Abstract
We analyze the fundamental quantum limit of the resolution of an optical imaging system from the perspective of the detection problem of deciding whether the optical field in the image plane is generated by one incoherent onaxis source with brightness \(\epsilon\) or by two \(\epsilon {\mathrm{/}}2\)brightness incoherent sources that are symmetrically disposed about the optical axis. Using the exact thermalstate model of the field, we derive the quantum Chernoff bound for the detection problem, which specifies the optimum rate of decay of the error probability with increasing number of collected photons that is allowed by quantum mechanics. We then show that recently proposed linearoptic schemes approach the quantum Chernoff bound—the method of binary spatialmode demultiplexing (BSPADE) is quantumoptimal for all values of separation, while a method using image inversion interferometry (SLIVER) is nearoptimal for subRayleigh separations. We then simplify our model using a lowbrightness approximation that is very accurate for optical microscopy and astronomy, derive quantum Chernoff bounds conditional on the number of photons detected, and show the optimality of our schemes in this conditional detection paradigm. For comparison, we analytically demonstrate the superior scaling of the Chernoff bound for our schemes with source separation relative to that of spatially resolved direct imaging. Our schemes have the advantages over the quantumoptimal (Helstrom) measurement in that they do not involve joint measurements over multiple modes, and that they do not require the angular separation for the twosource hypothesis to be given a priori and can offer that information as a bonus in the event of a successful detection.
Introduction
The influential Rayleigh criterion for imaging resolution,^{1} which specifies a minimum separation for two incoherent light sources to be distinguishable by a given imaging system, is based on heuristic notions. As pointed out by Feynman [ref. ^{2}, Sec. 30–4]: “Rayleigh’s criterion is a rough idea in the first place…” and a better resolution can be achieved “… if sufficiently careful measurements of the exact intensity distribution over the diffracted image spot can be made…” The fundamental measurement noise is the quantum noise necessarily accompanying any measurement. A more rigorous approach to the resolution measure that accounts for the quantum noise in ideal spatially resolved imageplane photon counting can be formulated using the classical CramérRao bound on the minimum estimation error for locating the sources.^{3,4} Very recently, using methods of quantum estimation theory,^{5,6} it was found that the estimation of the separation between two incoherent sources below the Rayleigh criterion can be drastically improved by measurements employing predetection linearoptic processing of the collected light, followed by photon counting.^{7,8,9,10,11,12,13,14,15,16,17,18,19,20}
Besides the minimum error of estimating the separation of two point sources, the resolving power of an imaging system can also be studied via the paradigmatic detection problem of deciding whether the optical field in the image plane is generated by one source or two sources.^{21,22,23,24,25} This detection perspective is especially relevant to the detection of binary stars and exoplanets^{23,26} and the detection of protein multimers with fluorescence microscopes.^{27} In a pioneering work,^{22} Helstrom obtained the mathematical description of the quantumoptimal measurement that minimizes the error probability for detecting one or two point sources emitting quasimonochromatic thermal light. Unfortunately, in addition to having no known physical realization, his method requires the separation between the two hypothetical sources to be given, though this separation is usually unknown in practice.
Here we investigate the performance of two practical quantum measurements for the detection of weak incoherent quasimonochromatic point light sources. We assess the performance of these measurements visavis direct imaging and the optimum quantum measurement using the asymptotic error exponent (or Chernoff exponent), which specifies the rate at which the error probability decreases exponentially as the observation time or number of received photons increases. We show that a binary spatialmode demultiplexing (BSPADE) scheme^{7} is quantumoptimal for all values of separations in the following two senses: (1) the asymptotic error exponent attains the maximum allowed by quantum mechanics, and (2) the error probability of a simple decision rule based on the observations of the BSPADE is close to the quantum limit. We also show that the scheme of superlocalization by image inversion interferometry (SLIVER)^{8,10} is nearoptimal for subRayleigh separations. The Chernoff exponents of both schemes are shown to be superior to that of ideal shotnoiselimited continuum direct imaging in the subRayleigh regime. In addition to their superiority over direct imaging, our methods do not require the capability to perform joint quantum measurements, do not require the twosource separation to be known a priori, can offer an accurate estimate of this separation in the event of a successful detection,^{7,8,9,10,11,12,13,14} and rely on methods that have been experimentally demonstrated in the context of parameter estimation.^{17,18,19,20} These advantages over the Helstrom measurement^{22} hold tremendous promise for practical detection applications in both astronomy^{23} and molecular imaging.^{27}
Results
One source versus two sources
The setup considered in this work is illustrated in Fig. 1. Under hypothesis H_{1}, we have a single thermal source of brightness (average photon number per temporal mode) \(\epsilon\) imaged at the origin of the image plane. Under hypothesis H_{2}, we have two thermal sources, each of strength \(\epsilon {\mathrm{/}}2\), located a distance d apart and imaged at the points ±d/2 = (±d/2, 0) in the image plane. To focus on the resolution power of an optical imaging system, the total brightness is assumed to be identical under the two hypotheses so that simple photon counting is ineffective as a decision strategy. Similarly, the sources are presumed to have identical frequency spectra so that spectroscopy cannot help to distinguish the hypotheses. A strategy for accepting one or the other hypothesis, known as a decision rule, is given by partitioning the space of observations \({\cal Z}\) (which is determined by the choice of measurement) into two disjoint regions \({\cal Z}_{\mathrm{1}}\) and \({\cal Z}_{\mathrm{2}}\); the onesource hypothesis H_{1} is accepted if the observation belongs to \({\cal Z}_{\mathrm{1}}\), and H_{2} is accepted otherwise. The performed quantum measurement can be described by a positiveoperatorvalued measure (POVM) \(\{ E(z)\} _{z \in {\cal Z}}\), where z denotes the outcome, and the E(z)’s are nonnegative operators resolving the identity operator as \({\int} {\kern 1pt} {\mathrm{d}}\mu (z)E(z) = 1\) with μ(z) being an appropriate measure on \({\cal Z}\).^{5,6,28} Define \(E_1 = {\int}_{z \in {\cal Z}_1} {\kern 1pt} {\mathrm{d}}\mu (z)E(z)\) and \(E_2 = {\int}_{z \in {\cal Z}_2} {\kern 1pt} {\mathrm{d}}\mu (z)E(z)\). Let ρ_{1} and ρ_{2} be the density operators for the fields arriving at the image plane per temporal mode under H_{1} and H_{2}, respectively. Assuming a flat emission spectrum over the bandwidth W, the probabilities of the falsealarm (accepting H_{2} when H_{1} is true) and miss (accepting H_{1} when H_{2} is true) errors for onesourceversustwo testing are given by \(\alpha \equiv {\mathrm{Tr}}\left( {E_2\rho _1^{ \otimes M}} \right)\) and \(\beta \equiv {\mathrm{Tr}}\left( {E_1\rho _2^{ \otimes M}} \right)\), respectively, where \(M \simeq WT\) with T being the observation time is the number of available temporal modes (also called the sample size). Assuming prior probabilities p_{1} and p_{2} for the respective hypotheses, the average probability of error is
which is widely used to assess the performance of a quantum decision strategy constituted by a quantum measurement and a classical decision rule.^{5} The minimum error probability optimized over all quantum decision strategies is given by the Helstrom formula^{5}
where \(\left\ A \right\_1 \equiv {\mathrm{Tr}}\sqrt {A^\dagger A}\) is the trace norm. The minimum error probability can be achieved by the Helstrom–Holevo test in which E_{2} is taken to be the projector onto the eigen subspace of \(p_2\rho _2^{ \otimes M}  p_1\rho _1^{ \otimes M}\) with positive eigenvalues.^{5,29} We refer to this optimal measurement as the Helstrom measurement henceforth.
While the Helstrom formula Eq. (2) allows exact computation of the optimum error probability in principle, it is difficult to physically implement the Helstrom measurement for several reasons. Firstly, the optimal measurement is a joint one over multiple samples.^{28} Secondly, this measurement depends on the separation between the two hypothetic point sources, which is often unknown in the first place. Lastly, the optimal measurement in general depends on the ratio of the prior probabilities of the two hypotheses, whose determination is often subjective. To circumvent these difficulties, we study the performance of two realizable measurements: BSPADE^{7} and SLIVER,^{8} originally introduced in the context of estimating the separation between two closelyspaced incoherent point sources.
BSPADE
Spatialmode demultiplexing refers to spatially separating the imageplane optical field into its components in any chosen set of orthogonal spatial modes.^{7} The binary version of spatialmode demultiplexing, BSPADE, uses a device that separates a specific spatial mode from all other modes orthogonal to it, and on–off detectors (that can only distinguish between zero and one or more photons) are placed at the two output ports. In our setup, the selected spatial mode is chosen to be that generated by the point source at the origin of the object plane. Such a separation of modes is always possible in principle for any given pointspread function (PSF), and various linearoptics schemes can be envisaged to realize it.^{30,31,32}
SLIVER
The second practical measurement we consider is SLIVER, which separates the optical field at the image plane into its symmetric and antisymmetric components with respect to inversion at the origin, followed by on–off photon detection in the respective ports.^{8} Here, we assume that the PSF is reflectionsymmetric in the yaxis, i.e., ψ(−x, y) = ψ(x, y), and consider a modified SLIVER for which the inversion operation is replaced by the reflection operation about the yaxis—this modification corresponds to the PixSLIVER scheme of ref. ^{10} with singlepixel (bucket) on–off detectors at the two outputs. For simplicity, we refer to this modified version as SLIVER henceforth. All photodetectors in both BSPADE and SLIVER are assumed free from dark counts, or at least that the darkcount rate is so far below the signalcount rate as to be negligible.
Asymptotic error (Chernoff) exponents
In realistic imaging situations, we usually deal with a large sample size \(M \gg 1\), which motivates using the asymptotic error exponent as a useful metric for comparing the performance of different measurement schemes against the Helstrom measurement. For any specific quantum measurement performed on each sample, it is known that the minimum error probability \(P_{{\mathrm{e}},{\mathrm{min}}}^{({\mathrm{meas}})}\) over all decision rules decreases exponentially in M as \(P_{{\mathrm{e}},{\mathrm{min}}}^{({\mathrm{meas}})}\sim {\mathrm{exp}}\left( {  M\xi ^{\left( {{\mathrm{meas}}} \right)}} \right)\). The asymptotic error exponent \(\xi ^{({\mathrm{meas}})} =  {\mathrm{lim}}_{M \to \infty }{\textstyle{1 \over M}}{\mathrm{log}}\,P_{{\mathrm{e}},{\mathrm{min}}}^{({\mathrm{meas}})}\) can be given by the Chernoff exponent (also known as Chernoff information or Chernoff distance),^{33,34,35} namely,
where Λ_{j}(z) = Tr[E(z)ρ_{j}] is the probability of obtaining the outcome z under the hypothesis H_{j}, and {E(z)} is the POVM for the measurement. On the other hand, the error probability P_{e,min} of the optimum quantum measurement (which is in general a joint measurement on the M samples) scales with the exponent ξ known as the quantum Chernoff exponent, which is given by:^{36,37,38,39,40}
Note that ξ is independent of the measurement and ξ ≥ ξ^{(meas)} holds for any measurement.
To calculate the Chernoff exponent, we need to know the characteristics of the imaging system. Without essential loss of generality, we suppose that the imaging system is spatially invariant and of unit magnification^{41} and is described by its 2D amplitude PSF ψ(r), where r = (x, y) is the transverse coordinate in the image plane \({\cal I}\). We take the PSF to be normalized, i.e., \({\int}_{\cal I} {\kern 1pt} {\mathrm{d}}x{\mathrm{d}}y\left {\psi (x,y)} \right^2 = 1\). For thermal sources, we show that the exact Chernoff exponents ξ^{(B−SPADE)} and ξ^{(SLIVER)} for BSPADE and SLIVER respectively and the quantum Chernoff exponent ξ are given by (see the Methods)
where the ddependent quantities
are defined in terms of the overlap function of the PSF for displacements along the x axis:
Moreover, we here assume that the overlap function (and hence \(\epsilon _ \pm\) and μ) is realvalued. This assumption is satisfied for inversionsymmetric PSFs, i.e., ψ(x, y) = ψ(−x, −y), and yaxis reflectionsymmetric PSFs, i.e., ψ(x, y) = ψ(−x, y).^{10,11}
It can be seen from Eq. (5) that the Chernoff exponent of the BSPADE is always equal to the quantum Chernoff exponent, meaning that BSPADE is asymptotically optimal. For SLIVER, the Chernoff exponent is in general not quantumoptimal but is close to quantumoptimal in the subRayleigh regime of small d, where μ is close to unity.
We consider three typical kinds of PSFs, corresponding to Gaussian apertures, rectangular hard apertures, and circular hard apertures. The PSFs can be, respectively, written as
where sinc(x) ≡ sin(x)/x, jinc(x) ≡ 2J_{1}(x)/x, and J_{1}(x) is the Bessel function of the first kind. The “characteristic lengths” σ, σ_{x}, σ_{y}, and σ_{c} are related to the features of apertures as follows. For a Gaussian aperture, which is commonly assumed in fluorescence microscopy,^{4} we have σ = λ/2πNA with λ being the freespace center wavelength and NA the effective numerical aperture of the system. For a D_{x} × D_{y} rectangular aperture, the characteristic length along the x and y directions are given by σ_{x} = λF/πD_{x} and σ_{y} = λF/πD_{y}, respectively, where F is the distance between the aperture plane and the image plane in a unit magnification system. For a Ddiameter circular hard aperture, we have σ_{c} = λF/πD. After some algebra, the overlap functions can be shown to be
using which the Chernoff exponents can be readily obtained.
We plot in Fig. 2 the Chernoff exponents of BSPADE and SLIVER for the above three PSFs in Eq. (9). We can see that in the subRayleigh regime the Chernoff exponents are insensitive to which PSF is used.
Weaksource model
To compare the performance of BSPADE and SLIVER with that of direct imaging, we introduce the weaksource model. In most applications in optical microscopy and astronomy, the source brightness \(\epsilon \ll 1\) photons per temporal mode.^{7,42,43,44} Then, ρ_{1} and ρ_{2} may be considered with high accuracy to be confined to the subspace consisting of zero or one photons, i.e.,
for i = 1, 2, where \(\left {{\mathrm{vac}}} \right\rangle\) denotes the vacuum state and η_{i} are the corresponding onephoton states obtained by neglecting \(O\left( {\epsilon ^2} \right)\) terms. This approximation enables us to simplify the theory in comparison with ref. ^{22} and still obtain similar results. Moreover, it will enable us to treat the case in which the sources in the twosource hypothesis have different brightnesses while demonstrating the advantages of BSPADE and SLIVER over direct imaging. Denote the brightnesses of the two point sources under H_{2} by \(\epsilon _1\) and \(\epsilon _2\), satisfying \(\epsilon _1 + \epsilon _2 = \epsilon\). The onephoton state for two hypothetical sources can be expressed as
where \(\left {x,y} \right\rangle\) are the onephoton Dirac kets satisfying 〈x, yx′, y′〉 = δ(x − x′)δ(y − y′), and \({\int}_{\cal I} {\kern 1pt} {\mathrm{d}}x{\mathrm{d}}y\left {x,y} \right\rangle \left\langle {x,y} \right = 1_1\) with 1_{1} being the identity operator on the onephoton subspace in the image plane field.^{7} We then have η_{1} = η(0) and η_{2} = η(d).
For the specific cases of rectangular and circular hard apertures, Helstrom has derived expressions for the minimum error probability for thermal light sources.^{22} However, these expressions are very complicated. Our weaksource model allows us to simplify the minimum error probability to
Here, \(\left( {\begin{array}{*{20}{c}} M \\ L \end{array}} \right)(1  \epsilon )^{M  L}\epsilon ^L\) is the probability of L photons arriving at the imaging plane and P_{e,minL} is the minimum probability of error conditioned on detecting L photons in the image plane. The form of Eq. (13) is due to the fact that the distinguishability between ρ_{1} and ρ_{2} lies in the onephoton sector and the zerophoton event is uninformative. It is implicitly assumed in Eq. (13) that the source flux is low enough that the on–off detectors’ recovery time is short compared to the average interarrival time of the photons. Either the conditional error probability of Eq. (14) or the unconditional one of Eq. (13) can be used as a figure of merit, depending on whether or not the number of the photons arriving at the image plane is measured. Helstrom in ref. ^{22} took the latter approach, and the performance was studied with respect to the average total number of photons \(N = M\epsilon\) detected over the observation interval. On the other hand, in fluorescence microscopy it is common practice to compare the performance of imaging schemes for the same number of detected photons L.^{3,4}
We can define a conditional Chernoff exponent \(\xi _{\mathrm{c}}^{({\mathrm{meas}})}\) satisfying \(P_{{\mathrm{e}},{\mathrm{min}}L}^{({\mathrm{meas}})}\sim {\mathrm{exp}}\left( {  L\xi _{\mathrm{c}}^{({\mathrm{meas}})}} \right)\), which is given by Eq. (3) with Λ_{j}(z) replaced by the probability of measurement outcomes conditioned on a photon being detected, i.e., Λ_{j}(z) = Tr[E(z)η_{j}]. Similarly, the optimum conditional error probability P_{e,minL} decays exponentially with L multiplied by the conditional quantum Chernoff exponent given by \(\xi _{\mathrm{c}} \equiv  {\mathrm{log}}\,{\mathrm{min}}_{0 \le s \le 1}{\mathrm{Tr}}\left( {\eta _1^s\eta _2^{1  s}} \right)\). It follows from Eq. (13) that the (unconditional) Chernoff exponents can be obtained via the relation \(e^{  \xi } = 1  \epsilon + \epsilon e^{  \xi _{\mathrm{c}}}\) and \(e^{  \xi ^{({\mathrm{meas}})}} = 1  \epsilon + \epsilon e^{  \xi _{\mathrm{c}}^{({\mathrm{meas}})}}\). This implies that the (unconditional) Chernoff exponent is monotonically increasing with the conditional one. Particularly, we have \(\xi ^{{\mathrm{meas}}} \simeq \varepsilon \xi _{\mathrm{c}}^{{\mathrm{meas}}}\) when \(\xi ^{{\mathrm{meas}}} \ll 1\). Therefore, we can use either ξ^{meas} or \(\xi _{\mathrm{c}}^{{\mathrm{meas}}}\) to compare the performance of quantum measurements.
The conditional Chernoff exponents are readily calculated in the weaksource model using Eq. (12):
Note that as d decreases, the SLIVER result converges to the BSPADE result, which can also be seen in Fig. 3.
The SLIVER measurement is independent of the PSF, while the BSPADE is adapted according to the PSF. It is remarkable that in the context of estimating the separation of two point optical source, Rehacek et al.^{13} showed there exist complete sets of modes that are optimal for any real symmetric PSF; nevertheless, the PSFadapted optimal measurement behaves much better than others when fewer modes are measured. In this work, with only two photon detectors, the PSFadapted BSPADE is optimal, while the SLIVER is suboptimal. It remains open whether there exists a complete set of modes that is optimal for the hypothesistesting scenario considered here, regardless of the details of the PSF. However, similarly to the estimation scenario, the performance of the SLIVER can be definitely enhanced when more modes in the symmetric port are sorted out and measured. This is because the quantum Chernoff exponent is monotonically nondecreasing under quantum operation,^{40} so that sorting out more modes to measure must result a Chernoff exponent not less than that for the SLIVER.
Direct imaging
Direct imaging (DI) using a chargecoupled device (CCD) camera is a standard detection technique in microscopy and telescopy.^{4} To compare our schemes to direct imaging, we make the conservative assumption of an ideal noiseless photodetector with infinite spatial resolution and unity quantum efficiency placed in the image plane. In the weaksource model, and conditional on a photon being detected in a given temporal mode, the observation consists of its position of arrival \((x,y) \in {\cal I}\). Using Eq. (12), the resulting probability densities for the observation are Λ_{1}(x, y) = ϒ(x, y;0) and Λ_{2}(x, y) = ϒ(x, y;d) under H_{1} and H_{2}, respectively, where
We show in the Methods that the conditional Chernoff exponent for ideal DI in the weaksource model scales as d^{4} in the interesting regime of small d:
where ϒ^{(n)}(x, y;d) denotes the nth order partial derivative of ϒ(x, y;d) with respect to d, and \({\cal I}^{\prime} {\equiv} \{ (x,y){\Upsilon} (x,y;0) {>} {0}\}\). In contrast, the conditional Chernoff exponents of BSPADE and SLIVER in the weaksource model are of order d^{2}, which can be seen by using Eqs. (15) and (16) and ∂δ(d)/∂d_{d=0} = 0.
The conditional Chernoff exponents for different measurements in the case of the Gaussian PSF are given in Table 1. Here, we have used a Taylor series expansion of Eq. (16) in d for SLIVER, and used Eq. (18) for estimating the Chernoff exponent for direct imaging. The characteristic scalings with respect to d of the conditional quantum Chernoff exponent and that of the three measurement schemes are shown in Fig. 3. We see that the Chernoff exponent of SLIVER agrees with the quantum limit for all practical purposes in the subRayleigh regime d ≤ 1.
Decision rule
In order to choose a hypothesis based on a sequence of BSPADE/SLIVER observations, we need to fix a decision rule. If the separation d is known, the optimal decision rule is given by the likelihoodratio test:^{34} For a given observation record (z_{1}, z_{2}, …, z_{M}), we choose H_{2} if \(\mathop {\prod}\nolimits_{j = 1}^M {\kern 1pt} {\mathrm{\Lambda }}_2\left( {z_j} \right){\mathrm{/\Lambda }}_1\left( {z_j} \right) > p_1{\mathrm{/}}p_2\), where p_{1} and p_{2} are the prior probabilities of H_{1} and H_{2}, respectively, and choose H_{1} otherwise. If the separation is unknown, one can use the generalizedlikelihoodratio test,^{45} which first estimates the separation and then does the likelihoodratio test with the estimated value.
Here, we propose a simplified decision rule that does not require the separation to be known or estimated. Observe that if the detector corresponding to the modes orthogonal to the first mode in the threemode basis (see Eqs. (23)–(25) in Methods) clicked for any sample, we can infer with certainty (in either source model) that two point sources are present, i.e., H_{2} is true. The simplified decision rule is then given by accepting H_{1} only if this detector does not click during the entire observation period.
From Table 2 in the Methods, under the simplified decision rule, the falsealarm probability for M samples is clearly α = 0 for both the BSPADE and SLIVER measurement. The miss error probability is the probability that the detector corresponding to H_{2} does not click, i.e.,
where Λ_{2}(·,·) is the probability of the measurement outcome under H_{2}. It then can shown that β^{(meas)} = exp(−Mξ^{(meas)}) for both the BSPADE and SLIVER measurement.
Discussion
We have examined the problem of discriminating one thermal source from two closely separated ones for a given diffractionlimited imaging system. Using the exact thermal state of the imageplane field, we have derived the quantum Chernoff exponent for the detection problem. We also have used the weaksource model of the imageplane field, which is very accurate in the optical regime due to the low brightness of a thermal source in each temporal mode, to obtain simple expressions for the Chernoff exponent. The persample BSPADE measurement that separates light in the PSF mode from the rest of the field was shown to attain the quantumoptimal Chernoff exponent for all values of twosource separation. Remarkably, it does so without the need for prior knowledge of the value of d, joint measurement over multiple modes, or photonnumber resolution in each mode. These properties are not shared by the quantumoptimal measurement elucidated by Helstrom,^{22} which is not a structured receiver. These advantages also adhere to the SLIVER measurement, which is nearquantumoptimal in the subRayleigh regime. Moreover, the experimental design of SLIVER is independent of the particular (reflectionsymmetric) PSF of the imaging system.
In fact, the simplified decision rules proposed here for BSPADE and SLIVER do not require resolving the arrival time of the detected photon or photons. To wit, only a single on–off detector without temporal resolution placed in the output corresponding to the modes orthogonal to the PSF (for BSPADE) or to the antisymmetric component (for SLIVER) is sufficient for achieving the error probability behavior derived here. Hypothesis H_{2} is accepted if and only if this detector clicks at any time during the observation period. If we need to simultaneously know the conditional error probability, then at least two photonnumberresolving detectors (or gated on–off detectors with sufficient temporal resolution) are required such that the total number of the photons arriving on the image plane can be obtained from the observation.
Although sophisticated optical microscopy techniques can help resolve multiple sources better than direct imaging,^{46} the manipulation of the source emission that they require is impossible in astronomical imaging for which the dominant detection technique is direct imaging. Our proof that the linearoptics schemes proposed here can yield Chernoff exponents that are orders of magnitude larger than that of direct imaging, coupled with the rapid recent experimental progress on similar schemes,^{17,18,19,20} holds out great promise for applications in astronomy and molecular imaging analysis in the near future.
Some remarks on the assumptions used in this work deserve to be mentioned here. First, the assumption of unit magnification of the imaging system is solely for omitting an irrelevant magnification factor in mathematical expressions and thus does not limit the validity of our result.^{41} Second, we assume the two sources has the same frequency spectra so that we can focus on the resolution power of an imaging system; otherwise, spectroscopy can also be used to distinguish the hypotheses. Third, we assume the brightness of the two sources are identical. For different brightnesses, calculating the quantum Chernoff exponent will become much more difficult, as the symmetric and antisymmetric field are no longer statistically independent, see Methods. However, with the weaksource approximation, we have showed that the Chernoff exponents of the BSPADE and the SLIVER, as well as their quantum limit, are the same as that for two equalstrength sources, if the optical axis of apparatus is perfectly aligned to the average position of the two point sources. Last, we assume that the optic axis of the devices are perfectly aligned such that the sources are symmetrically located about the optic axis. This may be the biggest potential issue for the practical application of the BSPADE/SLIVER scheme. Some of us have investigated a twostep hybrid scheme, where some of the collected photons are used for the centroid estimation by direct imaging, followed by the BSPADE/SLIVER measurement for the hypothesis testing. A similar scheme was studied for the scenario of estimating the separation of two sources.^{7,8} Such a hybrid configuration is sensible, considered from the following three aspects of direct imaging: (i) it is easy to implement; (ii) it is known to achieve a good centroid estimate; and (iii) it may utilize prior images, e.g., those generated by previous astroobservation. The preliminary simulations, which will be published in elsewhere, show that the hybrid scheme still gives a substantial enhancement for hypothesis testing if half of the photons are employed for centroid estimation and alignment. Besides these assumptions, another possible source of imperfection is dark counts in the photodetectors; this may affect the performance of the BSPADE and SLIVER schemes, especially those using the simplified decision rules. To improve the robustness against dark counts or extraneous background light, we may use feedback strategies, like those developed in the context of distinguishing between optical coherent states.^{47,48}
The quantum Chernoff method used in this work can be possibly extended to multiple sources. For multiple point sources, our method can be generalized in two ways, depending on testing binary or multiple hypotheses. In the first way, we can investigate the resolution power by testing whether there is a weak point light source between two strong point light sources or not. In such a case, the binary hypothesis testing and the Chernoff bound are still applicable. In the second way, we can consider the case where a set of light sources can have several possible configurations and we need to decide which configuration it is. For this problem, we must resort to the multiple Chernoff distance derived in the ref. ^{49}.
Methods
Density operators
To calculate the Chernoff exponent and error probabilities, we need to express the density operators ρ_{1} and ρ_{2} in an appropriate basis. We focus on a single temporal mode χ(t) of the imageplane field satisfying \({\int}_0^T \left {\chi (t)} \right^2{\mathrm{d}}t = 1\) on the observation interval [0, T]. The two mutually incoherent sources at ±d/2 under H_{2} are described by statistically independent zeromean circular complex Gaussian amplitudes A_{1} and A_{2} with the probability density
where \(\epsilon _1\) and \(\epsilon _2\) are the brightnesses of the two point sources. The imageplane field conditioned on (A_{1}, A_{2}) is described quantummechanically as a coherentstate, i.e., an eigenstate \(\left {\phi _{A_1,A_2}} \right\rangle\) of the positivefrequency field operator \(\hat E^{( + )}({\bf{r}},t)\) in the image plane: \(\hat E^{( + )}({\bf{r}},t)\left {\phi _{A_1,A_2}} \right\rangle = {\cal E}_{A_1,A_2}({\bf{r}})\chi (t)\left {\phi _{A_1,A_2}} \right\rangle\), where \({\cal E}_{A_1,A_2}({\bf{r}})\) is given by
and ψ(r) is the normalized PSF. The density operator under H_{2} is formally given by
Note that ρ_{2} depends on the separation d and is reduced to ρ_{1} when setting d = 0. Moreover, it can be seen from Eq. (21) that under H_{1} we have \({\cal E}_{A_1,A_2}({\bf{r}}) = \left( {A_1 + A_2} \right)\psi ({\bf{r}})\). Thus, it is evident that, while the relevant coherent states are defined on a complete set of transversespatial modes on \({\cal I}\), only three orthonormal modes are in excited (nonvacuum) states. These may be chosen as
where λ_{±} ≡ (1 ± δ(d))/2 and μ is given by Eq. (7). Using {ϕ_{1}(r), ϕ_{2}(r), ϕ_{3}(r)} as a spatialmode basis, we have
where \(A_ + = \sqrt {\lambda _ + } \left( {A_1 + A_2} \right)\) and \(A_  = \sqrt {\lambda _  } \left( {A_1  A_2} \right)\) are two complex random variables. When \(\epsilon _1 = \epsilon _2 = \epsilon {\mathrm{/}}2\), the random variables A_{+} and A_{−} are statistically independent.^{8} In such a case, we get
where \(\rho _{{\mathrm{th}}}(\epsilon ) = \mathop {\sum}\nolimits_n \left[ {\epsilon ^n{\mathrm{/}}(\epsilon + 1)^{n + 1}} \right]\left n \right\rangle \left\langle n \right\) is the singlemode thermal state with \(\epsilon\) average photons,^{50,51} U is a unitary beamsplitter transformation with transmissivity μ acting on the first two modes. The ddependent quantities \(\epsilon _ \pm\) and μ are given by Eq. (7). The transmissivity μ takes values in the range [−1, 1] and equals unit when d = 0. The beamsplitter implements the transformation
for input coherent states \(\left \alpha \right\rangle\) and \(\left \beta \right\rangle\), while for a number statevacuum input \(\left n \right\rangle \left 0 \right\rangle\), we have
Quantum Chernoff exponent
The quantum Chernoff exponent is given by \(\xi =  {\mathrm{log}}\,{\mathrm{min}}_{0 \le s \le 1}Q_s\) with \(Q_s \equiv {\mathrm{Tr}}\left( {\rho _1^s\rho _2^{1  s}} \right)\). Using Eqs. (27), (28) and (30), we get after some algebra:
where the coefficients are
It follows from \(0 \le \epsilon _ \pm \le \epsilon\) and \(\epsilon _ + + \epsilon _  = \epsilon\) that p ≤ 1 and q ≥ 1. Thus, Q_{s} takes its minimum at s = 0, i.e.,
from which we obtain the quantum Chernoff exponent in Eq. (5).
BSPADE and SLIVER
To calculate the Chernoff exponents as well as the error probabilities for BSPADE and SLIVER, it will be convenient to only focus on the effective action of the measurement on the relevant Hilbert subspace. Figure 4 illustrates the effective actions of BSPADE and SLIVER on the mode subspace spanned by ϕ_{1}(r), ϕ_{2}(r), and ϕ_{3}(r). The BSPADE measurement discriminates the first mode from the other two, while the SLIVER measurement discriminates the first two modes from the third, which is the sole excited antisymmetric mode. We emphasize that our schemes are very different from a detector that resolves each of the modes ϕ_{1}(r), ϕ_{2}(r), and ϕ_{3}(r), since implementing such a detector would require knowledge of d. Our schemes, on the other hand, work for any d. Using Eqs. (27) and (28) and the effective action of the measurements shown in Fig. 4, the probability distribution of measurement outcomes can be easily obtained as given in Table 2, on which the calculation of the Chernoff exponents is based.
For BSPADE, we get
where a and p are given in Eq. (32), \(\tilde b = \mu ^2\epsilon _ + {\mathrm{/}}\left( {1 + \epsilon _ +  \mu ^2\epsilon _ + } \right)\), and \(\tilde q = \epsilon \left( {1 + \epsilon _ +  \mu ^2\epsilon _ + } \right){\mathrm{/}}\mu ^2\epsilon _ + \ge 1\). It follows that Q_{s} is minimized over [0, 1] by taking s = 0, leading to the result of ξ^{(B−SPADE)} in Eq. (5). In the weaksource model, the probability distribution of a detected photon being at the two output ports is {1, 0} under H_{1} and {δ(d/2)^{2}, 1 − δ(d/2)^{2}} under H_{2}. Thus, one can easily obtain the result of \(\xi _{\mathrm{c}}^{({\mathrm{B}}  {\mathrm{SPADE}})}\) as shown in Eq. (15).
For SLIVER, the structure of ρ_{1} and ρ_{2} implies that the two detectors fire independently under both hypotheses. From these equations, the Chernoff exponent of SLIVER can be calculated (and corresponds to s = 0 as for BSPADE) with the result of Eq. (6). In the weaksource model, the probability distribution of a detected photon being at the two output ports is {1, 0} under H_{1} and {[δ(d) + 1]/2, [1 − δ(d)]/2} under H_{2}. Thus, one can easily obtain the result of \(\xi _{\mathrm{c}}^{({\mathrm{SLIVER}})}\) as shown in Eq. (16).
Leading term of Chernoff exponent
For a given measurement scheme, let ϒ(z;d) be the resulting probability density of a measurement outcome z, where d is the distance between the two hypothetic point sources. The Chernoff exponent of Eq. (3) for testing H_{2} (d > 0) against H_{1} (d = 0) can be written as \(\xi (d) =  {\mathrm{log}}\,{\mathrm{min}}_{0 \le s \le 1}Q_s(d)\) with \(Q_s(d) \equiv {\int} {\kern 1pt} {\mathrm{d}}\mu (z)\Upsilon (z;0)^{1  s}\Upsilon (z;d)^s\). Let us now focus on the leading term of ξ(d) for small separations d ≈ 0, where the optimal measurement performs much better than direct imaging. We expand Q_{s}(d) in a Taylor series as
where \({\cal I}^\prime \equiv \{ z\Upsilon (z;{0}) > {0}\}\). These coefficients Q_{s,k} are independent of d. Although in our model the separation d is nonnegative, the PSFs in Eq. (9) can be easily extended to real numbers and meanwhile assured to be smooth at d = 0. It then follows from Eq. (12) that ϒ(z;d) = ϒ(z; −d) and thus all odd derivatives of ϒ(z;d) with respect to d at d = 0 vanish for an arbitrary z. As a result, we have
where g^{(k)} denotes the kth derivatives of \(g(d) {\equiv} {\int}_{{\cal I}^{\prime} } {\kern 1pt} {\mathrm{d}}{\mu} (z){\Upsilon} (z;d)\) and
For the ideal direct imaging scenario in the weaksource model, the measurement outcome z is the coordinates of a detected photon in the image plane, i.e., \(z = (x,y) \in {\cal I}\). Suppose that the two point sources are aligned along the x axis, ϒ(z;d) is then given by Eq. (17) with z = (x, y). For all three typical kinds of PSFs considered in this work, we have g(d) = 1 for direct imaging. In such a case, the leading ddependent term in the Taylor series of Q_{s}(d) is of fourth order. It then follows that \(\xi _{\mathrm{c}}^{({\mathrm{DI}})} \simeq \frac{{d^4}}{{32}}{\cal K}\), which is Eq. (18). On the other hand, for BSPADE and SLIVER, it can be seen from Table 2 that g(d) = Λ_{2}(off, off) + Λ_{2}(on, off), so that g^{(2)}(0) is nonzero and the leading term in Q_{s}(d) is secondorder in d.
Data availability
Additinal data and analysis files are available from the authors on request.
References
Lord Rayleigh, F. R. S. Xxxi. investigations in optics, with special reference to the spectroscope. Philos. Mag. Ser. 5 8, 261–274 (1879).
Feynman, R., Leighton, R. & Sands, M. The Feynman Lectures on Physics:Volume I (AddisonWesley, Reading, MA, 1963).
Ram, S., Ward, E. S. & Ober, R. J. Beyond Rayleigh’s criterion: a resolution measure with application to singlemolecule microscopy. Proc. Natl Acad. Sci. USA 103, 4457–4462 (2006).
Chao, J., Ward, E. S. & Ober, R. J. Fisher information theory for parameter estimation in single molecule microscopy: tutorial. J. Opt. Soc. Am. A 33, B36–B57 (2016).
Helstrom, C. W. Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
Holevo, A. S. Probabilistic and Statistical Aspects of Quantum Theory (NorthHolland, Amsterdam, 1982).
Tsang, M., Nair, R. & Lu, X.M. Quantum theory of superresolution for two incoherent optical point sources. Phys. Rev. X 6, 031033 (2016).
Nair, R. & Tsang, M. Interferometric superlocalization of two incoherent optical point sources. Opt. Express 24, 3684–3701 (2016).
Tsang, M., Nair, R. & Lu, X.M. Quantum information for semiclassical optics. In Proc. SPIE ; Quantum and Nonlinear Optics IV (eds Gong, Q., Guo, G. & Ham, B.) Vol. 10029, 1002903 (2016).
Nair, R. & Tsang, M. Farfield superresolution of thermal electromagnetic sources at the quantum limit. Phys. Rev. Lett. 117, 190801 (2016).
Ang, S. Z., Nair, R. & Tsang, M. Quantum limit for twodimensional resolution of two incoherent optical point sources. Phys. Rev. A. 95, 063847 (2017).
Lupo, C. & Pirandola, S. Ultimate precision bound of quantum and subwavelength imaging. Phys. Rev. Lett. 117, 190802 (2016).
Rehacek, J., Paúr, M., Stoklasa, B., Hradil, Z. & SánchezSoto, L. L. Optimal measurements for resolution beyond the Rayleigh limit. Opt. Lett. 42, 231–234 (2017).
Tsang, M. Subdiffraction incoherent optical imaging via spatialmode demultiplexing. New J. Phys. 19, 023054 (2017).
Kerviche, R., Guha, S. & Ashok, A. Fundamental limit of resolving two point sources limited by an arbitrary point spread function. In 2017 IEEE International Symposium on Information Theory (ISIT) 441–445 (2017).
Yang, F., Nair, R., Tsang, M., Simon, C. & Lvovsky, A. I. Fisher information for farfield linear optical superresolution via homodyne or heterodyne detection in a higherorder local oscillator mode. Phys. Rev. A 96, 063829 (2017).
Tang, Z. S., Durak, K. & Ling, A. Faulttolerant and finiteerror localization for point emitters within the diffraction limit. Opt. Express 24, 22004–22012 (2016).
Yang, F., Taschilina, A., Moiseev, E. S., Simon, C. & Lvovsky, A. I. Farfield linear optical superresolution via heterodyne detection in a higherorder local oscillator mode. Optica 3, 1148–1152 (2016).
Tham, W.K., Ferretti, H. & Steinberg, A. M. Beating Rayleigh’s curse by imaging using phase information. Phys. Rev. Lett. 118, 070801 (2017).
Paúr, M., Stoklasa, B., Hradil, Z., SánchezSoto, L. L. & Rehacek, J. Achieving the ultimate optical resolution. Optica 3, 1144–1147 (2016).
Harris, J. L. Resolving power and decision theory. J. Opt. Soc. Am. 54, 606–611 (1964).
Helstrom, C. Resolution of point sources of light as analyzed by quantum detection theory. IEEE Trans. Inform. Theory 19, 389–398 (1973).
Acuna, C. O. & Horowitz, J. A statistical approach to the resolution of point sources. J. Appl. Stat. 24, 421–436 (1997).
Shahram, M. & Milanfar, P. Statistical and informationtheoretic analysis of resolution in imaging. IEEE Trans. Inform. Theor. 52, 3411–3437 (2006).
Dutton, Z., Shapiro, J. H. & Guha, S. Ladar resolution improvement using receivers enhanced with squeezedvacuum injection and phasesensitive amplification. J. Opt. Soc. Am. B 27, A63–A72 (2010).
Labeyrie, A., Lipson, S. G. & Nisenson, P. An Introduction to Optical Stellar Interferometry (Cambridge University Press, Cambridge, 2006).
Nan, X. et al. Singlemolecule superresolution imaging allows quantitative analysis of raf multimer formation and signaling. Proc. Natl Acad. Sci. USA 110, 18519–18524 (2013).
Hayashi, M. Quantum Information: An Introduction 1 edn (SpringerVerlag, Berlin/Heidelberg, 2006).
Holevo, A. Statistical decision theory for quantum systems. J. Multivar. Anal. 3, 337–394 (1973).
Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, P. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61 (1994).
Morizur, J.F. et al. Programmable unitary spatial mode manipulation. J. Opt. Soc. Am. A 27, 2524–2531 (2010).
Miller, D. A. B. Reconfigurable adddrop multiplexer for spatial modes. Opt. Express 21, 20220–20229 (2013).
Chernoff, H. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952).
Van Trees, H. L., Bell, K. L. & Tian, Z. Detection, Estimation, and Modulation Theory, Part I 2nd edn (Wiley, New York, NY, 2013).
Cover, T. & Thomas, J. Elements of Information Theory 2nd edn (Wiley, New York, NY, 2006).
Ogawa, T. & Hayashi, M. On error exponents in quantum hypothesis testing. IEEE Trans. Inf. Theory 50, 1368–1372 (2004).
Kargin, V. On the Chernoff bound for efficiency of quantum hypothesis testing. Ann. Stat. 33, 959–976 (2005).
Audenaert, K. M. R. et al. Discriminating states: the quantum Chernoff bound. Phys. Rev. Lett. 98, 160501 (2007).
Nussbaum, M. & Szkoła, A. The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Stat. 37, 1040–1057 (2009).
Audenaert, K., Nussbaum, M., Szkoła, A. & Verstraete, F. Asymptotic error rates in quantum hypothesis testing. Commun. Math. Phys. 279, 251–283 (2008).
Goodman, J. W. Introduction to Fourier Optics 3rd edn (Roberts and Company Publishers, Englewood, CO, 2005).
Goodman, J. W. Statistical Optics (John Wiley & Sons, New York, NY, 1985).
Gottesman, D., Jennewein, T. & Croke, S. Longerbaseline telescopes using quantum repeaters. Phys. Rev. Lett. 109, 070503 (2012).
Tsang, M. Quantum nonlocality in weakthermallight interferometry. Phys. Rev. Lett. 107, 270402 (2011).
Kay, S. M. Fundamentals of Statistical Signal Processing, Volume II: Detection Theory 1st edn (Prentice Hall, Upper Saddle River, NJ, 1998).
Weisenburger, S. & Sandoghdar, V. Light microscopy: an ongoing contemporary revolution. Contemp. Phys. 56, 123–143 (2015).
Dolinar, S. J. An optimum receiver for the binary coherent state quantum channel. MIT Res. Lab. Electron. Quart. Progr. Rep. 111, 115–120 (1973).
Geremia, J. Distinguishing between optical coherent states with imperfect detection. Phys. Rev. A 70, 062303 (2004).
Li, K. Discriminating quantum states: the multiple Chernoff distance. Ann. Stat. 44, 1661–1679 (2016).
Mandel, L. & Wolf, E. Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
Shapiro, J. H. The quantum theory of optical communications. IEEE J. Sel. Top. Quant. Elect. 15, 1547–1569 (2009).
Krovi, H., Guha, S. & Shapiro, J. H. Attaining the quantum limit of passive imaging. Preprint at arXiv:1609.00684 (2016).
Lu, X.M., Nair, R. & Tsang, M. Quantumoptimal detection of oneversustwo incoherent sources with arbitrary separation. Preprint at arXiv:1609.03025 (2016).
Acknowledgements
We thank Mankei Tsang for several useful discussions. This work was supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A050003, the National Natural Science Foundation of China under Grant Nos. 61871162 and 11805048, the Singapore National Research Foundation under NRF Grant No. NRFNRFF201107, the Singapore Ministry of Education Academic Research Fund Tier 1 Project R263000C06112, the Defense Advanced Research Projects Agency’s (DARPA) Information in a Photon (InPho) program under Contract No. HR001110C0159, the REVEAL and EXTREME Imaging program, and the Air Force Office of Scientific Research under Grant No. FA95501410052.
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X.M.L. performed the weaksource model analysis. H.K., S.G., J.H.S., and R.N. performed the thermalstate analysis. R.N. performed the SLIVER analyses. X.M.L. and R.N. wrote the manuscript. All the authors discussed extensively during the course of this work. This paper extends and unifies preliminary work in the preprints^{52} (thermalstate model) and^{53} (weaksource model).
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Lu, XM., Krovi, H., Nair, R. et al. Quantumoptimal detection of oneversustwo incoherent optical sources with arbitrary separation. npj Quantum Inf 4, 64 (2018). https://doi.org/10.1038/s415340180114y
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DOI: https://doi.org/10.1038/s415340180114y
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