In the original Larson and Saleh paper [1], a quantum model together with a quantum Fisher information measure is used to determine the ultimate sensitivity of estimating the separation between two point sources under conditions of partial coherence. In their Comment [2], Tsang and Nair use a different model based on a suboptimal classical measurement strategy together with a classical Fisher information measure, and obtain greater sensitivity. This contradiction led them to argue that there must be a fundamental error in the quantum model presented in Ref. [1]. In this Reply, we show that the discrepancy is instead attributed to an inconsistency in the Tsang–Nair model itself. We also respond to other issues raised in the Comment.

The principal issue with the model described in the Comment has to do with normalization. Based on Eq. (1) of the Comment, the mean total number of photons in the image plane is $2N_0[1+\mathrm{Re}(\gamma d)]$, when the mean number of photons emitted by the two sources is $2N_0$. Here, $\gamma$ is the degree of coherence of the emission, and $d$ is the overlap of the left- and right-shifted point-spread functions of the imaging system, $h(x\pm s/2)$, which depends on the two-point separation $s$. This means that power is gained or lost depending on the factor $[1+\mathrm{Re}(\gamma d)]$. Clearly, this is inconsistent with a passive lossless imaging system represented by a unitary transformation.

The authors of the Comment incorrectly argue that the total power, or average photon number $ϵ$, in the image plane “can depend on the parameters because of interference.” This is of course true only for the optical intensity at any point, but not for the total power. This is potentially dubious when $\gamma =-1$ in the limit $s\to 0$, i.e., $d=1$, for which the factor $[1+\mathrm{Re}(\gamma d)]\to 0$ and the mean power drops to 0. The inconsistency disappears if $\gamma =0$, i.e., if the sources are incoherent, so that their earlier results [3] are valid.

Another problematic implication of the Tsang–Nair model is the dependence of the mean total number of photons in the image plane on the separation $s$, which is the very parameter to be estimated. If the estimation procedure is constrained by a fixed mean power in the image plane, then their model suggests that the optical sources emit at a rate that depends on the emitter’s own proximity (and also on their degree of partial coherence)—an unviable supposition. With fixed total optical power, $ϵ=2N_0$ and the second term in Eq. (7) of the Comment is necessarily zero. Thus, the assertion that the calculations in Ref. [1] underestimated the Fisher information predictions, is false, and the model in the Comment must be appropriately corrected.

It is ironic, however, that the expression for the density operator of the weak thermal model in the Comment, which is written as a weighted sum $\rho =(1-ϵ){\rho }_0+ϵ{\rho }_1$ of a vacuum state ${\rho }_0$ and a single-photon state ${\rho }_1$, is in fact consistent with, and requires, appropriate single-photon normalization. This is because conservation of probability requires that $\mathrm{Tr}(\rho )=1$. Given $\mathrm{Tr}({\rho }_0)=1$, it is hence implied that $\mathrm{Tr}({\rho }_1)=1$. Surprisingly, the authors of the Comment are critical of the fixed-power constraint used in our model [1].

Our model [1], based on a fixed equal power in both the source and image planes, namely, a single photon, is indeed self-consistently normalized, as it should be.

The second issue raised in the Comment brings up the salient point that one can instead parameterize the estimation problem in terms of the complex degree of coherence $\gamma$, rather than the parameter $p$, as was done in Ref. [1]. Since there is a one-to-one correspondence between these two parameters, i.e., every non-zero value of $\gamma$ corresponds to a unique non-zero value of $p$ (see Fig. 2 of the Comment), the results are expected to be similar. In Ref. [1], we elected to use the parameter $p$ because it represents the simplest perturbation to the original state describing a single photon in a mixed quantum state. Nevertheless, because $\gamma$ and $p$ are nonlinearly related, it is indeed interesting to reformulate the multiparameter estimation problem based on the correct model in Ref. [1], with $\gamma$ as a parameter. This is accomplished by rewriting $\rho$ in Eq. (10) of [1] using Eq. (13) of [1], and calculating the quantum Fisher information matrix (QFIM) in terms of $s$ and $\gamma$, rather than $s$ and $p$. The results are shown in Fig. 1.

 

Fig. 1. Precision of estimates of the separation $s$ based on calculation of the quantum Fisher information assuming an exactly known degree of coherence $\gamma$. Results are plotted for negative $\gamma$ (solid) and positive $\gamma$ (dashed).

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Fig. 2. Precision of estimates of the separation $s$ based on the QFIM, assuming concurrent estimation of an unknown degree of coherence $\gamma$ for $\gamma =1,2/3,1/3,0,-1/3,-2/3$, and 1. In all cases, the sensitivity approaches zero as the separation $s$ approaches zero (Rayleigh’s curse).

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As shown in Fig. 1, Rayleigh’s curse is indeed avoided for $-1<\gamma <1$, as it is in the Tsang–Nair model. However, our model predicts greater sensitivity, particularly for negative $\gamma$. In the limiting case of $\gamma =+1$, the curse resurges, as it also does in the Tsang–Nair model. The other limiting case, $\gamma =-1$, exhibits anomalous behavior for which the high sensitivity at large negative $\gamma$ drops sharply as $\gamma \to -1$, and the curse emerges. This result is contrary to the Tsang–Nair model, which predicts high sensitivity at $\gamma =-1$. It can be shown that with appropriate normalization, the Tsang–Nair model leads to zero sensitivity for $\gamma =-1$ at $s=0$, contrary to Fig. 1 of the Comment.

It is essential to point out that the foregoing conclusions are valid under the assumption that the parameter $\gamma$ is known precisely. If $\gamma$ is unknown and is estimated concurrently with $s$, then the optimal sensitivity drops to zero as $s$ approaches zero, so that Rayleigh’s curse is exhibited for all values of $\gamma$, as shown in Fig. 2. Unsurprisingly, the results are nearly identical to those in Ref. [1], which use $p$ as a parameter.

In many of the multiparameter treatments of this estimation procedure that have been studied since the early conception of the problem [3], the introduction of a second parameter that is not precisely known causes a resurgence of Rayleigh’s curse. As an example, Rehacek et al. [4] find that if the incoherent sources are allowed to have an unequal emission ratio, and the ratios are not exactly known, then the sensitivity drops to zero as $s$ approaches zero for every unequal emission ratio. The emphasis being that exact knowledge of each newly introduced parameter is not practical and that the multiparameter treatment must be used to model real estimation problems.

Finally, we disagree with the statement in the Comment that “spatial coherence of the sources is unlikely to be significant in key applications of incoherent imaging.” As amply discussed in many textbooks (see, e.g., [5,6]), spatial coherence plays a key role in the design of optical imaging systems even with spatially incoherent illumination sources, since the optical field in the object plane becomes partially coherent.