1. INTRODUCTION

Estimation and detection theory, formulated originally as a useful tool for signal analysis and efficient parameter estimation, became indispensable in quantum information processing, where the effects are subtle, signals are sparse, and any wasting of information is detrimental. However, these well-established techniques can be used even in classical detection schemes, with robust signals pushing the resolution to ultimate limits that have not yet been fully explored.

Recent research pioneered by Tsang and collaborators, and inspired by a reconsideration of the classical Rayleigh criterion for the resolution of optical instruments such as telescopes or microscopes, has received considerable attention in the optical community (see [1] and references therein). The problem can be paraphrased: How well can we distinguish two bright spots? The celebrated Rayleigh arguments suggest that it can be done up to the distances when two blurred spots start to overlap. This rule of thumb can be justified by an analysis of the Fisher information for the intensity pattern, and one finds that the Fisher information vanishes for zero separation. As shown by Tsang and coworkers [2] and demonstrated experimentally [3], this behavior can be avoided if quantum estimation theory is adopted for the estimation of geometrical parameters, namely the transversal separation and the centroid positions of two equally bright spots with known intensities. In this context, the Fisher information refers to quantum measurements and becomes the quantum Fisher information (QFI) upon optimizing over all thinkable measurement schemes.

As shown in [4], however, the model used in [2] is not robust with respect to the inclusion of other parameters. When the intensities of the bright spots are considered as estimated parameters, together with the separation and the centroid, the QFI remains constant only if the two intensities are equal, but it drops to zero for unequal intensities. The unphysical situation of exactly equal intensities is singular and exhibits anomalous features.

The ongoing research on the estimation of optical effects also addresses the possible coherence of optical signals, and a recent discussion did not reach a consensus [5–7]. Whereas the paper [5] claims that the presence of coherence yields a QFI that vanishes for zero separation, the comment [6] shows by explicit calculations that this result need not be so. The argument somehow paradoxically sticks to Rayleigh’s reasoning for incoherent image processing instead of applying the Sparrow resolution limit [8] and its modifications [9–11], which is the appropriate tool for quantifying the performance of (partially) coherent systems. According to the Sparrow criterion, two point sources can just be resolved when the second derivative of the image intensity vanishes at the point midway between the overlapping images of the two points. Particularly remarkable is the argumentation in favor of using an “anti-phase” superposition [10]: “Since the amplitude impulse response is an even function, zero intensity results at the mid-point between the two images whatever is the value of the separation. This suggests that, under ideal conditions, infinite resolution is approached.”

In this Letter, we explain the reasons for these misunderstandings on the basis of simple physical arguments and explicit calculations for an elementary model of a coherent superposition. Our central observation is quite simple: When coherence effects are taken into account, the Fisher information itself is no longer a meaningful measure of accuracy because the channels exhibiting interference are not equivalent with respect to the strength of the signal. Indeed, the (quantum) Fisher information $F$ quantifies the content of information per registered particle; the Crámer–Rao inequality (here for a single parameter $\theta$),

We recall that the Crámer–Rao bound on the precision in (1) is subject to two specific assumptions: (i) The estimator is unbiased; and (ii) the detection events are uncorrelated; they are independent and identically distributed (i.i.d.) random events. As a consequence of assumption (i) we have the unit numerator, while assumption (ii) is crucial for the product $nF$ in the denominator—the single-event Fisher information is multiplied by the number of detection events. One needs to verify that both assumptions are true in the situation of interest. Further, estimation is always model-dependent; therefore, one must check the ingredients of the model that is used. We take for granted that all these verifications have been done.

2. METHODS AND RESULTS

Let us now elaborate on the argumentation for an ideal equal-weight superposition of symmetrically displaced sources. We phrase what follows in a quantum parlance, so a wave of complex amplitude $U(x)$ can be assigned to a ket $|U⟩$, such that $U(x)=⟨x|U⟩$, where $⟨x|$ is the bra for a point-like source at $x$. The quantum formulation (using these bra and ket symbols) facilitates the optimization since the intensity detection (and the corresponding complex amplitudes) need not represent an optimal scheme. More specifically, we denote by $\mathrm{\Psi }(x)=⟨x|\mathrm{\Psi }⟩$ the amplitude of the (generic) point-spread function (PSF) of the coherent spatially invariant imaging system. The coherence matrix relevant for the discussion is

It is important to note that we are not dealing with a genuine quantum problem. We are using the quantum formalism for classical optics. The PSF amplitude $\mathrm{\Psi }(x)$ is not a probability amplitude but a classical quantity, such as a component of the electric field. Therefore, $\mathrm{\Psi }(x)$ is real, and the corresponding distribution for $P$ is even, so that $⟨f(P)⟩=⟨\mathrm{\Psi }|f(P)|\mathrm{\Psi }⟩=⟨f(-P)⟩$ for all functions of $P$. In particular, then, $⟨P⟩=0$ and there is no difference between $⟨P^2⟩$ and the momentum variance of the PSF, ${(\mathrm{\Delta }P)}^2=⟨P^2⟩-{⟨P⟩}^2$.

The QFI for the parameter $s$ can be calculated from the rank-1 expression $F=2\mathrm{Tr}[{({\partial }_s\rho )}^2]$, which yields

To understand the role of coherence, the moment expansion for a small displacement $s\to 0$ is essential. A complication arises since $|\mathrm{\Phi }⟩=0$ when both $s=0$ and $\phi =\pi$, so that $\rho$ is ill-defined in this limiting situation of destructive interference at vanishing separation, and it matters whether the limit $\phi \to \pi$ succeeds or precedes the limit $s\to 0$. For $s=0$ and $\phi \ne \pi$, one obtains

For $\phi =\pi$, we have

 

Fig. 1. Dependence of the QFI on the displacement $s$ for the phase values $\phi =0$, $\frac{1}{4}\pi$, $\frac{1}{2}\pi$, $\frac{3}{4}\pi$, and $\pi$. The QFI diverges for $\phi \to \pi$. The plot is for a Gaussian PSF with the variances ${(\mathrm{\Delta }X)}^2={\sigma }^2$ and ${(\mathrm{\Delta }P)}^2=1/(4{\sigma }^2)$. The displacement is in units of $\sigma$, and $F$ in units of ${\sigma }^{-2}$.

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It is amusing to note that the QFI for the coherent superposition with $\phi =\pi /2$ equals exactly the limit of incoherent mixtures. This illustrates nicely Goodman’s observation [12] that “when (coherent) sources are in quadrature, the image intensity distribution is identical to that resulting from incoherent point sources.”

3. DISCUSSION

Understanding the behavior of the QFI for $\phi =\pi$ is essential for the correct interpretation of the role of coherence in estimation problems. The QFI clearly exhibits a singularity when $s\to 0$ in this situation of destructive interference. Physically speaking, we are detecting a signal on a dark fringe, where the intensity is extremely low. If the norm $N=⟨\mathrm{\Phi }|\mathrm{\Phi }⟩$ is taken as a weighting factor into the definition of the precision $H$ in (1), the singularity disappears. There are plausible and sound physical arguments for the inclusion of such a weight: constructive and destructive coherence is always manifested by an enhancement or a suppression of the emerging signal. This represents a valuable resource that should be taken into account. This argument can be supported by an exact calculation of the cost of preparing the superposition in (2). The analysis can be linked to state-of the art technology [13] for the deterministic generation of these superpositions.

We generate the superposition in (2) from an entangled state,

 

Fig. 2. Dependence of the total QFI on the displacement $s$ for both constructive and destructive interference channels detected independently for phases $\phi =0$, $\frac{1}{4}\pi$, and $\frac{1}{2}\pi$. This properly normalized QFI is always limited by its value for an incoherent superposition. Note that the upper bound is saturated for large displacements, and also for zero displacement for all the phases except when $\phi =0$ or $\pi$. The plot is for a Gaussian PSF with variances $(\mathrm{\Delta }x)={\sigma }^2$ and ${(\mathrm{\Delta }P)}^2=1/(4{\sigma }^2)$.

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In view of this argumentation, it is clear that the coherence, although the QFI may diverge for one sub-ensemble, does not provide any improvement over an incoherent source, if the cost of generating such a signal is properly taken into account. Nothing is gained by an increase of the factor $F$ in the product $nF$ in (1) if the value of $n$ decreases even more.

Other sorting schemes than scheme (S) can also be realized with the option of having situations intermediate between the fully coherent and the completely mixed sub-ensembles. In the case of partial coherence, the explicit form of the partially coherent state matters for the QFI of this sub-ensemble. The properly weighted total QFI for any sorting scheme, however, cannot exceed the upper bound set by $F_{\mathrm{ent}}=F_{\mathrm{inc}}$. This observation supports the arguments used in [5], and we believe settles the discussion in [5–7].

For simplicity, the discussion above deals with the estimation of a single parameter, the separation $s$, which is sufficient for demonstrating the case; namely, that the sub-ensembles carry weights, and these weights enter the total QFI in (9). When the data from an actual experiment are evaluated, however, the multiparameter situation of asymmetrically displaced sources, with unequal intensity and partial coherence, matters. Then, the sub-ensembles are not specified by pure states like those in (8), but by rank-2 states of the generic form,

Finally, concerning the so-called “Rayleigh curse,” a term coined in [2], where the value of the QFI at vanishing separation is by itself regarded as a significant measure for distinguishability, and $F(0)=0$ is the poor “classical” resolution (yes curse) while $F(0)>0$ is the superior “quantum” resolution (no curse), we observe a few points. First, the estimator for the displacement $s$ usually exhibits a substantial bias when $s$ is small, and then the Crámer–Rao bound of (1) does not apply without the necessary modification. Second, the product $nF$ is relevant in (1), not just the Fisher information, and nothing is gained by an increase of $F$ if it is compensated for by a decrease of $n$; while an individual QFI in the sum in (9) can easily exceed $F_{\mathrm{inc}}$, the properly weighted sum cannot.

4. CONCLUDING REMARKS

The simple model studied here is sufficient to make the point that the QFI is but one ingredient and that there is no genuine advantage of coherent over incoherent sources when all aspects are considered. The model is good enough to explain the discrepancies in the analysis of coherent effects in [5–7]. The model, however, has obvious limitations in that only one parameter is considered (the separation). Therefore, any analysis of a realistic situation must deal with at least two more parameters; namely, the centroid position and the relative intensity of the two sources. While there could be more parameters of relevance, such as the degree of coherence, certainly these three must be estimated jointly from the data. A realistic analysis must also pay close attention to how the parameters are estimated from the data; the biases and the mean-square errors (or any other measure of accuracy) of the estimators actually used matter in practice, not the Crámer–Rao bound for optimal unbiased estimators. Clearly, much more work is needed before the community can reach a definite conclusion about the benefits of coherent sources or coherent procedures for data acquisition for the resolution of optical instruments.