Attaining quantum limited precision of localizing an object in passive imaging

Aqil Sajjad,1, ∗ Michael R Grace,1, † Quntao Zhuang,2, 1 and Saikat Guha1, 2, ‡

1James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 857212Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721

We investigate our ability to determine the mean position, or centroid, of a linear array of equally-bright incoherent point sources of light, whose continuum limit is the problem of estimating thecenter of a uniformly-radiating object. We consider two receivers: an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) Spatial-mode Demultiplexing(SPADE) in the image plane, prior to shot-noise-limited photon detection. We compare the FisherInformation (FI) for estimating the centroid achieved by these two receivers, which quantifies theinformation-accrual rate per photon, and compare those with the Quantum Fisher Information(QFI): the maximum attainable FI by any choice of measurement on the collected light allowed byphysics. We find that focal-plane direct imaging is strictly sub-optimal, although not by a largemargin. We also find that the HG mode sorter, which is the optimal measurement for estimatingthe separation between point sources (or the length of a line object) is not only suboptimal, but itperforms worse than direct imaging. We study the scaling behavior of the QFI and direct imaging’sFI for a continuous, uniformly-bright object in terms of its length, and find that both are inverselyproportional to the object’s length when it is sufficiently larger than the Rayleigh length. Finally, wepropose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.

I. INTRODUCTION

Rayleigh’s criterion for the resolution of two inco-herent light sources [1] remains one of the most impor-tant results in optical imaging. Based on diffractioneffects, it tells us that we cannot resolve two objectswhose angular separation is less than λ/D, where λ isthe wavelength and D is the size of the receiver’s aper-ture. This relies on a somewhat heuristic argument.A more rigorous estimate for the maximum estima-tion precision for the separation between the twopoint-like sources, when imaged by an ideal direct-detection focal plane array, can be obtained from theclassical Cramer-Rao bound [2], expressed in termsof the Fisher Information (FI). This analysis stillshows that the FI, whose inverse gives a lower boundon the variance of any unbiased estimator, and there-fore serves as a measure for precision, sharply dropsand approaches zero as the separation between thetwo sources falls below λ/D, approaching zero. Inqualitative terms, both Rayleigh’s criterion and theCramer-Rao bound are essentially telling us the samecommon sense thing: if two objects are too close toeach other, then it is hard to tell them apart, andhence determine their separation, if their images—blurred by the point-spread function (PSF) of theaperture—overlap so much that it is hard to tell if

∗ aqilsajjad@optics.arizona.edu† michaelgrace@email.arizona.edu‡ saikat@arizona.edu

the image is that of a single point source or that oftwo closely-spaced sources. This also translates intoour inability to resolve any features of the objectthat are too small compared to λ/D. Obtaining rele-vant information about objects in the sub-Rayleighregime has therefore been a major topic of interest fora wide variety of fields ranging from astronomy [3, 4]to biological imaging [5, 6].

A. Super-resolution imaging usingpre-detection mode sorting

Recent findings based on quantum estimation the-ory show that it is possible to build new imaging de-vices that surpass Rayleigh’s limit. One useful tool inquantum estimation theory is the quantum Cramer-Rao bound (QCRB), introduced by Helstrom [7], ex-pressed in terms of the quantum Fisher information(QFI): the QFI is an upper bound to the maximumFI attainable with any physically-allowed measure-ment scheme. Thus, if we find that the FI for a givenmeasurement (the physical device that detects theinformation bearing light producing an electrical sig-nal) is equal to the QFI, then we know that the saidmeasurement is optimal, and no other measurementin its place will generate an estimate of the param-eter of interest with a lower variance. The inversestatement also holds for estimating a single scalarparameter: if we find that there is a gap betweenthe QFI, and the FI of a specific measurement, thenit means that we can do better by employing some

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other measurement whose FI is equal to the QFI.

It turns out that for estimating the separation be-tween two incoherent point sources, the QFI remainsconstant instead of shrinking to zero as the separationreduces to zero [8], proving thereby that Rayleigh’scriterion—in its commonly stated form—is an arti-fact of direct, i.e., intensity, detection in the imageplane (corrupted by the fundamental Poisson shotnoise), which discards valuable information in thephase of the field. For the case of a Gaussian PSF,Tsang et al. showed that the QFI-attaining (opti-mal) measurement can be realized by an image-planeHermite-Gaussian (HG) spatial-mode demultiplexer(SPADE), followed by shot-noise-limited photon de-tection on those sorted modes. Along the same lines,Kerviche et al. [9] and Rehacek et al. [10] indepen-dently showed that the optimal measurement for ahard aperture pupil (sinc-function PSF) is realizablewith an image-plane sinc-Bessel SPADE followed byphoton detection. Moreover, for the same problemwith an arbitrary aperture function, both these workspropose measuring in a basis comprising the PSF andits orthogonalized derivatives, with the latter prov-ing that such a measurement attains the quantumoptimal performance provided the PSF is an evenfunction. In [11], Dutton et al. generalized the workof Tsang et al. to the case of estimating the angu-lar extent of M > 1 equidistant equally-bright pointsources. In the limit where the number of points goesto infinity, this equates to the problem of estimatingthe length of a continuous line-shaped object withuniform brightness. It was shown that an image-plane HG SPADE is again the optimal measurementfor estimating the length of such a uniform-brightnessline object [11]. In [12], the results of Tsang et al.were extended to 2-dimensions, showing that the 2-dHG basis is quantum optimal. A 3-dimensional gen-eralization was the subject of [13] where the so-calledZernike basis functions were shown to attain the QFI.

These mode sorters, however, need to be pointedexactly at the centroid of the incoherent point sourcesin order to obtain an FI for estimating the separationthat equals the QFI. In fact, Tsang et al. showedthat even a somewhat small misalignment of theSPADE can result in a large drop in its performance,especially in the regime where the separation is muchsmaller than the Rayleigh limit. In this limit, itis even possible to simultaneously estimate the sep-aration and centroid of two (not necessarily equalbrightness) point sources optimally; however, such ameasurement likewise needs to be spatially alignedwith respect to the intensity-weighted centroid of thesources in order to avoid a significant loss of perfor-

mance [14]. This means that if the centroid is notperfectly known a priori, we first need to estimateit before we carry out the SPADE measurement todetermine the separation. Based on this intuition,a two-stage optimization scheme was proposed byGrace et al. in [15]. This receiver first estimatesthe centroid—a nuisance parameter—using direct de-tection, and once a good enough estimate has beenobtained, the system then switches to the SPADEfor finding the separation. In addition to the abovemotivation for estimating the centroid first when weare mainly interested in estimating the separation,there is also the fact that finding the location of anobject is an important problem in its own right. Wecan for instance be interested in locating a knownobject whose size and features we already know.

All the above-mentioned works on estimating theseparation generally assume that the centroid canbe determined fairly accurately from direct imaging.This intuition in part stems from the fact that Hel-strom showed that for a single point source, idealimage-plane direct detection is the optimal measure-ment to estimate its position [7]. Moreover, Tsang etal. showed that the Fisher information for the cen-troid for two equally bright sources, while less thanthe QFI, is not suboptimal by a very large difference.

B. Main results

Since SPADE-like measurements are so sensitiveto misalignment, even small improvements in esti-mating the centroid can be beneficial. With that inmind, in this paper, we present a thorough study ofthe optimal measurement to estimate the centroid fortwo or more equally bright incoherent point sources,assuming a Gaussian PSF, including the case of an in-finite number of equally-bright equally-spaced pointsources in a straight line of sub-Rayleigh length. Theinfinite case is naturally of special interest because wewant to be able to estimate the position of continuumobjects, such as localizing a star or planet (in astro-nomical imaging) or localizing a cellular structure(in biological imaging).

We investigate the performance of both direct imag-ing and the HG SPADE for centroid estimation. Theinterest in the latter arises from it being the optimalmeasure for the separation, and it is therefore worthstudying whether it can also be useful for finding thecentroid. This is also of interest from the perspec-tive of a two-stage detection scheme such as the oneproposed in [15], where first the centroid would bedetermined using direct imaging or some other more

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optimal measurement, and once a reasonable esti-mate has been made for it, the device would switchautomatically to the HG SPADE measurement forfinding the separation. We show that for two or moresources with equal separation, direct imaging is infact not an optimal measurement for estimating thecentroid. We also find that the HG SPADE is moresub-optimal than direct imaging. Moreover, the bet-ter the HG SPADE is aligned with the centroid, theworse its ability to determine the centroid.

Finally, we present a two-stage adaptive modalmeasurement strategy that achieves the QFI for esti-mating the centroid of a constellation of n equally-bright equally-spaced point sources. The strategy wepresent applies to finding the QFI-attaining receivermeasurement for any n-point constellation.

C. Organization of the paper

In section 2, we introduce the overall set up forcentroid estimation and the underlying assumptionsof the model, give a brief overview of the classical andquantum Cramer-Rao bounds, and summarize thekey findings of [8, 11] that are most relevant for ourstudy of centroid estimation. In section 3, we presentour results on centroid estimation for different num-bers of equi-distant, uniformly bright emitters placedin a single line, including the infinite case of a uni-formly bright object. We compare the performanceof direct imaging and the HG SPADE with the QFI,and show that direct imaging is sub-optimal for cen-troid estimation, but not by a substantial amount.We then go on to discuss the performance of the HGSPADE, and show that it is mostly worse than directimaging for locating an object, even though it givesus the QFI-attaining measurement for finding the sizeor end-to-end diameter. Finally, we present a two-stage measurement scheme that attains the QFI forcentroid estimation in the limit of large integrationtime. In section 4, we present our conclusions. Wealso include several appendices at the end of the pa-per, proving important results and explaining knowncalculation methods but with additional details andclarifications that have been generally skipped in theliterature and may be helpful for the reader.

II. THE PHYSICAL SET UP

For the set up and our basic assumptions aboutthe physics, we closely follow the framework laid outin [8], except that we generalize it to more than two

incoherently-radiating point sources. For simplicity,we assume that the object and image planes areone-dimensional with unit magnification and thatour light sources emit nearly monochromatic lightwith paraxial waves [1]. We also make the standardassumption—valid for optical-frequency radiation—that the average number of photons ε per temporalmode arriving at the image plane is much less thanone, requiring many photons to be detected overa large number of temporal modes, to extract anyuseful information [16–20].

A. Quantum model for imaging scene made upof incoherently radiating point emitters

Let λ be the center wavelength, W (measured inHz) the spectral bandwidth of the collected light(around λ), and T (measured in seconds) the inte-gration time. In that time-bandwidth window, thereare roughly M ≈WT mutually-orthogonal temporalmodes. We take N = Mε to be the mean numberof photons received over the integration time, whereε is the mean number of photons collected per tem-poral mode. We now write the density operator ofthe photon field in a single temporal mode over theinfinite number of mutually-orthogonal spatial modesspanning the receiver telescope’s entrance pupil’s spa-tial extent. In the (conventional) image plane, thisdensity operator can be written as:

ρ = (1− ε)ρ0 + ερ1 +O(ε2), (1)

where ρ0 = |0〉〈0| is the zero-photon or “vacuum”state and ρ1 is a single-photon quantum state, bothof a single temporal mode over some infinite-basis ofspatial modes. O(ε2) denotes higher order terms inε, which we will ignore, since at visible frequencies,ε� 1. The quantum state of all the collected lightduring the integration is given by ρ⊗M .

The one-photon state ρ1 is a mixed state: anincoherent mixture of states |ψs〉, a pure state ofone photon—spread over an infinite basis of spatialmodes—of the image plane field, corresponding to thes-th point source making up the overall scene. For ascene comprised of n equally bright point sources,

ρ1 =1

n

n∑s=1

|ψs〉〈ψs|, (2)

with,

|ψs〉 =

∫ ∞−∞

dxψ(x− xs)|x〉, (3)

4

where ψ(x) is our coherent PSF, xs is the positionof the s-th point source, and |x〉 = a†(x)|0〉 is the(un-physical) state of one photon localized exactlyat the spatial point x in the image plane, where theannihilation and creation operators obey the delta-function commutator:

[a(x), a†(x′)

]= δ(x−x′). We

will consider a Gaussian PSF:

ψ(x) =1

(2πσ2)1/4exp

[−x2/(4σ2)

], (4)

where σ = 1/(2∆k) = λ/(2πNA), with ∆k2 ≡∫∞−∞[∂ψ(x)/∂x]2dx, λ the center wavelength, and

NA the effective numerical aperture.

Now, let us consider the model for ideal directimaging in the traditional image plane: an infinite-size continuum active surface (i.e., infinitely smalldetector pixels with unity fill factor), where eachof those pixels is a unity quantum efficiency shot-noise-limited photon-number-resolving detector withan infinite bandwidth, no read noise, and no deadtime. This ideal continuum detector array generates aspatio-temporal photo-current process that—for theaforesaid model of collected light—is characterizedby a space-time Poisson point process with a rategiven by the squared-magnitude photon-unit field inthe image plane. We remind the reader that, perour model, for each of the M temporal modes, atmost one photon can be detected, since ε� 1. Giventhere is a photon in a particular temporal mode, thespatial probability density of its detection is givenby:

Λ(x) =1

n

n∑s=1

|〈x|ψs〉|2 =1

n

n∑s=1

|ψ(x− xs)|2. (5)

Therefore, over a single temporal mode, the expecta-tion value of the number of photons being detectedin a region of width dx around x is then given by aPoisson distribution with the mean of εΛ(x) [18, 21–23]. Over M temporal modes, the average numberof photons detected (over the entire detector array)becomes N = Mε, with an average of NΛ(x)dx pho-tons in a region of width dx around x in the imageplane.

In general, if we instead measure the received op-tical field by some other receiver (e.g., homodynedetection), which is associated with an observable

Y, then the probability distribution of measurementoutcomes is given by P (Y) = 〈Y|ρ1|Y〉, where Y ∈ Ris a particular value of the observable and |Y〉 is theeigenket associated with that value.

B. Quantum Fisher Information and theCramer-Rao Bound

Let us say we are presented with N copies of aquantum state, i.e., ρ⊗N , and we wish to estimate aset of parameters {θµ} embedded in ρ by measuring

Y on each copy of ρ. In other words, we have theclassical estimation theory problem, wherein we wishto estimate parameters {θµ} embedded in a randomvariable Y , by N i.i.d. samples of Y , each drawn fromthe distribution P (Y) = 〈Y|ρ|Y〉. Consider a set of

estimators θµ(Y) and the error covariance matrix:

Σµν ≡∫dYP (Y)

[θµ(Y)− θµ

] [θν(Y)− θν

]. (6)

If θµ(Y) is an unbiased estimator, then it obeys theCramer-Rao bound on the covariance matrix

Σ ≥ J−1, (7)

where for N measurements,

Jµν ≡ N∫dY 1

P (Y)

∂P (Y)

∂θµ

∂P (Y)

∂θν(8)

is the Fisher Information matrix [2] associated withthis specific chosen receiver measurement. Moreover,if θµ(Y) is the maximum likelihood estimator, thenwe saturate the inequality in (7) for large N . TheFI thus quantifies the performance of a measurementin determining the parameter we are interested in.Therefore, ideally, we want to choose a measurementthat maximizes the FI.

The quantum Cramer-Rao bound gives us the max-imum possible FI that any physically-permissiblemeasurement scheme could achieve. In other words,

Σ ≥ J−1 ≥ K−1, (9)

where K is the quantum Fisher information (QFI)matrix [7]. For ρ⊗N encoding parameters of interest{θµ}, the QFI matrix is given by:

Kµν(ρ⊗N ) ≡ N tr (ρ{Lµ(ρ), Lν(ρ)}) , (10)

where Lµ(ρ) is the symmetric logarithmic derivative(SLD) of ρ with respect to the parameter θµ. It is aHermitian operator that is defined by the relation:

∂ρ

∂θµ=

1

2(ρLµ(ρ) + Lµ(ρ) ρ) . (11)

If ρ =∑j Dj |ej〉〈ej | is the decomposition of ρ in

terms of its eigenvalues Dj and eigenvectors |ej〉,

5

then the SLD is given by:

Lµ(ρ) =∑

j,k;Dj+Dk 6=0

2

Dj +Dk〈ej |

∂ρ

∂θµ|ek〉|ej〉〈ek|.

(12)Note that the QFI matrix does not depend on a par-ticular choice of measurement, but is a property of thequantum state ρ. If the FI for a chosen measurementscheme is equal to the QFI, then we know that it isthe best possible way to estimate the parameter(s)of interest.

C. Estimating geometrical parameters of alinear point source constellation

Where possible, it is convenient to work in termsof parameters that give a diagonal QFI and FI. Forour physical system of a linear constellation of equi-distant uniformly bright light sources, it turns outthat the QFI and the FI for direct imaging are bothdiagonal in terms of the centroid

θ1 =

∑ns=1 xsn

, (13)

and the separation between the first and last pointsource

θ2 = xn − x1. (14)

In terms of these two parameters, the individualpositions of the point sources are given as:

xs = θ1 −θ2

2+

(s− 1)θ2

n− 1, 1 ≤ s ≤ n. (15)

This diagonality of the QFI and the direct imaging FIin terms of θ1 and θ2 arises from the symmetry of ourphysical set up around the centroid, and we prove thisin Appendix A. It is also worth noting that due to thephysical symmetry around the centroid, the directimaging FI and QFI matrices will be independent ofθ1.

Tsang et al. studied the problem of estimating theseparation between two equally bright point sources,i.e., n = 2 in the above notation, assuming the cen-troid θ1 is known apriori [8]. The Fisher informationfor estimating the separation using direct imagingapproaches zero as the separation goes to zero (seeFig. 1). This is a manifestation of the so-called“Rayleigh’s curse”, since the two sources become un-resolvable when they are very close to each other, astheir image-plane fields have a width comparable to

FIG. 1. QFI (solid) and direct-imaging FI (dashed) forestimating the separation θ2 between n point sources,plotted as a function of θ2. We show two sets of plots:one for n = 2 point sources, and the other correspondingto n→ ∞, which corresponds to a continuous line-shapedobject. A Gaussian PSF with width σ is assumed.

their separation. However, Tsang et al. showed thatthe QFI for estimating θ2 is a non-zero constant evenwhen the separation approaches zero: the same con-stant the direct-imaging FI approaches when θ2 →∞(see Fig. 1). This means that the so-called Rayleigh’scurse is only an artifact of direct imaging rather thanbeing a fundamental limit imposed by the PSF ofthe imaging system. They go on to show that if wecarry out a measurement of the image-plane fieldusing a Hermite Gaussian (HG) basis spatial-modedemultiplexer (SPADE) that is aligned perfectly withthe centroid, and assuming that the centroid itselfis perfectly known apriori, the FI attained by thismeasurement for estimating the separation θ2 equalsthe QFI, and hence being a quantum optimal mea-surement scheme for estimating the separation.

These results were generalized by Dutton et al. tothe problem of finding the end-to-end separation foran arbitrary number of point sources (n ≥ 2) in [11],including the n → ∞ case of a line-shaped objectof length θ2. They again found that the QFI doesnot fall to zero even as the separation becomes small,and that an HG SPADE aligned with the centroidattains the QFI.

We show these results in Figure 1, where we repro-duce the plots reported in Tsang et al. for the QFIand the FI for direct imaging for finding the separa-tion between 2 incoherent emitters [8]. We also showthe same quantities for the continuous line source,reproducing the results of Dutton et al. [11]. The

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QFI curves in both cases also represent the FI for theHG SPADE, since it attains the QFI. Additionally,we note that a “binary SPADE” measurement—inwhich only the zeroth (or the first) image-plane HGmode is detected, by separating it from the rest of thelight (which is also detected using a bucket detector)—attains the QFI in the limit of θ2 → 0. This wasshown for n = 2 in [8, 9] and for n ≥ 2 in [11].

It is however important to emphasize that all theseresults are strongly contingent on the alignment ofthe SPADE with the centroid. In fact, in the limitwhere θ2 → 0 and θ1 � θ2, where the SPADE isaligned to the position x = 0, the FI for the HGSPADE drops all the way from attaining the QFIto being 0. This is for instance discussed in theAppendix D of [8] as well as in [15].

Grace et al. studied the performance of a binary-SPADE measurement to estimate the separation oftwo point sources, when their centroid is not knownapriori [15]. They consider a two-stage adaptive re-ceiver, where image-plane direct imaging is employedin the first segment of the optical integration timeto obtain a estimate of the centroid, and the receiverthen dynamically switches over to a second stagewhere a binary HG SPADE is employed whose centeris aligned with respect of the (noisy) estimate of thecentroid obtained from the first stage. Grace et al.developed an algorithm for that dynamic switchingand the ensuing parameter estimation, which wouldenable 10 to 100 fold reduction in the integrationtime needed to obtain a desired (small) mean squarederror in estimating θ2 despite no prior informationof θ1 is assumed, compared to the scenario whenimage-plane direct detection is used for the entireintegration time [15].

The choice of image-plane direct detection to ob-tain a pre-estimate of the centroid during the firststage of the aforesaid adaptive receiver was drivenby intuition. Image-plane direct imaging is quantumoptimal (attains QFI) for localizing a single pointsource [7], but suboptimal when it comes to esti-mating the centroid of two point sources [8]. Giventhe performance of SPADE-like measurements is ex-tremely sensitive to misalignment, even small im-provements in estimating the centroid can be verybeneficial. This is our motivating reason to studythe problem of centroid estimation.

In this paper, we investigate the performance ofboth direct imaging and the HG SPADE for esti-mating the centroid θ1 of n ≥ 2 point sources in aline spanning an angular length of θ2. The formerbecause it is the simplest and the most obvious mea-surement, and the latter because it is worth asking

if the SPADE can again outperform direct imagingin some region of parameter space for finding thecentroid, just as it did for the separation. We alsocalculate the quantum limit (QFI) of centroid estima-tion to quantify the gaps to the FIs attained by theaforesaid two measurements. Finally, we describe anadaptive two-stage receiver design that would attainthat QFI in the limit of long integration time.

III. QUANTUM LIMIT OF LOCALIZINGAN OBJECT IN PASSIVE IMAGING

A. The QFI and FI of direct imaging for alinear constellation of point sources

We now consider the behavior of the direct imagingFI and the QFI for estimating the centroid θ1 of alinear array of n ≥ 2 equally-spaced point emittersspanning a total angular extent θ2. First, it is worthnoting that the QFI and direct imaging FI shouldboth be independent of the value of the centroidθ1 due to the assumption of a linear, shift-invariantphysical imaging system. In the calculation of the FIfor direct imaging, i.e., from the samples drawn fromthe spatial probability density Λ(x) of photon clicksas in Eq. (5), this appears in the form of the shiftsymmetry of the variable of integration in (21) fromx to x− θ1, which removes θ1 from the integrand.

In the case of a single light source, Helstrom showedthat direct imaging is quantum optimal [7]. The QFIand direct imaging FI for this case can be calculatedeasily as we show in Appendix B, and we find thatthey are both equal for any arbitrary PSF:

K1−pt = J1−pt = 4N∆k2, (16)

where

∆k2 ≡∫ ∞−∞

dx

[∂ψ(x)

∂x

]2

. (17)

For our Gaussian PSF defined in (4), this yields N/σ2

for the QFI and the direct imaging FI. This is a resultthat we will regularly use throughout the rest of thispaper since all the cases involving 2 or more pointsalso have special limiting points where the FI andthe QFI will approach this value.

For 2 point sources, the QFI has been workedout analytically by Tsang et al. [8]. We describetheir calculation in Appendix C, and simply statethe result here. For the diagonal component of theQFI matrix involving the centroid, i.e., the QFI for

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estimating the centroid, they obtain:

K11 = 4N(∆k2 − γ2), (18)

where ∆k2 was defined in (17), and

γ =

∫ ∞−∞

dx∂ψ(x)

∂xψ(x− θ2). (19)

It is worth noting that γ goes to zero when θ2 goes tozero or infinity for any symmetric PSF. When θ2 goesto zero, the derivative of ψ(x) is anti-symmetric, sothe integral in (19) tends to zero. On the other hand,when θ2 becomes large, then ψ(x−θ2) and ∂ψ(x)/∂xoverlap very little with each other for PSFs ψ(x) thatfall off to zero away from the origin. Therefore, again,γ goes to zero. Consequently, the QFI approaches4N∆k2 in these two limits, that is, the result for thesingle point-source case. In between, however, thereis a region where γ is not zero, and we get a smallerQFI than that for a single point source. It is thisregime where direct imaging is unable to attain theQFI for estimating the centroid.

The physical explanation for this behavior is thatwhen the separation is very small, the impulse re-sponse of two point sources, i.e., their aperture-blurred fields in the image plane, each of width σ, looklike the impulse response of a single point source atorigin. Therefore, the problem of centroid estimationin this limit should reduce to that of finding the loca-tion of a single point source, for which image-planedirect detection is known to be quantum optimal [7].For a slightly larger separation, the images of twopoint sources no longer overlap as much, and theeffect of diffraction is to cause a decrease in the QFI.However, when the separation becomes significantlylarger than the width of the PSF, the images of thetwo point sources fully separate with no overlap, inwhich regime their individual positions can be esti-mated separately—again quantum-optimally usingimage-plane direct detection—treating the two assingle point sources. Each point source now emitshalf the light, and therefore the QFI for its locationis 2N∆k2, but the total sum is still 4N∆k2. For thespecific case of our Gaussian PSF (4), we get

K11 =N

σ2− Nθ2

2

4σ4exp

(− θ2

2

4σ2

), (20)

which approaches the single point-source result ofN/σ2 in the θ2 → 0 and θ2 → ∞ limits with a dipin between as discussed above (see Fig. 2A).

The FI of the centroid from direct imaging for the 2point-source case has a somewhat similar qualitative

behavior with the same physical intuition, exceptthat its dip between the two limiting cases of θ → 0and θ2 →∞ is deeper. It is given by:

J11 = N

∫dx

1

Λ(x)

(∂Λ(x)

∂θ1

)2

, (21)

where Λ(x) is the probability density given in (5).We are unable to do this integral analytically, andtherefore use numerical integration. The result wasplotted along with the QFI in [8], and we reproduce itin Fig. 2 along with our results for when n > 2 pointemitters constitute the scene. We see that the directimaging FI approaches the QFI for small and largeseparation, as expected from the aforesaid intuitiveexplanation, for all n ≥ 2. But, there is a gap in theregion between these two limiting regimes. This gapis not too large, especially deep in the sub-Rayleighregime. In particular, in the small θ2 regime forn = 2, we can see this explicitly by Taylor expandingthe QFI (20) and also Taylor expanding the integrandof the direct imaging FI (21) in the θ2 → 0 limit andintegrating term by term. The resulting limitingbehavior for the QFI and direct imaging FI are givenby

K11 =N

σ2− Nθ2

2

4σ2+Nθ4

2

16σ6− Nθ6

2

128σ8+O(θ8

2) (22)

and

J11 =N

σ2− Nθ2

2

4σ2+Nθ4

2

16σ6− Nθ6

2

64σ8+O(θ8

2) (23)

from which we see that the two quantities vary onlyin 6th order in θ2. Tsang et al. argued [8] that weshould be able to obtain a reasonable estimate forthe centroid from direct imaging in order to correctlyalign the SPADE for estimating the separation, anintuition that was validated in the adaptive two-stagereceiver designed and analyzed by Grace et al. [15].

Importantly, the above discussion using FI andQFI as performance benchmarking tools does notaddress the fact that if we are not in the regimeθ2/σ � 1, the maximum likelihood estimator of θ1—either with direct detection or the quantum-optimalmeasurement as the receiver choice—would in generalalso depend on the true (apriori unknown) values ofθ1 and θ2. The fact that with θ1 known apriori, theoptimal measurement and estimator to estimate θ2 isthe HG SPADE and is independent of the estimateof θ2, was a happy coincidence.

We now come to generalizing the result in [8] for the2-point-source case to a general number of incoherent

8

A. B.

FIG. 2. A. QFI (solid lines) and FI of direct imag-ing (dashed lines) for estimating the centroid of n pointsources (Blue: n = 2; Gold: n = 3; Red: n = 4; Cyann = 6; Green: continuous line) with separation θ2 givena Gaussian PSF with width σ. B. QFI (solid line) andFI of direct imaging (dashed line) for a continuous linesource with a constant photon flux per unit length of thesource.

point sources. For n = 3 or more emitters, calculat-ing the QFI becomes increasingly complicated as itinvolves diagonalizing larger and larger matrices. Wetherefore perform these diagonalizations numerically.The detailed procedure we employ for this purposeis described in Appendix D, and here we focus onthe results. Figure 2A shows the plots of QFI anddirect imaging FI against the end-to-end separationθ2 for n = 2, 3, 4 and 7 emitters, as well as for acontinuous line (n→∞). We see that as one wouldexpect, both the QFI and direct imaging FI go toN/σ2 when θ2 approaches zero, for any n as well asthe continuous line case.

As discussed above for 2 sources, even for n ≥ 2sources, as θ2 increases from 0, the QFI and directimaging FI fall from N/σ2 due to the diffraction-induced overlap among nearby point sources, andhence our ability to estimate the centroid decreases.However, the performance of direct imaging falls morerapidly than the QFI, and we see a small gap betweenthe two. For any finite n, as θ2 increases to theextent that the n point sources no longer significantlyoverlap, they essentially all become totally separatepoint sources, and their locations can be estimatedindividually as totally separate single emitters, justas we argued for 2 points, in which regime the QFI isattainable with direct imaging. As a result, the QFIand direct imaging FI rise back towards the N/σ2

value for a single emitter as θ2 becomes sufficientlylarge. However, as n the number of point sourceskeeps increasing, θ2 must increase further for thepoints to become “totally separate”. Therefore wesee that the QFI for 3 sources has a minimum at a

larger θ2 compared to that for the 2 source case beforeit starts increasing again; and the direct imaging FIbehaves the same way. Further increasing the numberof point sources augments this effect, with QFI anddirect imaging FI having their minima at even largervalues of θ2 and requiring more and more separationfor the QFI and FI to rise back toward the respectivevalues for totally separated points. For a continuousline source, i.e., n = ∞, since there is an infinitenumber of points next to each other, the QFI anddirect imaging FI both monotonically decrease as weincrease the length because the “constituent pointsources” comprising the uniformly-radiant object cannever be “totally separated”. In this case, increasingθ2 only makes it more and more difficult to estimatethe location of the centroid for a given total meanintegrated photon number N .

In fact, it turns out that both the QFI and directimaging FI for the continuous line scale as 1/θ2 forlarge θ2. This can be seen as follows. Instead ofassuming a fixed total number of photons N , let usconsider the case when the total brightness of theobject is proportional to its length. In other words,the total number of photons is Nθ2, which amountsto simply multiplying the QFI and direct imagingFI values for N photons, by θ2. These results areshown in Figure 2B as solid and dashed curves respec-tively. We see that they asymptote to constant valuesof about 1.95N/σ2 and 1.80N/σ2, for the QFI anddirect imaging FI, respectively, which means that(1) the scaling behavior for constant total bright-ness (that does not scale with length) is indeed 1/θ2,and (2) there is a constant-factor gap between directimaging FI with QFI, in the large θ2 limit. It is alsopossible to see this scaling behavior analytically, eventhough we cannot carry out the full calculations forthe QFI and direct imaging FI analytically and haveto resort to numerical methods. We describe thisin Appendix E, where we outline the calculation forthe continuum case where the sums over the infinitenumber of emitters, as in (2) and (5) are replaced byintegrals.

To get a clearer picture about how the perfor-mance of direct imaging compares with the quantum-optimal measurement for estimating the centroid, weconsider the ratio of the direct imaging FI to the QFI.We plot this ratio against the end-to-end separationθ2 for different number of point sources as well asfor the continuous line in Fig. 3. We see that overallthe ratio is generally not too low, though in someplaces it goes down below 50 %. The minimum isabout 73%, 60%, 53%, and 45% for n = 2, 3, 4 and6 points, respectively. For the continuous line, the

9

FIG. 3. Ratio between the QFI and the direct imagingFI for estimating the centroid of n point sources (Blue:n = 2; Gold: n = 3; Red: n = 4; Cyan n = 6; Green:continuous line) with end to end separation θ2 given aGaussian PSF with width σ.

ratio continuously falls, asymptoting to a constantvalue of around 92.5%.

All these results mean that while direct imagingis sub-optimal, it is not significantly worse than theQFI. Therefore, it is not possible to get a signifi-cant improvement over direct imaging by using anyother measurement. Yet, the fact that it falls downto about 72% for 2 point sources and even slightlybelow 50% for a few more points, suggests that thereis some room for improvement, especially in a situa-tion where simply collecting more photons to get thesame improvement is not the most desirable option.It is also worth mentioning that since the gap be-tween direct imaging and QFI is generally small forthe continuum case than for a small, finite numberof emitters, direct imaging is closer to the optimalscheme for locating a full, uniformly bright objectrather than one that has internal structure and/orsparsity. Finally, these results and our methods out-lined above apply to any two- or three-dimensionalpoint-source constellations, and it is possible that thegap between QFI and direct-imaging FI is higher forother more general constellations of point-emittersor continuous objects.

B. Comparison with the HG SPADE’sperformance, and an interesting duality

We now consider an image-plane HG mode SPADEmeasurement [8, 24], but for centroid estimation. Letus take the central position of this SPADE to be x =0, such that the optical axis of the SPADE defines the1D Cartesian coordinate system. The question wewill consider is whether this can allow us to determinethe centroid θ1 more efficiently than direct imaging.Let us begin with writing the quantum state of asingle temporal mode of the collected image planefield with exactly one photon in the q-th HG mode.In other words, a single photon Fock state of the q-thHG mode is given by:

|φq〉 =

∫ ∞−∞

dxφq(x) |x〉, (24)

with q = 0, 1, . . ., and

φq(x) =(2πσ2)−1/4

√2qq!

Hq

(x√2σ

)exp

(− x2

4σ2

),

(25)where Hq are the Hermite polynomials and |x〉 isthe unphysical 1-photon Fock state of the perfectlylocalized (delta-function) spatial mode at positionx. Now, recalling (2), it is straightforward to seethat if we pass the image-plane field through a HGmode sorter and detect photons on each mode, theprobability of finding the photon in mode q is:

P (q) =1

n

n∑s=1

Ps(q), (26)

where Ps(q) is the probability for a photon fromsource s to be found in mode q:

Ps(q) ≡ |〈φq|ψs〉|2 =

∣∣∣∣ ∫ ∞−∞

dxφq(x)ψ(x− xs)∣∣∣∣2.(27)

For our Gaussian PSF defined in (4), this gives

Ps(q) = exp (−Qs)Qqsq!, (28)

where

Qs =x2s

4σ2. (29)

It is now a straightforward exercise to obtain theFisher information,

JHG,11 =

∞∑q=0

N

P (q)

(∂P (q)

∂θ1

)2

10

=

∞∑q=0

(∑i

Qi(qQq−1i −Qqi )

2e−2Qi

+∑i 6=j

2Qij(qQq−1i −Qqi )(qQ

q−1j −Qqj)e

−Qi−Qj

)

× N

nσ2q!∑i exp (−Qi)Qqi

, (30)

where Qij ≡ (xixj)/(4σ2) =

√QiQj .

For the continuous line case of the number of pointsbecoming infinite, we replace the sum in (26) by anintegral. We show in Appendix F that

JHG,l,11 = N

∞∑q=0

(exp (−Q+)

Qq+q! − exp (−Q−)

Qq−q!

)2

θ2

∫ θ2/2y=−θ2/2 exp (−Q(y)) Q(y)q

q! dy

(31)where

Q± =(θ1 ± θ2/2)2

4σ2. (32)

Note that this clearly has a dependence on θ2 otherthan the 1/θ2 factor, and therefore does not obeythe same type of scaling behavior for large θ2 thatwe found for the QFI and the direct imaging FI.

It is not clear how to do the sum over the FIcontributions for the individual HG modes in (30) or(31) analytically, so we have to do this numerically.However, the series sum does simplify nicely for afew special cases:

1. When θ2 approaches zero but θ1 does not.This is essentially the limiting case where all theemitters effectively merge into a single one, butthe SPADE is mis-aligned with the centroid bya constant amount. For this, the FI approachesthe N/σ2 value, the QFI for a single emitter asdiscussed in appendix B 3. But then, for thisθ2 → 0 case, we also know that the performanceof direct imaging approaches the QFI [7].

2. When θ1 approaches zero but θ2 does not.This is the case when the SPADE is almostperfectly aligned with the centroid, and theend-to-end separation is a constant. In thiscase, we get zero for the FI. This means, theHG SPADE yields tending-to-zero informationabout the centroid as the SPADE’s alignmentwith the true centroid approaches near perfect.

To see this, recall (26), (28) and (29) and con-sider the partial derivative of P (q):

∂P (q)

∂θ1=

1

n

n∑s=1

∂Ps(q)

∂θ1

=

n∑s=1

xs2nσ2

exp (−Qs)(qQq−1

s

q!− Qqs

q!

). (33)

Now, if we have an even number of emitters,then all their locations come in pairs of theform xs = θ1 ± |cs|θ2/2, where cs is a constnatfactor whose exact value depends on s. Whenθ1 = 0, these become xs = ±|cs|θ2/2, and thecorresponding Q values (which are proportionalto x2

s), are then equal for each pair. The sum in(33) then becomes zero since the contributionsfrom the two points in each pair cancel due tothe xs in front. If we have an odd number ofsources, then all of them are in similar pairs,except the middle one at x = θ1. But thisbecomes zero for θ1 = 0, and therefore, we stillget zero for the sum in (33).

This result has important implications for thetwo-stage set up of the kind being proposed inRef. [15], in which we first obtain an estimateof the centroid from direct imaging in order toalign an HG mode sorter in the second stage forestimating the separation or object size. Oncewe switch to the HG SPADE with a reasonablydecent alignment close to the centroid, we willnot be able to get any improvement in ourestimate for the centroid. Our entire estimationprecision of the centroid will therefore be basedon the first measurement stage alone.

We now compare the performance of the HGSPADE with direct imaging and the QFI for cen-troid estimation. For this purpose, we focus on the2-emitter and the continuum cases as our two exam-ples that illustrate the overall pattern. For 2 sources,the plots of the QFI, direct imaging FI and the HGSPADE for different fixed values of the mis-alignmentare given in Figure 4A. The same comparison for thecontinuum case is shown in Figure 4B. The QFI anddirect imaging curves in these figures are of coursethe same as those shown in Figure 2A, and for the2-emitter case, the QFI and direct imaging curves arethe same as those shown in Ref. [8], but that refer-ence does not compare these with the HG SPADE’sFI. We see that the performance of the HG SPADEis mostly worse than direct imaging for these graphs.

11

It does however tend to converge with direct imag-ing from below in the θ2 →∞ limit. But it is neverhigher than the direct imaging curve, except in a verysmall region for the θ1 = 2σ curve for the 2-emittercase. This is a special region between the θ2 → 0and large θ2 extremes where the direct imaging FIdrops sufficiently, and the FI for the HG SPADE risesenough to achieve a very small amount of superiority.It is worth noting that this is a region where the HGSPADE has enough mis-alignment with the centroidand is also pointed sufficiently away from either ofthe two emitters. We can also find some other suchregions where this happens and the SPADE performsbetter than direct imaging, but the improvement isvery small, and for most of the parameter space, thelatter outperforms the former by a bigger margin.

Therefore, the overall conclusion is that for all prac-tical purpose, direct imaging is better than the HGSPADE for estimating the centroid, and that there isa small gap between the HG SPADE’s performanceand the ultimate quantum limit in the intermediaryrange of the object’s length.

This highlights an interesting duality and com-plementarity between direct imaging and the HGSPADE. Unless the separation is large, direct imag-ing performs poorly for estimating the separation,especially in the sub-Rayleigh regime, whereas theHG SPADE, if aligned perfectly with the centroid, al-lows us to attain the QFI. But the SPADE generallydoes not perform very well for estimating the centroid,whereas direct imaging comes closer to attaining theQFI. We should however qualify this statement bypointing out that this complementarity is not totallyperfect; it certainly holds when the SPADE pointsexactly at the centroid, but there are regions of theparameter space for non-zero alignment where thisrelationship is no longer true. The small region wesaw in the 2-emitter case where the SPADE with amis-alignment from the centroid of 2σ slightly out-performs direct imaging is an example of this. Thereis also a duality in the sense that a perfectly alignedSPADE attains the QFI for separation estimation butcompletely fails in determining the centroid, whereasa mis-aligned SPADE generally tends to be better forestimating the centroid rather than the separation.

C. The quantum-optimal measurement schemefor estimating the centroid

Having found that direct imaging is not optimal forcentroid estimation in general, and the HG SPADE’sperformance is mostly worse than direct imaging, we

A.

B.

FIG. 4. A. QFI, direct detection FI, and HG-SPADEFI for estimating the centroid of two point sources withseparation θ2 and a centroid of θ1 from the origin (i.e.,the SPADE alignment axis). B. Same as panel A, but fora continuous line source.

now discuss a scheme for surpassing it and asymptot-ically attaining the QFI. In general, the projectivequantum measurement given by the eigenbasis of theSLD for a given quantum state defines a physicallyallowed measurement that achieves the QFI [25–27].This SLD measurement would translate to a SPADE(not the HG SPADE), followed by photon detectionon the sorted spatial modes. This is because anyprojective measurement on a quantum state of onephoton in many (spatial) modes—which is the casefor the quantum description of the state of a singletemporal mode of collected light in our problem—isalways realizable by a passive linear optical transfor-mation followed by photon detection [28].

However, since in this case the SLD depends on

12

the true value of the centroid itself, this eigenbasis,and hence the aforesaid quantum-optimal SPADE todetect each temporal mode of the collected field, alsodepends upon the centroid. Therefore, we cannotcarry out a measurement in this basis unless wealready know the centroid, which is the variable weare trying to estimate in the first place. To get aroundthis problem, we use a 2-stage adaptive measurementscheme proposed by [26], applied to our problem:

1. Recall that N = Mε is the mean photon num-ber of the total collected field, where M is thenumber of temporal modes and ε � 1 is themean photon number per mode. The receiver’sfirst stage uses a small proportion of the inte-gration time, worth Nα mean photon number,with any 0 < α < 1, to obtain an initial max-

imum likelihood estimate θ1 for the unknowncentroid parameter θ1. The measurement usedfor this stage can be direct imaging, and doesnot have to be an optimal choice in any sense.The only requirement on this measurement isthat it has a non-zero FI for estimating thecentroid, θ1.

2. Based on the estimate θ1 obtained during thefirst stage, we carry out a measurement oneach temporal mode of the remaining collectedphotons (of mean photon number N − Nα)using the eigenbasis of the SLD of θ1, evaluated

at θ1. Based on this measurement, we obtaina maximum likelihood estimate for θ1.

We would like to mention that using any other QFIattaining measurement in place of the SLD eigenbasisin the second stage will give the same performancein theory. Since the SLD basis is not necessarilythe only measurement that attains the QFI, theremay be other alternatives too; for example, the linearinterferometric approach put forward by [29] is worthinvestigating for this purpose.

The above procedure prescribes a measurementthat asymptotically reaches the efficiency of the QFIas N tends to infinity. This two-stage scheme is alsodescribed in [30, 31] and section 6.4 of [32], with thespecific choice of α = 1/2, and with the conditionthat the FI for the first stage should be non-zero.

While we refer the reader to the above referencesfor detailed derivation of why this two-stage approachshould attain the efficiency of the QFI in the largeN limit even though the SLD depends upon the a-priori -unknown parameter, here is a short summaryof the argument. If N is sufficiently large, Nα or√N in particular will also be large. And therefore,

the variance of the estimate θ1 scales as 1/(J1Nα),

where J1 > 0 is the FI of this stage-one measurement,hence approaching zero as N →∞, around the true(a-priori-unknown) value of θ1. Now, the Fisherinformation in stage-two should be (N −Nα)K, if wemeasure in the eigenbasis of the SLD based on theexactly true value of θ1. But in reality, since we willcarry out this measurement at the estimated value

θ1 = θ1 + θ1,err, we must replace K by the FI for theSLD eigenbasis measurement evaluated at this valuerather than the true θ1. We can express this FI as aTaylor expansion around the true value θ1 as

JSLD(θ1+θ1,err) = JSLD(θ1) + θ21,err

∂2JSLD(θ1)

∂θ21

+ . . .

(34)Here we do not have a first derivative term becauseJSLD has a maximum at θ1 equal to K, so the firstderivative must be zero. Moreover, since this is amaximum, the second derivative will be a negativeconstant with respect to θ1,err. Therefore, we canrewrite the stage-two FI as

JSLD(θ1 + θ1,err) =K(1−O

(θ2

1,err

))=K

(1−O

(1

NαJ1

))(35)

where in the last step, we have used the fact that themean squared error of the initial centroid estimateis approximately equal to the inverse of the FI forstage-one. The total FI accumulated over stage-two

is therefore (N −Nα)K(

1−O(

1NαJ1

)). And since

when N is large

1

(N −Nα)K(

1−O(

1NαJ1

)) ≈ 1

NK, (36)

the variance approaches that of the optimal measure-ment. Based on the choice of the stage-one measure-ment, and its FI J1, one could optimize the choiceof α such that the overall FI attained at the endof stage-two is maximized, for a given fixed N . Amulti-stage adaptive quantum estimation algorithmis given in [33] where the result of each stage is usedas input for the next one, which could lead to furtherimproved performance in this non-asymptotic setting.However, finding the quantum optimal measurementfor finite N is being left open for future work. Notethat in the context of this adaptive measurement, Nindicates the number of photons that are dedicatedto estimating the centroid. If the ultimate goal is toestimate the separation, there also will be photons

13

set aside for the estimation of the separation (e.g.,using a SPADE aligned to the centroid estimate) [15].

We can therefore apply this 2-step procedure forestimating the centroid if we can calculate the eigen-vectors of the SLD. For 2 point sources of light, thenon-zero entries of the SLD are given in Appendix C.For more than 2 light sources, including the case of acontinuous line, we can obtain the SLD numerically,calculating it in the basis of HG-basis 1-photon Fockstates, as described in Appendix D. It is importantto note here that the SLDs not only depend on thecentroid, but also on the separation (or, the objectlength, in case of a continuous line). Therefore, ifwe already know the separation, then we only needto estimate θ1 in stage one of our two-step adaptivescheme. But if we do not know the separation, then

we also need to extract an initial estimate θ2 for theseparation from the measurement in the first stage.We can then switch to the second stage where we

calculate the eigenbasis of the SLD in terms of θ1

and θ2 to get a good estimate for the centroid whosequality approaches the QFI for a large number ofintegrated photons.

IV. CONCLUSION

We have carried out a detailed, systematic study ofour ability to estimate the centroid of a linear arrayof incoherent light sources as well as a line-shaped ob-ject with uniform brightness. Our approach is easilyextensible to estimating the centroid of more com-plex objects. We calculated the QFI for estimatingthe centroid and compared it with the FI for directimaging as well as an image-plane HG SPADE. Wedescribed a two-stage readily-realizable measurementthat would attain the QFI of centroid estimation.

Our key conclusions can be summarized as follows:Direct imaging in the image plane is strictly sub-

optimal compared to the QFI for centroid estimation,though the ratio of the QFI to the Direct imaging FIis less than an order of magnitude. Therefore, directimaging should offer a fairly good estimate of thelocation of an object, as intuitively expected.

The gap between the performance of direct imagingand QFI is generally less for the continuum objectsthan for a constellation involving a small number ofpoint-like emitters. This suggests that direct imagingperforms closer to the optimal scheme for locatinga full, uniformly bright object rather than one thathas more internal features.

The performance of the HG SPADE is mostlyworse than direct imaging for centroid estimation,

except for some special limiting regimes where itsFI approaches the direct imaging FI from below,or some small regions of parameter space where itmarginally surpasses direct imaging. However, theseregions where it slightly outperforms direct imagingare negligible, and the performance improvement isalso too little to be of any practical significance.

We have also found that the HG mode performsvery poorly when it is nearly aligned with the cen-troid. This means that if we employ a two-stageprocedure for determining θ1 and θ2, where we firstestimate θ1 and then use it to align the SPADE fordetermining θ2 in stage 2 as in [15], then we wouldnot be able to extract much additional informationabout the centroid from the SPADE measurement instage 2.

There is a complementarity between direct imagingand the HG SPADE. Direct imaging has a fairly goodperformance for centroid estimation, even though itis not the optimal measurement. But it performsvery poorly for determining the separation in thesub-Rayleigh regime, and the Fisher information forthe separation goes to zero when the separation ap-proaches zero. The HG SPADE, on the other hand,is the optimal measurement for finding the separa-tion, but it performs very poorly for determining thecentroid, when it is nearly aligned with the centroid.

We have found an interesting scaling behavior forthe QFI and direct imaging for the continuous linefor large θ2, with a constant-factor gap. Specifically,we have found that the QFI and the direct imagingFI both scale as 1/θ2 in this region. This makesvery good intuitive sense: the larger the length of acontinuous object, the greater the portion of the ob-ject that has spatially constant irradiance, and hencefewer information-bearing photons are available toestimate the centroid.

V. ACKNOWLEDGEMENTS

The authors thank Mankei Tsang and Ran-jith Nair for valuable discussions. This workwas supported by a Defense Advanced ResearchProjects Agency (DARPA) Defense Sciences Of-fice (DSO) seedling project awarded under contractnumber W911NF2010039. QZ acknowledges theDARPA Young Faculty Award (YFA), Grant numberN660012014029.

14

Appendix A: The diagonality of the FI and QFIfor our centroid and separation parameters

1. The FI for Direct imaging

First, consider direct imaging. For our choice ofa Gaussian PSF, the density probability functionΛ(x) defined in (5) is an even function around thecentroid x = θ1. It is a straightforward exercise to

see that ∂Λ(x)∂θ1

is an odd function around the centroid,

whereas ∂Λ(x)∂θ2

is even. Their product is therefore anodd function around θ1, and the integral over x from−∞ to ∞ is therefore zero.

To see why ∂Λ(x)/∂θ1 is odd and ∂Λ(x)/∂θ2 iseven, note that if we have an even number of points,they come in pairs of the form xs = θ1 ± csθ2, inwhich cs is a factor that only depends on s. If wehave an odd number of points, then we have a pointin the middle at x = θ1, and all the other points againcome in such pairs with the same distance on eitherside of the centroid. The partial derivative of Λ(x)will therefore also contain pairs with contributionsof the form

∂|ψ(x− θ1 ± csθ2)|2

∂θ1

= −2|ψ(x− θ1 ± csθ2)|∂|ψ(x− θ1 ± csθ2)|∂(x− θ1 ± csθ2)

(A1)

Now, since |ψ(x−θ1±csθ2)| is symmetric around θ1∓cs,

∂|ψ(x−θ1±csθ2)|∂(x−θ1±csθ2) is anti-symmetric. And therefore,

the sum over both elements of the pair with ±csθ2

is an anti-symmetric function around θ1. In case thenumber of points is odd, then the partial derivativeof the middle point will also be an anti-symmetricfunction. So ∂Λ(x)/∂θ1 is odd around the centroid.

In contrast

|ψ(x− θ1 ± θ2)|2

∂θ2

= ±2cs|ψ(x− θ1± csθ2)|∂|ψ(x− θ1 ± csθ2)|∂(x− θ1 ± csθ2)

(A2)

So now the two members of the pair have opposite

signs. This, along with the fact that ∂|ψ(x−θ1±csθ2)|∂(x−θ1±csθ2) is

anti-symmetric around θ1∓ csθ2, means that the twomembers of each pair combine to give a symmetricfunction around θ1. If the total number of points isodd, then there is also an additional point located

in the middle at the centroid. But since its positiondoes not depend on θ2, it does not contribute to∂Λ(x)/∂θ2. Therefore, we conclude that ∂Λ(x)/∂θ2

is an even function around θ1.

2. The QFI

For the QFI, the argument is somewhat similar,but now we need to think in terms of the densityoperator and its partial derivatives. Recalling (2),the 1-photon part of the density operator is

ρ1 =1

n

n∑s=1

|ψs〉〈ψs| =1

n

n∑s=1

|ψ(x− xs)〉〈ψ(x− xs)|

(A3)This again comes in pairs, and the density operatoris symmetric in each pair. Now, consider the partialderivative

∂ρ1

∂θ1=

1

n

n∑s=1

(∂|ψ(x− xs)〉

∂θ1

〈ψ(x− xs)|+ |ψ(x− xs)〉∂〈ψ(x− xs)|

∂θ1

)(A4)

But |ψ(x− xs)〉 =∫dxψ(x− xs)|x〉, and therefore,

it has the same even-odd parity as ψ(x− xs). Like-wise, the partial derivatives of this ket will have thesame even-odd parity as the partial derivatives of thefunction ψ(x − xs). Therefore, like ∂Λ(x)/∂θ1 and∂Λ(x)/∂θ2, ∂ρ1/∂θ1 and ∂ρ1/∂θ2 will be odd andeven, respectively. And therefore, the off-diagonal en-tries of the QFI evaluated in terms of the parametersθ1 and θ2 will be zero.

3. The HG SPADE

For the HG mode sorter, our physical set up isnot symmetric unless the SPADE is perfectly alignedwith the centroid. Therefore, the above-mentionedsymmetry arguments no longer hold, and the FI fora measurement in the HG basis is not diagonal ingeneral. For the particular case of the SPADE beingperfectly aligned with the centroid, we saw in sectionsection III B that ∂P (q)/∂θ1 becomes zero due tocancellations of the contributions from each memberof the symmetric pair. Therefore, the FI is not onlydiagonal, but all its entries other than the diagonalone corresponding to θ2 are zero.

15

However, when the SPADE is mis-aligned, we willnot get a diagonal QFI matrix in general. To seethis, consider the specific example of the 2 point case.The probability function is

P (q) =1

2q!(exp (−Q1) Qq1 + exp (−Q2) Qq2) (A5)

where Q1 = (θ1+θ2/2)2

4σ2 and Q2 = (θ1−θ2/2)2

4σ2 . The twoterms are clearly not symmetric or anti-symmetric,since they have different Gaussian decay factors aswell as (θ1 ± θ2)q which will be different for bothpoints.

Appendix B: The single point case

1. The FI for direct imaging

For a single point, the probability function Λ(x)defined in (5) becomes

Λ1(x) = |ψ(x− θ1)|2 (B1)

The Fisher information for direct imaging is then

J1−pt =

∫dx

N

Λ(x)

(∂Λ(x)

∂θ1

)2

=4N

∫dx

(∂|ψ(x− θ1)|

∂θ1

)2

=4N∆k2

(B2)

For our Gaussian PSF defined in (4), the result is

J1−pt =N

σ2(B3)

2. The QFI

When we only have one point, the single-photonpart of the density matrix is simply one-dimensional

ρ1 = |ψ(x− θ1)〉〈ψ(x− θ1)| (B4)

The partial derivative of this is

∂ρ1

∂θ1=

(∂|ψ(x− θ1)〉

∂θ1

〈ψ(x− θ1)|+ |ψ(x− θ1)〉∂〈ψ(x− θ1)|∂θ1

)

= −|ψ′(x− θ1)〉〈ψ(x− θ1)|− |ψ(x− θ1)〉〈ψ′(x− θ1)|(B5)

since ∂ψ(x− θ1)/∂θ1 = −ψ′(x− θ1) where ψ′ is thederivative of ψ. It is straightforward to see that|ψ(x− θ1)〉 and |ψ′(x− θ1)〉 are orthogonal states forany symmetric ψ(x):

〈ψ(x− θ1)|ψ′(x− θ1)〉 =

∫dxψ(x− θ1)

∂ψ(x− θ1)

∂θ1

= 0 (B6)

We thus have an orthogonal basis and only need tonormalize ∂|ψ(x− θ1)〉/∂θ1. Our orthonormal basisis thus

|e1〉 ≡ |ψ(x− θ1)〉 (B7)

|e2〉 ≡1

∆k|ψ′(x− θ1)〉 (B8)

where ∆k =√∫

dx|ψ′(x− θ1)|2 is a normalization

factor and is equal to the square root of ∆k2, definedin (17). The density operator in this eigen basis issimply ρ1 = |e1〉〈e1| with eigen values D1 = 1 andD2 = 0 corresponding to |e1〉 and |e2〉. Recalling(12) and (B5), the symmetric logarithmic derivativeis then

L = 2∆k (|e2〉〈e1|+ |e1〉〈e2|) (B9)

The QFI is then

K1−pt = Ntr(ρL2

)= 4N∆k2 (B10)

We see that this is equal to the FI for direct imagingin (B2) for any PSF.

3. The HG SPADE FI

The probability function for the qth HG mode forlight coming from a single point with a GaussianPSF is given by

Pq = exp (−Q)Qq

q!(B11)

where Q =θ21

4σ2 . The FI for the qth mode is then

Jq,HG,1−pt =N

Pq

(∂Pq∂θ1

)2

(B12)

16

It is a straightforward exercise to calculate this andcarry out the sum over the whole series in q, and theresult is

JHG,1−pt =N

σ2(B13)

which is equal to the QFI as well as the FI for directimaging.

Appendix C: The QFI for 2 points

This calculation has been explained by Tsang etal. in their paper. Therefore, we will only summarizetheir method while clarifying one or two points.

From (2), the single-photon part of the densitymatrix for the 2 point case is

ρ1 =1

2(|ψ1〉〈ψ1|+ |ψ2〉〈ψ2|) (C1)

However, for the QFI, we need to work in an orthonor-mal eigenbasis that spans the whole space spanned by|ψ1〉 and |ψ2〉 as well as their partial derivatives withrespect to θ1. And it turns out that while |ψ1〉 and|ψ2〉 individually have norm 1, they are not mutuallyorthogonal in general

δ ≡ 〈ψ1|ψ2〉 = 〈ψ2|ψ1〉 6= 0 (C2)

for a real valued ψ(x). Therefore, we first need toexpress ρ1 in an orthonormal basis

ρ1 = D1|e1〉〈e1|+D2|e2〉〈e2| (C3)

To find the eigenvalues Di and the eigenstates |ψi〉,we write down a 2× 2 matrix of the inner products〈ψi|ρ|ψj〉 (

1 δδ 1

)(C4)

The normalized eigenvectors of this matrix give usan orthogonal set of functions, and the square rootsof the eigenvalues give us the normalization factors.We find that the eigenvalues are 1 ± δ, with theeigenvectors 1√

2(1,±1). Therefore, our orthonormal

basis of states spanning |ψ1〉 and |ψ2〉 is

|e1〉 =1√

2(1− δ)(|ψ1〉 − |ψ2〉) (C5)

|e2〉 =1√

2(1 + δ)(|ψ1〉+ |ψ2〉) (C6)

It is a straightforward exercise to see that these arealso the eigenstates of our density operator ρ1, andthat the eigenvalues Di of ρ1 are the correspond-ing eigenvalues of the matrix of inner products (C4)divided by 2:

D1 =1− δ

2(C7)

D2 =1 + δ

2(C8)

This division by 2 is simply the 1/2 factor in frontof |ψ1〉〈ψ1|+ |ψ2〉〈ψ2|.

However, ∂ρ1∂θ1

also contains the derivatives of |ψ1〉and |ψ2〉. Therefore, we need to extend our eigenbasisto span these states too. We therefore include thederivatives of |ψi〉 and carry out an orthogonalizationprocedure. This gives us the following additionalstates

|e3〉 =1

c3

[∆k√

2(|ψ11〉+ |ψ22〉)−

γ√1− δ

|e1〉],

(C9)

|e4〉 =1

c4

[∆k√

2(|ψ11〉 − |ψ22〉) +

γ√1 + δ

|e2〉],

(C10)where ∆k2 and γ were defined in (17) and (19), whichare reproduced here for the reader’s convenience

∆k2 =

∫ ∞−∞

dx

[∂ψ(x)

∂x

]2

(C11)

and

γ =

∫ ∞−∞

dx∂ψ(x)

∂xψ(x− θ2) (C12)

The other quantities defined here are

|ψ11〉 ≡1

∆k

∫dx

∂ψ(x− x1)

∂x1|x〉, (C13)

|ψ22〉 ≡1

∆k

∫dx

∂ψ(x− x2)

∂x2|x〉, (C14)

c3 ≡(

∆k2 + b2 − γ2

1− δ

)1/2

, (C15)

c4 ≡(

∆k2 − b2 − γ2

1 + δ

)1/2

, (C16)

17

b2 ≡∫dx

∂ψ(x− x1)

∂x1

∂ψ(x− x2)

∂x2, (C17)

and δ was defined in (C2).Since ρ1 = D1|e1〉〈e1|+D2|e2〉〈e2|, we get

D3 = D4 = 0 (C18)

Having found all the eigenbasis states and the eigen-values of ρ1, it is now a simple exercise to use theformula (12) for the SLD and compute the QFI, withthe results given in (C26) and (C27). For reference,the non-zero entries of the SLD with respect to thecentroid in the |ei〉 (i = 1 . . . 4) basis. The SLD withrespect to the centroid has the non-zero entries areas follows:

L1,12 =2γδ√1− δ2

(C19)

L1,14 =2c4√1− δ

(C20)

L1,23 =2c3√1 + δ

(C21)

The non-zero entries of the SLD with respect to theseparation are

L2,11 = − γ

1− δ(C22)

L2,13 = − c3√1− δ

(C23)

L2,22 =γ

1 + δ(C24)

L2,24 = − c4√1 + δ

(C25)

From these, it is a straightforward exercise to calcu-late the QFI. For the centroid, we obtain

K11 = 4N(∆k2 − γ2) (C26)

and for the separation,

k22 = N∆k2 (C27)

Appendix D: Calculating the QFI for more than2 points

Calculating the QFI analytically for more than2 points by employing Tsang et al’s procedure for2 points becomes a very complicated process, sinceit requires diagonalizing larger and larger matricesas the number of points is increased. Even doingthis numerically is a very involved process, and infact we soon start running into floating point errorswhen we go to about 10 or so points. It also doesnot allow us to calculate the QFI for a continuousline. We therefore follow the more efficient numericalapproach employed in [11]. The idea is that insteadof working with states |ψi〉 and their derivatives andcarrying out a laborious diagonalization process, wework in the HG basis. We only need to considerthe first few HG modes, since higher order modeshave diminishing contributions. The HG SPADEcalculations in this paper have been carried out with50 HG modes, and we have checked that this ismore than enough for the results to converge for theparameters being considered.

Specifically, we express our states ψ(x− xs) in theHG basis, which gives

ψ(x− xs) =

∞∑q=0

exp

(− x2

s

8σ2

)xqs√q!φq(x) (D1)

where φq(x) are the HG functions. We then expressour density matrix ρ1 and its derivative ∂ρ1/∂θ1 inthis basis. Since φq(x) does not depend on the loca-tion of the individual points or the centroid, the par-tial derivatives do not change the basis, and therefore,we do not have to carry out any orthogonalizationprocedure to find our additional basis states. Wesimply numerically calculate the eigenvectors andeigenvalues for the ρ1 in the basis of the first 50 HGmodes (or whatever other number we decide to con-sider). We then use the formula (12) to obtain theSLD, and calculate the QFI from N tr

(ρL2

).

Appendix E: The QFI and the direct imaging FIfor the continuum case and its scaling behavior

Here we show why the QFI and direct imagingFI for the centroid scale as 1/θ2 for large θ2 in thecontinuum case of an infinite number of emitters.First, consider direct imaging and recall the definitionof the probability function Λ(x) in (5). This is anaverage over m points, and for a continuous line, we

18

replace it by an integral

Λline(x) =

∫ θ2/2y=−θ2/2 |ψ(x− θ1 − y)|2dy

θ2(E1)

For θ2 sufficiently large compared to σ, this, beingan integral of a sharply peaked but smooth function,should give a nearly flat function that is constantover the length of the line and zero elsewhere, butwith smooth edges with width of order σ. For ourGaussian PSF, we get

Λline(x) =1

2θ2

(erf

(x− θ1 + θ2/2√

2σ

)

− erf

(x− θ1 − θ2/2√

2σ

))(E2)

which is indeed nearly 1/θ2 over the line and almostzero elsewhere, but has smoothly falling edges. Theother ingredient we need for the FI is the partialderivative of this with respect to θ1, which can eas-ily be computed using the fundamental theorem of

calculus. Since ∂|ψ(x−θ1−y)|2∂θ1

= ∂|ψ(x−θ1−y)|2∂y , we ob-

tain ∂Λline(x)∂θ1

by removing the integral and evaluating

|ψ(x− θ1 − y)|2 at the y values of the end points:

∂Λline(x)

∂θ1=

1

θ2

(|ψ(x− θ1 − θ2/2)|2

− |ψ(x− θ1 + θ2/2)|2)

(E3)

If θ2 is sufficiently large compared to σ, this is the dif-ference between two non-overlapping sharply peakedbut smooth functions at the edges of the line. Thesquare of this will therefore be a sum of two evenmore sharply peaked functions at the edges of theline. The integral over x to evaluate FI thereforeonly gets noticeable contributions near the edges ofthe line, and hence the distance between the edgesdoes not have any bearing on it. This leaves the1/θ2 factor in front as the main θ2 dependent partin (E3). This, along with the 1/θ2 scaling of Λ(x) in(E2), means that the FI should scale as 1/θ2. It isworth noting that this reasoning should hold equallyfor any other PSF that has a high peak in the centerand quickly but smoothly falls to zero away from it.

A somewhat similar argument can be made for thescaling of the QFI in the large θ2 region. The sum in(2) for the density matrix gets replaced by an integral

over the line. Recalling the definition of |ψs〉 in (3),this integral is

ρ1 =1

θ2

∫dx dx′

∫ θ2/2

y=−θ2/2dy ψ(x− θ1 − y)

× ψ(x′ − θ1 − y)|x〉〈x′|(E4)

The partial derivative of this with respect to θ1 canagain be obtained from the fundamental theorem ofcalculus by removing the integral over y and evaluat-ing this at the end points:

ρ1 =

∫dx dx′

1

θ2

[ψ(x− θ1− θ2/2)ψ(x′− θ1− θ2/2)

− ψ(x− θ1 + θ2/2)ψ(x′ − θ1 + θ2/2)]|x〉〈x′| (E5)

For θ2 sufficiently larger than σ, the two terms willsharply peak when both x and x′ are simultaneouslyequal to the end point locations of the line at θ1+θ2/2and θ1 − θ2/2. Elsewhere they will be nearly zero.The height and width of these peaks will not dependon θ2, and therefore the scaling of ∂ρ/∂θ1 in termsof θ2 arises almost entirely from the 1/θ2 factor infront.

As for the scaling of ρ, we can return to (E4).Again, focusing on θ2 sufficiently larger than σ, wenote that with ψ(x−y) and ψ(x′−y) sharply peakedat y = x and y = x′, performing the integral over ywill give us a function that sharply peaks at x = x′

provided x and x′ lie somewhere on our line between−θ2/2 and θ2/2. For our Gaussian point spreadfunction, we get

ρ1 =1

θ2

∫dx dx′

exp(− (x−x′)2

8σ2

)2(

erf

(θ + x+ x′)

23/2σ

)− erf

(−θ + x+ x′

23/2σ

))(E6)

in which we indeed have a sharply peaked Gaussianinvolving (x− x′), and the erf functions are simplystep functions with smooth edges, and hence do notcontribute to the scaling in terms of θ2. This leavesthe 1/θ2 in front as the only factor that contributesto the scaling. Since the QFI is given as K = tr(ρL2)and satisfies the relation (11) it is clear that theoverall scaling is essentially of two powers of ∂ρ/∂θ1

and an inverse power of ρ, so overall we get 1/θ2.Like the argument for the direct imaging FI, thisreasoning for the scaling of the QFI should also holdfor any other smooth PSF that has a sufficientlysharp peak.

19

Appendix F: HG SPADE FI for the continuumcase

For the continuum case, we replace the sum in (26)by an integral. Recalling (28), we get

Pl(q) =1

θ2

∫ θ2/2

y=−θ2/2dy exp (−Q(y))

Q(y)q

q!(F1)

where

Q(y) =θ1 + y

4σ2(F2)

And the partial derivative with respect to θ1 can beobtained, according to the fundamental theorem ofcalculus, by just removing the integral over y andevaluating at the end points:

∂Pl(q)

∂θ1=

1

θ2

(exp (−Q+)

Qq+q!− exp (−Q−)

Qq−q!

)(F3)

where

Q± =(θ1 ± θ2/2)2

4σ2(F4)

It is now straightforward to write down the FI con-tribution for each HG mode, and summing over allthe modes gives us the total FI

JHG,l,11 = N

∞∑q=0

(exp (−Q+)

Qq+q! − exp (−Q−)

Qq−q!

)2

θ2

∫ θ2/2y=−θ2/2 dy exp (−Q(y)) Q(y)q

q!

(F5)

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