inherent in the formulation of the two-point separation quantum estimation problem is the assumption that the centroid of the two points is precisely known and their intensities are equal. If this is not the case, then the problem can be formulated as a multiparameter quantum estimation problem . It was shown, for example, that for point sources with unknown intensity ratio, the precision bound on estimates of the separation falls to zero as the separation diminishes . Nevertheless, enhancement of the precision can be gained by use of optimal quantum measurement. Other multiparameter estimation problems were considered in this context, including estimations of the Cartesian components of the separation as a vector in the source plane, and estimation of moments of the spatial distribution of an extended source [27,32].
In this paper, we expand the scope of this quantum estimation problem further by considering the effect of partial coherence on the two-point resolution. In the earlier years of the development of the theory of optical coherence, it was shown that both the magnitude and the phase of the degree of partial coherence of the source play important roles in determining the distribution of the image and its statistical properties [33,34]. It was found that, based on direct measurement of the image intensity, a greater degree of coherence corresponds to greater resolvability of pairs of point sources, i.e., greater accuracy of the binary decision on whether the illumination originates from one or two point sources. However, in the context of classical estimation of the separation between a pair of point sources, coherence has the opposite effect, namely, lowering the estimation precision. The question arises as to whether the quantum precision bound on estimates of the two-point separation is also lowered under conditions of partial coherence. We show here that this is indeed the case. This aspect of the resolution problem is important since the optical fields produced by two point sources are correlated when they share a common origin, e.g., when point scatterers are illuminated by a common extended source .
By formulating the quantum estimation problem as one of estimating two parameters—the separation and the degree of coherence of the two-point source—we show that the quantum bound on the precision of the separation estimator drops to zero at small separations, even for a very small, but nonzero, degree of coherence. This is remarkable since it means that the noted success of quantum measurements in breaking Rayleigh’s curse for incoherent sources is vulnerable to the smallest correlation between the two sources. Although the optimal quantum measurement offers some benefit over direct intensity measurement, even when an unknown degree of partial coherence is present, it ultimately falls victim to Rayleigh’s curse. It is evident that the limitation caused by partial coherence is more fundamental than the limitation imposed by direct image intensity detection.
2. CLASSICAL AND QUANTUM MODELS
A. Coherent Imaging
We first compare the imaging of two point sources for coherent classical light and pure-state single-photon light. Consider a coherent shift-invariant imaging system with a symmetric amplitude point spread function (PSF) that satisfies the condition . For a source comprised of two emitters located at in the object plane and having equal amplitudes and equal (in-phase) or opposite (out-of-phase) phases, the optical field in the image plane is the superposition
Now, consider another source generating a single photon in a pure quantum state:3) guarantees that , where 2) and (6) become identical if we assume that the total power of the classical source is unity, i.e., , the analogy between the classical (coherent) and the single-photon (pure state) cases is evident. It is important, however, to note the difference in the physical interpretation: for classical imaging, is the optical intensity, while for single-photon imaging, is the probability density function of the position at which the photon is detected.
B. Partially Coherent Imaging
We now generalize this paradigm to a partially coherent classical source and an analogous single-photon source in a mixed state. In the classical case, we assume that the amplitudes of the two emitters are random variables and so that the total optical field
An analogous quantum source is a single photon in a mixed state assumed to be a statistical combination of a coherent component—the pure state in Eq. (3)—and a maximally mixed state with density operator
For this quantum state, the probability density of detecting the photon at position is :11) is identical to Eq. (8) for a real-valued . In the limit , , and , corresponding to the coherent case for which , discussed before. In the other limit , , and , corresponding to the incoherent case , which was previously considered in the literature in the context of two-point resolution . Note that in our model, is real. The signs denote the in-phase and out-of-phase cases, which correspond to positive and negative degrees of coherence, hereafter called correlated and anti-correlated, respectively. A key assumption in both the classical model and the quantum, single-photon model treated here is that the optical power is fixed at the sources, and equal to that in the image plane. Physically, this means that there is an assumed known rate of emission of the point sources, and that all power generated by the point sources reaches and is measured in the image plane.
C. Two-Point Resolution
We are concerned here with the resolution of the system, viz., the accuracy of estimating the separation parameter , and the role of coherence in this process. For classical imaging, the optical intensity is typically measured and used to estimate . To assess the accuracy of such estimation, a model for the measurement noise is necessary. For an ideal detector, the accuracy is ultimately limited by the inherent Poisson noise (or shot noise) in the detection process, which depends on the intensity level. For quantum imaging with a single photon, the location of the photon in the image plane is measured, repeatedly, and the probability density is determined, from which the separation is estimated. Since is a probability density function depending on the unknown parameter , we may directly determine bounds on the estimation accuracy by calculating the classical Fisher information (CFI) and its inverse—the Cramér—Rao bound (CRB) .
When is known and we wish to estimate , the Cramér–Rao theory states that the variance of estimates based on measurement of a probability distribution is bounded by the inverse of the Fisher Information, , where36], the variance becomes bounded by . For our calculation, we assume that every photon is collected and measured perfectly.
As an example, we consider a Gaussian PSF with width , given by1 as a function of for several values of for the correlated and anti-correlated cases. Based on these plots, the following observations can be made. (i) For any fixed , the precision is a monotonic decreasing function of the coherence parameter so that the highest precision is attained in the incoherent case (). (ii) The correlated case corresponds to precision greater than that afforded by the anti-correlated case, for the same and . (iii) In all cases, the precision drops to 0 (i.e., the variance becomes infinite), as approaches 0, in which case finding an unbiased estimate of becomes impossible. This is Rayleigh’s curse.
This problem is further compounded when is not known since we must instead look at the multi-parameter CRB. To derive this bound, we determine elements of the classical Fisher information matrix (CFIM):2 as functions of for several values of in the correlated and anti-correlated cases. The following observations are noted: (i) is a monotonic increasing function of exhibiting Rayleigh’s curse, and (ii) greater is attained in the anti-correlated case.
These direct detection schemes, both classical and quantum, may be improved if the optical field in the image plane is processed by some prescribed system before the measurement of the optical intensity or the photon probability density is taken. Certain systems may offer enhanced resolution. As an example, it has been shown both theoretically and experimentally that projection onto a subset of spatial modes of the optical field offers measurements that are less susceptible to Rayleigh’s curse than direct imaging [23,25,28,37].
A third paradigm is to seek the best possible measurement system. Although it is not generally possible to find such system, the quantum Cramér–Rao bound (QCRB) always allows us to determine a bound on the precision for that optimal quantum measurement for a given quantum state . Such calculations have been made [22,26] for the incoherent system described by in Eq. (9). We determine these resolution bounds here for light in the partially coherent state described by the density operator in Eq. (10). As was the case for the calculation of the classical CRB, if is unknown, then we are faced with a multi-parameter estimation problem and its associated QCRB. This calculation proceeds in the next section.
3. QUANTUM FISHER INFORMATION RESOLUTION
To determine the ultimate precision that a measurement in the imaging plane can obtain, we calculate the multi-parameter quantum Fisher information matrices (QFIMs) corresponding to the states for the correlated (+) and anti-correlated (−) cases . For estimation of either the separation or the coherence parameter when neither is known, the elements of the QFIM, , are given by18) for the SLD operators and . As shown in Supplement 1, these operators have a commutator that has an expectation value of zero, so that 39,40]. However, since the off-diagonal terms and do not necessarily vanish, Eqs. (21) and (22) indicate that, although and are simultaneously estimatable, the lack of precise knowledge of one parameter will degrade the precision of the estimate of the other [30,41].Supplement 1, the four vectors , , , and form an orthonormal basis that spans the support of both and for both . When these operators, represented in this four-dimensional basis, are used in Eq. (18) the following expressions of the SLDs are obtained for : 17) and accounting for the zero matrix elements of and , we obtain the following expressions for elements of the QFIM: 21) and (22) to calculate the precisions and for any PSF .
We now use the Gaussian PSF described by Eq. (15) as an example to determine the dependence of and on the normalized separation and the coherence parameter . In this example, the parameters in Eqs. (35)–(39) are
Estimation of separation with known coherence parameter. If is perfectly known, i.e., is the only unknown parameter, then the variance of the estimate of is simply bounded by the QFIM element so that . This precision bound is plotted in Fig. 3 as a function of for several values of . In the limit , which corresponds to the incoherent case, has no functional dependence on , and hence the precision bound has a constant value extending to the limit , so that it is in principle always possible to make a measurement that gives an unbiased estimate with non-zero precision. This limiting result has recently led to the announcement that Rayleigh’s curse has been broken . The graphs in Fig. 3 show that for a source with positive degree of coherence, the precision bound at drops as (or ) increases, and ultimately vanishes when , resurrecting Rayleigh’s curse. It is interesting, however, that for a source with negative degree of coherence, with even the smallest magnitude, the curse is revived for any (or ).
Concurrent estimation of separation and coherence parameter. In this case, the dependence of the precision bounds and on and are shown in Fig. 4. Remarkably, for both the correlated and anti-correlated cases, as , for any so that if there is any degree of correlation between the point sources, no matter how small, Rayleigh’s curse resurges.
Comparison between direct imaging and quantum-optimal measurement. It is revealing to compare the precision afforded by estimation based on optimal field measurement together with quantum Fisher information to the precision bound based on direct intensity measurements together with CFI. This comparison is depicted in Fig. 5, showing that the optimal quantum measurement offers significant improvement in precision over direct intensity measurement, although neither estimator beats Rayleigh’s curse when .
Precision bounds on the coherence parameter. While the principal focus of this paper is on the precision of estimates of the separation when the coherence parameter is either known or estimated, an important byproduct of the analysis is bounds on the precision of estimates of when is either known or estimated. Estimating the degree of coherence can be useful in applications for which the correlation between two emitting or scattering sources is to be assessed. The results are displayed in Fig. 6 for estimates based on direct intensity measurement and optimal quantum measurement.
In all cases, higher values of are estimated with greater precision. Of crucial importance here is whether the sources are correlated or anti-correlated. The precision is always higher for anti-correlated sources and also has a stronger dependence on . This is to be expected for direct intensity measurement since close anti-correlated sources create an intensity distribution with a visible dip at the center, and the depth of the dip is greater for larger . No such dip exists for correlated sources, and if the separation is small, it is difficult to discern the effect of coherence, so that the precision is low and practically independent of . Also, for both optimal direct intensity measurement and optimal quantum measurement, the precision of estimates of drops to zero in the correlated case as the separation , while it remains constant and finite in the anti-correlated case. In both cases, however, the precision is orders of magnitude greater for optimal quantum measurement compared to optimal direct intensity measurement, and this is particularly so for the correlated case at small separations, where direct intensity measurement is not precise.
Conventional imaging systems rely on direct measurement of the image-plane optical intensity, which provides only a portion of the information about the object that is carried by the optical field. For such systems, the two-point resolution is limited by diffraction, which diminishes the precision of estimating two-point separation as the separation is reduced, and the system succumbs to Rayleigh’s curse. With optimal quantum measurement of the optical field, the curse is broken and the separation may be estimated with finite precision no matter how small the separation is. What we have shown in this paper is that this is true only when the emissions from the two points are completely uncorrelated, or incoherent. The introduction of any correlation, positive or negative, between the emissions has a detrimental effect on the precision of estimates of the separation, and this effect is particularly strong for small separations so that even optimal quantum measurements, which offer unsurpassable precision, fail to defeat Rayleigh’s curse. This effect is similar to ill-posed inverse problems for which solutions exist, but are highly sensitive to the slightest uncertainty in the measured data.
One reason for the resurgence of the curse in the presence of correlation may be attributed to the fact that the degree of coherence is a new unknown parameter that must be estimated jointly with the separation. But we have shown that even if this parameter is known a priori, if the correlation is negative, then the curse remains. The presence of known positive correlation does break the curse, however. It should be noted that these findings are applicable when the degree of partial coherence is real, positive, or negative. For a complex , the analysis is more involved, and it is possible that at certain values of , the curse is avoided, much like when .
Source correlations cannot be ignored, particularly at small separations, which is exactly the region for which Rayleigh’s curse is manifest. A Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength with positive correlation at small separations, and light gains transverse coherence as it propagates . Consequently, Rayleigh’s curse endures as a fundamental dictum.
See Supplement 1 for supporting content.
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