Abstract
Quantum imaging can beat classical resolution limits, imposed by the diffraction of light. In particular, it is known that one can reduce the image blurring and increase the achievable resolution by illuminating an object by entangled light and measuring coincidences of photons. If an nphoton entangled state is used and the nthorder correlation function is measured, the pointspread function (PSF) effectively becomes \(\sqrt{n}\) times narrower relatively to classical coherent imaging. Quite surprisingly, measuring nphoton correlations is not the best choice if an nphoton entangled state is available. We show that for measuring (nāāā1)photon coincidences (thus, ignoring one of the available photons), PSF can be made even narrower. This observation paves a way for a strong conditional resolution enhancement by registering one of the photons outside the imaging area. We analyze the conditions necessary for the resolution increase and propose a practical scheme, suitable for observation and exploitation of the effect.
Introduction
Diffraction of light limits the spatial resolution of classical optical microscopes^{1,2} and hinders their applicability to life sciences at very small scales. Quite recently, a number of super resolving techniques, suitable for overcoming the classical limit, have been proposed. The approaches include, for example, stimulatedemission depletion microscopy^{3}, super resolving imaging based on fluctuations^{4} or antibunched light emission of fluorescence markers^{5}, structured illumination microscopy^{6,7}, and quantum imaging^{8,9,10,11,12}.
Quantum entanglement is known to be a powerful tool for resolution and visibility enhancement in quantum imaging and metrology^{8,9,10,11,12,13,14,15,16}. It has been shown that using n entangled photons and measuring the nth order correlations, one can effectively reduce the width of the pointspread function (PSF) \(\sqrt{n}\) times^{9,14,15,17} and beat the classical diffraction limit. The increase of the effect with the growth of n can naively be explained as summing up the "pieces of informationā carried by each photon when measuring their correlations. Such logic suggests that being given an nphoton entangled state, the intuitively most winning measurement strategy is to maximally exploit quantumness of the illuminating field and to measure the maximal available order of the photon correlations (i.e., the nth one).
Surprisingly, it is not always the case. First, it is worth mentioning that effective narrowing of the PSF and resolution enhancement can be achieved with classically correlated photons^{9} or even in the complete absence of correlations between fields emitted by different parts of the imaged object (as it is for the stochastic optical microscopy^{4}). Moreover, the maximal order of correlations is not necessarily the best one^{18,19}. In this contribution, we show that, for an entangled nphoton illuminating state, it is possible to surpass the measurement of all nphoton correlations by loosing a photon and measuring only nāāā1 remaining photons. According to our results, measurement of (nāāā1)thorder correlations effectively leads to \(\sqrt{2(n1)/n}\) times narrower PSF and better resolution, quantified by Fisher informationbased approach, than for commonly considered nphoton detection. It is even more strange in view of the notorious entanglement fragility^{20}: if even just one of the entangled photons is lost, the correlations tend to become classical.
The insight for understanding that seeming paradox can be gained from a wellestablished ghostimaging technique^{21,22,23,24,25,26,27,28} and from a more complicated approach of quantum imaging with undetected photons^{29,30,31,32}. In both approaches, one of the entangled photons gains information about the object and "transmitsā it through entanglement to the other photon, detected by a positionsensitive detector, by modifying its state. In our case, detecting nāāā1 photons and ignoring the remaining nth one effectively comprises two possibilities (see the imaging scheme depicted in Fig. 1): the nth photon can either fly relatively close to the optical axis of the imaging system towards the detector or go far from the optical axis and fail to pass through the aperture of the imaging system. In the first case, the photon can be successfully detected and provide us its piece of information. In the second case, it does not bring us the information itself, but effectively modifies the state of the remaining nāāā1 photons (as in refs.ā^{33,34,35,36,37,38,39,40}). It effectively produces positiondependent phase shift, thus performing wavefunction shaping^{37} and leading to an effect similar to structured illumination^{6,7}, PSF shaping^{41}, or linear interferometry measurement^{42}, and enhancing the resolution. We show that for nā>ā2 photons, the sensitivityenhancement effect leads to higher information gain than just detection of the nth photon, and measurement of (nāāā1)photon correlations surpasses nphoton detection. We refer to such (nāāā1)photon detection as a "lost photonā case since one of the photons is ignored during the measurement.
The discussed sensitivityenhancement effect can be used to increase resolution in practical imaging schemes. One can devise a conditional measurement setup by placing a bucket detector outside the normal pathway of the optical beam (e.g., near the lens outside of its aperture) and postselecting the outcomes when one photon gets to the bucket detector and the remaining nāāā1 ones successfully reach the positionsensitive detector, used for the coincidence measurements. We show that such a postselection scheme indeed leads to an additional increase of resolution relatively to (nāāā1)photon detection. This is again an nphoton detection technique, but now inspired by the "lost photonā considerations and being more efficient than the traditional measurement of the nth order correlations. Resolution enhancement by postselecting the more informative field configuration is closely related to the spatial mode demultiplexing technique^{43,44,45}. However, in our case, the selection of a more informative field part is performed by detection of a photon while all the remaining photons are detected in the usual way rather than by filtering the beam itself. Also, our technique bears some resemblance to the multiphoton ghostimaging^{23}.
Analysis of the PSF represents a fruitful approach for drawing useful conclusions based on our intuition, but, generally, it does not represent an accurate measure of resolution (see e.g., refs.ā^{18,19}). For drawing quantitative conclusions about the resolution enhancement of the proposed technique relatively to traditional measurements of n and (nāāā1)photon coincidences, we employ the Fisher information, which has already proved itself as a powerful tool for analysis of quantum imaging problems and for meaningful quantification of resolution^{18,19,41,43,44,46,47,48,49}. Our simulations show that for imaging a set of semitransparent slits (i.e., for multiparametric estimation problem), one indeed has a considerable increase in the information per measurement and the corresponding resolution enhancement. While the genuine demonstration of the discussed effects requires at least 3 entangled photons, which can be generated by a setup with complex nonlinear processes (e.g., cascaded spontaneous parametric downconversion (SPDC)^{50}, a combination of SPDC with upconversion^{51}, cascaded fourwave mixing^{52}, or the thirdorder SPDC^{53,54,55}), a relatively simple biphoton case is still suitable for observing resolution enhancement for a specific choice of the region where the nth photon (here, the second one) is detected.
Results
Imaging with entangled photons
We consider the following common model of a quantum imaging setup (Fig. 1). An object is described by a transmission amplitude A(s), where s is the vector of transverse position in the object plane. For simplicityās sake, we consider a commonly encountered case of an object with a realvalued transmission amplitude 0āā¤āA(s)āā¤ā1 (see e.g., refs.ā^{9,10,11,12,23,25,28}). It is illuminated by linearly polarized monochromatic light in an nphoton entangled quantum state
where a^{+}(k) and a^{+}(s) are the operators of photon creation in the mode with the transverse wavevector k and at transverse position s, respectively. An optical system with the PSF (Greenās function) h(s,ār) maps the object onto the image plane, where the field correlations are detected.
Features of the field passing through the analyzed object (and, thus, the object parameters) can be inferred from the measurement of intensity correlation functions accomplished by simple coincidence photocounting. The detection rate of the nphoton coincidence at a point r is determined by the value of the nthorder correlation function (see āMethodsā section for details):
The signal, described by Eq. (2), includes the nth power of the PSF, which is \(\sqrt{n}\) times narrower than the PSF itself. At least for the object of just two transparent pointlike pinholes, such narrowing yields \(\sqrt{n}\) times the better visual resolution of the object than for imaging with coherent light (see e.g., refs.ā^{8,15}).
Alternatively, one may try to ignore one of the photons and measure correlations of the remaining (nāāā1) ones. The rate of (nāāā1)photon coincidences is described by the (nāāā1)thorder correlation function:
Here, the 2(nāāā1)th power of the PSF is present. For nā>ā2, the resolution enhancement factor \(\sqrt{2(n1)}\) is larger than the factor \(\sqrt{n}\) achievable for nphoton detection. For nā=ā2, we get nā=ā2(nāāā1) and recover the common result (see also Fig. 2f below): imaging with biphotons yields practically the same effective width of PSF as one would have for incoherent (thermal light) imaging (however, biphotons are advantageous if one is interested in the phase information about the object)^{8,9,56,57}. Notice that the resolution for imaging with biphotons (nā=ā2) is better than for the standard coherent imaging with uncorrelated photons (nā=ā1)^{8,9,10,11}.
The result obtained looks quite counterintuitive: each photon carries some information about the illuminated object while discarding one of the photons leads to additional information gain. This seeming contradiction is just a consequence of applying classical intuition to the quantum dynamics of an entangled system. Due to quantum correlations, an entangled photon can affect the state of the remaining ones and increase their "informativityā even when it is lost without being detected. Here we show that in our imaging scheme such an enhancement by loss is indeed taking place. Moreover, an additional resolution increase can be achieved through conditioning by detecting the photon outside the aperture of the imaging system (see Fig. 1).
Effective state modification
Let us consider the change of the (nāāā1) photons state depending on the "fateā of the nth photon in more detail. We follow the approach discussed in ref.ā^{35}, which consists of splitting the description of an nphoton detection process into 1photon detection, density operator modification, and subsequent (nāāā1)photon detection for the modified density operator. If we detect (nāāā1)photon coincidence, the nth photon can be: (1) transmitted through the object into the imaging system aperture, (2) transmitted through the object outside the imaging system aperture, or (3) absorbed by the imaged object. Here, the "numberingā of photons has solely operational meaning: we are not aiming at distinguishing them, and the "nth photonā is not a particular photon, rather than just the last one remaining after nāāā1 photons have already been considered.
For the first possibility, the nth photon can reach the detector and potentially be registered at a certain point \({{{\bf{r}}}}^{\prime}\). The effective state of the remaining photons is (see āMethodsā section):
If we are interested in the detection of all the n photons (i.e., post select the cases when the nth photon is successfully detected at the position \({{{\bf{r}}}}^{\prime} ={{{\bf{r}}}}\)), the information gain due to the nth photon detection results from the factor h(s,ār) introduced into the effective state (Eq. 4) of the remaining photons. It forces the photons to pass through the particular part of the object, which is mapped onto the vicinity of the detection point r, and effectively reduces the image blurring.
If, according to the second possibility, the nth photon goes outside the aperture of the imaging system and has the transverse momentum component k, the affective state of the remaining photons is
An important feature of Eq. (5) is the factor e^{ikā s}, which effectively introduces the periodic phase modulation of the field, illuminating the object, and leads to a similar effect as intensity modulation for the structured illumination approach^{6,7}.
To take into account possible absorption of the nth photon by the imaged object, one can introduce an additional mode and model the object as a beamsplitter (see e.g., ref.ā^{29}). Similarly, to the two previously considered cases, the following expression can be derived for the effective (nāāā1)photon density operator:
By averaging over the three discussed possibilities (see āMethodsā section), one can obtain the following expression for the effective state of the remaining (nāāā1) photons:
which is a separable (nonentangled, classically correlated) one: a mixture of states with (nāāā1)photon excitations of spatial modes.
The detailed derivation of the result, while being quite trivial from a formal point of view, helps us to get to the following physical conclusions:

The effective state of nāāā1 photons (and the (nāāā1)th order correlation function) is modified, even if the nth photon is not detected by the observer, and depends on its "fateā (the way the photon actually passes). The "which pathā information is generated due to the photonās interaction with its surrounding (the object, the detector, etc.), but maybe unavailable to the observer unless the two conditions are satisfied: (i) the photon successfully gets to the detector and (ii) the observer is measuring nphoton coincidences instead of ignoring the nth photon.

The effective state of nāāā1 photons might be changed in a way, which provides the object resolution enhancement.

When nphoton coincidences are measured, the nth photon detection actually leads to the postselection due to the discarding of possibilities leading to the photon loss.

For nā>ā2, the advantage gained from registering more photon coincidences with the nth photon detection does not compensate for the information loss caused by discarding outcomes corresponding to the strongly modified (nāāā1)photon state, which is more sensitive to the object features (see Eqs. (2) and (3)).
Further, we discuss how the advantageous outcomes can be postselected, instead of being discarded, for resolution enhancement.
Model example
To gain a better understanding of the processes of resolution enhancement by a photon loss and postselection, let us consider a standard model object illuminated by an nphoton entangled state and consisting of two pinholes, which are separated by the distance 2d and positioned at the points d and ād (Fig. 2a). If the pinholes are small enough, the light passing through the object can be decomposed into just two field modes, corresponding to the spherical waves emerging from the two pinholes and further denoted by the indices "+ā and "āā for the upper and the lower pinhole, respectively. For simplicityās sake, we assume that the PSF is a realvalued function and that the light state directly after the object has the form of a NOONstate of the discussed modes "+ā and "āā:
The nthorder correlation signal contains separate contributions from the single pinholes and a crossterm, caused by constructive interference and leading to an additional blurring of the image (Fig. 2b):
While the scheme in Fig. 1 with the object from Fig. 2a may resemble the classical doubleslit interference experiment, it contains a lens, which effectively ensures nearfield imaging by compensating any phase difference introduced by the spatial separation of the considered pinholes. In the resulting image, the interference is governed by the phases of the input light only and remains constructive for any separation d if the light state is given by Eq. (8).
The (nāāā1)thorder correlations include only separate singlepinhole signals and produce a sharper image for nā>ā2 (Fig. 2c):
Let us interpret these results in terms of detecting nāāā1 photons conditioned by the nth photon detection. According to Eq. (4), if the photon is detected at the point r of the image plane, it transforms the state of the remaining photons into
The state coherence is preserved, while the blurring, caused by constructive interference, is slightly reduced due to certain "which pathā information provided by the nth photon detection.
If the nth photon is characterized by the transverse momentum k, \( {{{\bf{k}}}}\, > \,{k}_{\max }\), and does not get into the imaging system aperture, the effective modified state of the remaining photons is (see Eq. (5)):
Now, the phase shift between the two modes depends on k and can lead to destructive interference, which enhances the contrast of the image. For example, when kāā ādā=āĻ/2, one has maximally destructive interference and the detected signal is proportional to ā£h^{(nā1)}(d,ār)āāāh^{(nā1)}(ād,ār)ā£^{2} with 100% visibility of the gap between the two peaks (Fig. 2d). Such an advantageous situation can be postselected by placing an additional detector in the farfield outside the aperture in the direction k/ā£kā£ from the object.
Discarding the information about the nth photon (measuring G^{(nā1)}) corresponds to averaging over the possibilities to have the photon passing to the detector and missing it, and yields the following mixed state of the remaining photons:
I.e., the crossterms with constructive and destructive interference cancel each other, and the resulting mixed state allows for some resolution gain over the pure nth photon NOON state.
Application to quantum imaging
To illustrate the possible application of the ideas to practical quantum imaging, we consider an object represented by a set of semitransparent slits (Fig. 3a, c). The resolution of the modeled optical system is limited by diffraction at the lens aperture, which admits only the photons with the transverse momentum k not exceeding \({k}_{\max }\): \( {{{\bf{k}}}} \le {k}_{\max }\).
We compare the following three strategies: (i) measuring G^{(n)}(r) along the direction perpendicular to the slits (the signal is described by Eq. (2)); (ii) measuring G^{(nā1)}(r) at the same points (Eq. (3)); (iii) measuring coincidence signal G^{(nā1,ā1)}(r,āĪ©) of nāāā1 photons detected at the point r of the detection plane and the nth photon being anywhere in certain region Ī© outside the lens aperture. For the latter case, the signal can be written as
where
Integration in Eq. (15) corresponds to detection of the nth photon by a bucket detector, similarly to multiphoton ghost imaging^{23,24,25}. The difference is that the remaining nāāā1 photons do also pass through the investigated object before getting to the positionresolving detector in our case. The scheme can also be considered as a generalization of hybrid nearfield and farfield imaging when the entangled photons are analyzed partially in position space and partially in momentum space. Note, that G^{(nā1,ā1)} does not turn into G^{(n)} even in the limiting case when the region Ī© shrinks to a single point: the first case corresponds to farfield detection of the nth photon (i.e., Ī© defines a point in kspace, not a particular position r), while in the latter case all the n photons are localized in the position space by the detector.
Simulated images are shown in Fig. 3b, d. One can clearly see that (nāāā1)photon detection yields better visual resolution than measurement of the nthorder coincidences, while G^{(nā1,ā1)} provides additional contrast enhancement. While using a narrower region Ī© may additionally increase the contrast of the image and the information content per a single detection event (see Fig. 4 below), it also reduces the number of detected coincidences (see Fig. 5). On the other hand, even a relatively large bucket detector, used for the shown simulations, is sufficient for noticeable enhancement of resolution.
Of course, the effective narrowing of the PSF by measuring correlation functions does not necessarily mean a corresponding increase in precision of inferring of the analyzed parameters (positions of the object details, channel characteristics, etc.)^{18,19}. However, at least for certain imaging tasks (such as, for example, a cornerstone problem of finding a distance between two point sources in the farfield imaging), narrowing of the PSF can indeed lead to an increase of the informational content per measurement, and to the potentially unlimited resolution with increasing of n^{19}.
To describe the resolution enhancement in a quantitative and more consistent way, we employ Fisher information. Let the transmission amplitude of the object be decomposed as A(s)ā=āā_{Ī¼}Īø_{Ī¼}f_{Ī¼}(s), where the basis functions f_{Ī¼}(s) can represent e.g., slitlike pixels for the considered example^{47}. Then the problem of finding A(s) becomes equivalent to the reconstruction of the unknown decomposition coefficients Īø_{Ī¼}. If one has a certain signal S(r), sampled at the points {r_{i}}, Fisher information matrix (FIM)^{58,59}, normalized by a single detection event, can be introduced as
Here, the "detection eventā corresponds to the registration of a coincidence signal of n or nāāā1 photons (depending on the measurement type) within the specified time frame, rather than the detection of every single photon.
CramĆ©rRao inequality^{59,60} bounds the total reconstruction error (the sum of variances of the estimators for all the unknowns {Īø_{Ī¼}}) by the trace of the inverse of FIM:
where N is the number of registered coincidence events. When the size of the analyzed object features (e.g., the slit size d in Fig. 3a) tends to zero, the bound in Eq. (17) diverges (the effect is termed "Rayleighās curseā). The achievable resolution can, therefore, be determined by the feature size d, which \({{{\rm{Tr}}}}{F}^{1}\) starts growing rapidly with the decrease of d. A more rigorous definition can be given by specifying a certain reasonable threshold \({N}_{\max }\) for the maximal required number of registered coincidence events N (e.g., we take \({N}_{\max }=1{0}^{5}\) for further examples) and imposing the restriction \({{{\rm{Tr}}}}{F}^{1}\le {N}_{\max }\) (see āMethodsā section).
The dependence of the predicted reconstruction error on the normalized object scale d/d_{R} (where \({d}_{{{\mbox{R}}}}=3.83/{k}_{\max }\) is the classical Rayleigh limit for the considered optical system) is shown in Fig. 4. The sampling points for the signal are taken with the step d/2 along a line perpendicular to slits. As expected, ignoring nth photon and measuring (nāāā1)photon coincidences brings about larger achievable information per measurement, and correspondingly lower errors in object parameter estimation, yielding (10āĆ·ā20)% better resolution for nā=ā3 and 4. According to the theoretical predictions, for nā=ā2 no resolution increase is observed.
Also, our results confirm that the proposed hybrid scheme is indeed capable of increasing the resolution for nā=ā3 and 4 by conditioned detection of the nth photon. Additional information that can be gained relative to the measurement of (nāāā1)photon correlations is about (10āĆ·ā15)%. However, that gain vanishes in the regime of deep superresolution (dāā²ā0.2d_{R}): the black solid and red dotdashed lines intersect with the green dashed one for small d/d_{R} in Fig. 4. The reason for such behavior is that the effective phase shift, introduced by detection of the nth photon with the transverse momentum k outside of the aperture of the imaging system, becomes insufficient for the resolution enhancement for ā£kā£dāāŖā1.
For nā=ā2, the scheme also can give certain advantages (Fig. 4e, f), which, however, are not so prominent because they do not stem from the fundamental requirement of having a better resolution for G^{(nā1)} than for G^{(n)}. Still, taking into account the difficulties in the generation of 3photon entangled states^{50,51,52,53,54,55}, an experiment with biphotons can be proposed for initial tests of the approach.
The plots, shown in Fig. 4, represent information per single coincidence detection event. Therefore, certain concerns about the rates of such events may arise: waiting for a highly informative, but very rare event can be impractical. Fig. 5 shows the ratio of the overall detection probabilities p_{nā1,1}/p_{n}, where p_{nā1,1} corresponds to (nāāā1)photon coincidence, conditions by detection of a photon outside the aperture, and p_{n} describes the traditional measurement of nphoton coincidences. For a signal S(r_{i}), the overall detection probability is defined as pā=āā_{i}S(r_{i}) and represents the denominator of Eq. (16). When plotting Fig. 5, we do not include the measurement of G^{(nā1)} in the comparison, because the ratio of probabilities for (nāāā1) and nphoton detection events strongly depends on details of a particular experiment, such as the efficiency of the detectors.
The rate of (nāāā1)photon coincidences, conditioned by the detection of a photon outside of the aperture, is indeed 3āĆ·ā20 times smaller than the rate of nphoton coincidences. However, for the considered multiparametric problem, the "Rayleigh curseā leads to a very fast decrease of information when the slit size d becomes smaller than the actual resolution limit, and the effect of rate difference is almost negligible. For example, for nā=ā4, the object, shown in Fig. 3a, and the threshold \({{{\rm{Tr}}}}{F}^{1}\le 1{0}^{5}\), the minimal slit width d for successful resolution of transmittances equals 0.212d_{R} for the measurement of G^{(n)}, 0.170d_{R} for the measurement of G^{(nā1,ā1)} with the nth photon detected in the region \({{\Omega }}=\{{{{\bf{k}}}}:{k}_{\max }\le  {{{\bf{k}}}} \le 2{k}_{\max }\}\), and 0.177d_{R} for the same measurement of G^{(nā1,ā1)} when the reduced detection rate is taken into account. Notice, that all the mentioned values of the resolved feature size d are quite far beyond the classical resolution limit d_{R}.
At the first glance, the reported percentage of the resolution enhancement does not look very impressive or encouraging. However, one should keep in mind that the increase of the number of entangled photons from nā=ā2 to nā=ā3, while requiring significant experimental efforts, leads to the effective PSF narrowing just by 22% for the measurement of G^{(n)}. The transition from a 3photon entangled state to a 4photon one yields only 15% narrower PSF. Moreover, the actual resolution enhancement is typically smaller than the relative change of the PSF width^{18,19}, especially for high orders of the correlations, where it may saturate completely. The proposed approach provides a similar magnitude of the resolution increase on the cost of adding a bucket detector to the imaging scheme, which is much simpler than changing the number of entangled photons.
A similar concern about the soundness of the results may be elicited by recalling a simple problem of resolving twopoint sources, commonly investigated theoretically^{41,43,48}. For such a simple model situation, the error of inferring the distance d between the sources scales as Īdāāād^{ā1}N^{ā1/2}^{41}, where N is the number of detected events. Therefore, to resolve twice smaller separation of the two sources with the same error, one just needs to perform a 4 times longer experiment and collect 4N events. The situation becomes completely different when a more practical multiparametric problem is considered^{47}: the achievable resolution becomes practically insensitive to the data acquisition time (as soon as the number of detected events becomes sufficiently large). For example, for the situation described by the solid black line in Fig. 4a, a 100fold increase of the acquisition time leads to a 14% larger resolution. Moreover, as one can see from the doubledotdashed purple lines in Fig. 4b, c, d, traditional coherentlight imaging with intensity detection does not provide sufficient information about the object features in the superresolution regime (dāā²ā0.4d_{R}) even for 10^{8}times increase of the number of detected photons. That observation indicates that appropriately used entangled photons sources can outperform classical light sources, which are brighter even by many orders of magnitude. The effect is especially important for biological samples, vulnerable to photodestruction.
Discussions
We have demonstrated how to enhance the resolution of imaging with an nphoton entangled state by loosing a photon and measuring the (nāāā1)thorder correlation function instead of the nthorder coincidence signal. The resolution gain occurs despite the breaking of entanglement as a consequence of the photon loss. We have explained the effect in terms of the effective modification of the remaining photons state when one of the entangled photons is lost. Measurement of (nāāā1)photon coincidences for an nphoton entangled state not only discards some information carried by the ignored nth photon but also makes the resulting signal more informative in the considered imaging experiment. The latter effect prevails for nā>ā2 and leads to an increase of the information per measurement and to decrease of the lower bounds for the object inference errors.
The (nāāā1)photon detection represents a mixture of different possible outcomes for the discarded nth photon, including its successful detection at the image plane (resulting in the nphoton coincidence signal). The information per a single (nāāā1)photon detection event is averaged over the discussed possibilities and, for nā>ā2, is larger than the information for a single nphoton coincidence event. It means that certain outcomes for the nth photon provide more information per event than the average value, achieved for G^{(nā1)}. We prove that proposition constructively by proposing a hybrid measurement scheme, which provides resolution increase relatively to detection of (nāāā1)photon coincidences. Intentional detection of a photon outside the optical system, used for imaging of the object, introduces an additional phase shift and increases the sensitivity of the measurement performed with the remaining photons. Our simulations show that the effect can be observed even for nā=ā2, thus making its practical implementation much more realistic. We believe that our observation will pave a way for practical exploitation of entangled states by devising a super resolving imaging scheme conditioned on detecting photons not only successfully passing through the imaging system, but also those missing it.
Methods
Expressions for field correlation functions
For the imaging setup, shown in Fig. 1, the positivefrequency field operators E(r) at the detection plane are connected to the operators E_{0}(s) of the field illuminating the object as
The nthorder correlation function for the nphoton entangled state (Eq. 1) is calculated according to the following standard definition:
where \({E}^{()}({{{\bf{r}}}})={[{E}^{(+)}({{{\bf{r}}}})]}^{+}\) is the negativefrequency field operator. By substitution of Eq. (18) into Eq. (19), one can obtain the expression (Eq. 2) in the Results section.
The (nāāā1)thorder correlation function is calculated according to the expression
which yields Eq. (3) after substitution of Eq. (18).
Effective (nāāā1)photon state
The density operator, describing the effective (nāāā1)photon state averaged over the possible "fatesā of the nth photon, discussed in the main text, is
where the operators \({\rho }_{n1}^{(k)}\) are indexed according to the introduced possibilities and normalized in such a way that \({{{\rm{Tr}}}}{\rho }_{n1}^{(k)}\) is the probability of the kth "fateā.
According to the approach, discussed in ref.ā^{35}, detection of the nth photon at the position \({{{\bf{r}}}}^{\prime}\) of the detector effectively modifies the states of the remaining (nāāā1) photons in the following way:
Substitution of Eqs. (1) and (18) yields Eq. (4).
If we ignore the information about the position \({{{\bf{r}}}}^{\prime}\) of the photon detection, the contribution to the averaged density operator (Eq. 21) is
For the possibility, described by Eq. (5) and corresponding to the nth photon passage outside the aperture of the imaging system, the contribution to the averaged density operator (Eq. 21) is
where \({k}_{\max }\) is the maximal transverse momentum transferred by the optical system: \({k}_{\max }=kR/{s}_{o}\); k is the wavenumber of the light, R is the radius of the aperture, and s_{o} is the distance between the object and the lens used for imaging.
Calculating integrals in Eqs. (23), (24), and (6), and taking into account the connection between the PSF shape and \({k}_{\max }\) (see e.g., ref.ā^{8}), one can obtain Eq. (7) for the effective (nāāā1)photon state.
Model of the pointspread function
For the simulations, illustrated by Figs. 3, 4, and 5, we assume for simplicity that the magnification of the optical system is equal to 1, neglect the phase factor in PSF, and use the expression
where somb(x)ā=ā2J_{1}(x)/x, J_{1}(x) is the firstorder Bessel function and \({k}_{\max }\) is the maximal transverse momentum transferred by the optical system.
Quantification of resolution
Let us assume that a reasonable number of detected coincidence events N in a quantum imaging experiment is limited by the value \({N}_{\max }\) and the acceptable total reconstruction error (see Eq. (17)) is Ī^{2ā}ā¤ā1. Therefore, Eq. (17) implies the following threshold for the trace of the inverse of FIM:
Therefore, one can define the spatial resolution, achievable under the described experimental conditions, as the minimal feature size d, for which the condition (Eq. 26) is satisfied.
Data availability
The datasets generated and analyzed during this study are available from the corresponding author on reasonable request.
References
Abbe, E. BeitrĆ¤ge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung. Arch. fĆ¼r. Mikroskopische Anat. 9, 413ā468 (1873).
Rayleigh, L. XXXI. Investigations in optics, with special reference to the spectroscope. Lond. Edinb. Dublin Philos. Mag. J. Sci. 8, 261ā274 (1879).
Hell, S. W. & Wichmann, J. Breaking the diffraction resolution limit by stimulated emission: stimulatedemissiondepletion fluorescence microscopy. Opt. Lett. 19, 780ā782 (1994).
Dertinger, T., Colyer, R., Iyer, G., Weiss, S. & Enderlein, J. Fast, backgroundfree, 3D superresolution optical fluctuation imaging (SOFI). Proc. Natl Acad. Sci. USA 106, 22287ā22292 (2009).
Schwartz, O. et al. Superresolution microscopy with quantum emitters. Nano Lett. 13, 5832ā5836 (2013).
Classen, A., von Zanthier, M. O., Scully, J. & Agarwal, G. S. Superresolution via structured illumination quantum correlation microscopy. Optica 4, 580ā587 (2017).
Classen, A., von Zanthier, J. & Agarwal, G. S. Analysis of superresolution via 3D structured illumination intensity correlation microscopy. Opt. Express 26, 27492ā27503 (2018).
Shih, Y. An Introduction To Quantum Optics: Photon And Biphoton Physics (CRC press, 2018).
Giovannetti, V., Lloyd, S., Maccone, L. & Shapiro, J. H. SubRayleighdiffractionbound quantum imaging. Phys. Rev. A 79, 013827 (2009).
Xu, D.Q. et al. Experimental observation of subRayleigh quantum imaging with a twophoton entangled source. Appl. Phys. Lett. 106, 171104 (2015).
UnternĆ¤hrer, M., Bessire, B., Gasparini, L., Perenzoni, M. & Stefanov, A. Superresolution quantum imaging at the Heisenberg limit. Optica 5, 1150ā1154 (2018).
Toninelli, E. et al. Resolutionenhanced quantum imaging by centroid estimation of biphotons. Optica 6, 347ā353 (2019).
Boto, A. N. et al. Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit. Phys. Rev. Lett. 85, 2733ā2736 (2000).
Rozema, L. A. et al. Scalable spatial superresolution using entangled photons. Phys. Rev. Lett. 112, 223602 (2014).
Giovannetti, V. & Lloyd, L. M. S. Quantumenhanced measurements: beating the standard quantum limit. Science 306, 1330ā1336 (2004).
Polino, E., Valeri, M., Spagnolo, N. & Sciarrino, F. Photonic quantum metrology. AVS Quantum Sci. 2, 024703 (2020).
Tsang, M. Quantum imaging beyond the diffraction limit by optical centroid measurements. Phys. Rev. Lett. 102, 253601 (2009).
Pearce, M. E., Mehringer, T., von Zanthier, J. & Kok, P. Precision estimation of source dimensions from higherorder intensity correlations. Phys. Rev. A 92, 043831 (2015).
Vlasenko, S. et al. Optimal correlation order in superresolution optical fluctuation microscopy. Phys. Rev. A 102, 063507 (2020).
Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865ā942 (2009).
Pittman, T. B., Shih, Y. H., Strekalov, D. V. & Sergienko, A. V. Optical imaging by means of twophoton quantum entanglement. Phys. Rev. A 52, R3429āR3432 (1995).
Strekalov, D. V., Sergienko, A. V., Klyshko, D. N. & Shih, Y. H. Observation of twophoton "ghostā interference and diffraction. Phys. Rev. Lett. 74, 3600 (1995).
Agafonov, I. N., Chekhova, M. V., Iskhakov, T. S. & Wu, L.A. Highvisibility intensity interference and ghost imaging with pseudothermal light. J. Mod. Opt. 56, 422ā431 (2009).
Chan, K. W. C., OāSullivan, M. N. & Boyd, R. W. Highorder thermal ghost imaging. Opt. Lett. 34, 3343ā3345 (2009).
Chen, X.H. et al. Arbitraryorder lensless ghost imaging with thermal light. Opt. Lett. 35, 1166 (2010).
Bai, Y. & Han, S. Ghost imaging with thermal light by thirdorder correlation. Phys. Rev. A 76, 043828 (2007).
Erkmen, B. I. & Shapiro, J. H. Unified theory of ghost imaging with Gaussianstate light. Phys. Rev. A 77, 043809 (2008).
Moreau, P.A. et al. Resolution limits of quantum ghost imaging. Opt. Express 26, 7528 (2018).
Lemos, G. B. et al. Quantum imaging with undetected photons. Nature 512, 409ā412 (2014).
Lahiri, M., Lapkiewicz, R., Lemos, G. B. & Zeilinger, A. Theory of quantum imaging with undetected photons. Phys. Rev. A 92, 013832 (2015).
Wang, L. J., Zou, X. Y. & Mandel, L. Induced coherence without induced emission. Phys. Rev. A 44, 4614 (1991).
Zou, X. Y., Wang, L. J. & Mandel, L. Induced coherence and indistinguishability in optical interference. Phys. Rev. Lett. 67, 318 (1991).
Skornia, C., von Zanthier, J., Agarwal, G. S., Werner, E. & Walther, H. Nonclassical interference effects in the radiation from coherently driven uncorrelated atoms. Phys. Rev. A 64, 063801 (2001).
Thiel, C. et al. Quantum imaging with incoherent photons. Phys. Rev. Lett. 99, 133603 (2007).
Bhatti, D., Classen, A., Oppel, S., Schneider, R. & von Zanthier, J. Generation of N00Nlike interferences with two thermal light sources. Eur. Phys. J. D. 72, 191 (2018).
Cabrillo, C., Cirac, J. I., GarciaFernandez, P. & Zoller, P. Creation of entangled states of distant atoms by interference. Phys. Rev. A 59, 1025 (1999).
Brainis, E. Quantum imaging with Nphoton states in position space. Opt. Express 19, 24228ā24240 (2011).
Opatrny`, T., Kurizki, G. & Welsch, D.G. Improvement on teleportation of continuous variables by photon subtraction via conditional measurement. Phys. Rev. A 61, 032302 (2000).
Olivares, S., Paris, M. G. A. & Bonifacio, R. Teleportation improvement by inconclusive photon subtraction. Phys. Rev. A 67, 032314 (2003).
Ourjoumtsev, A., Dantan, A., TualleBrouri, R. & Grangier, P. Increasing entanglement between Gaussian states by coherent photon subtraction. Phys. Rev. Lett. 98, 030502 (2007).
PaĆŗr, M. et al. Tempering Rayleighās curse with PSF shaping. Optica 5, 1177ā1180 (2018).
Lupo, C., Huang, Z. & Kok, P. Quantum limits to incoherent imaging are achieved by linear interferometry. Phys. Rev. Lett. 124, 080503 (2020).
Tsang, M., Nair, R. & Lu, X.M. Quantum theory of superresolution for two incoherent optical point sources. Phys. Rev. X 6, 031033 (2016).
Tsang, M. Subdiffraction incoherent optical imaging via spatialmode demultiplexing. N. J. Phys. 19, 023054 (2017).
Len, Y. L., Datta, C., Parniak, M. & Banaszek, K. Resolution limits of spatial mode demultiplexing with noisy detection. Int. J. Quantum Inf. 18, 1941015 (2020).
Motka, L. et al. Optical resolution from Fisher information. Eur. Phys. J. 131, 130 (2016).
Mikhalychev, A. B. et al. Efficiently reconstructing compound objects by quantum imaging with higherorder correlation functions. Comm. Phys. 2, 134 (2019).
PaĆŗr, M. et al. Reading out Fisher information from the zeros of the point spread function. Opt. Lett. 44, 3114ā3117 (2019).
Datta, C. et al. SubRayleigh resolution of two incoherent sources by array homodyning. Phys. Rev. A 102, 063526 (2020).
HĆ¼bel, H. et al. Direct generation of photon triplets using cascaded photonpair sources. Nature 466, 601ā603 (2010).
Keller, T. E., Rubin, M. H., Shih, Y. & Wu, L.A. Theory of the threephoton entangled state. Phys. Rev. A 57, 2076 (1998).
Wen, J., Oh, E. & Du, S. Tripartite entanglement generation via fourwave mixings: narrowband triphoton w state. J. Opt. Soc. Am. B 27, A11āA20 (2010).
Corona, M., GarayPalmett, K. & UāRen, A. B. Experimental proposal for the generation of entangled photon triplets by thirdorder spontaneous parametric downconversion in optical fibers. Opt. Lett. 36, 190ā192 (2011).
Corona, M., GarayPalmett, K. & UāRen, A. B. Thirdorder spontaneous parametric downconversion in thin optical fibers as a photontriplet source. Phys. Rev. A 84, 033823 (2011).
Borshchevskaya, N. A., Katamadze, K. G., Kulik, S. P. & Fedorov, M. V. Threephoton generation by means of thirdorder spontaneous parametric downconversion in bulk crystals. Laser Phys. Lett. 12, 115404 (2015).
Abouraddy, A. F., Saleh, B. E. A., Sergienko, A. V. & Teich, M. C. Entangledphoton fourier optics. J. Opt. Soc. Am. B 19, 1174ā1184 (2002).
Saleh, B. E. A., Teich, M. C. & Sergienko, A. V. Wolf equations for twophoton light. Phys. Rev. Lett. 94, 223601 (2005).
Fisher, R. A. Theory of statistical estimation. In Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 22, 700ā725 (Cambridge University Press, 1925).
Rao, C. R. Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81ā91 (1945).
Cramer, H. Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946).
Acknowledgements
All the authors acknowledge financial support from the King Abdullah University of Science and Technology (grant 4264.01), A.B.M., I.L.K., and D.S.M. also acknowledge support from the EU Flagship on Quantum Technologies, project PhoG (820365).
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The theory was conceived by A.B.M. and D.S.M. Numerical calculations were performed by A.B.M. The project was supervised by D.S.M., D.L.M., and A.B.M. All the authors participated in the manuscript preparation, discussions, and checks of the results.
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Mikhalychev, A.B., Novik, P.I., Karuseichyk, I.L. et al. Lost photon enhances superresolution. npj Quantum Inf 7, 125 (2021). https://doi.org/10.1038/s41534021004654
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DOI: https://doi.org/10.1038/s41534021004654