Lost Photon Enhances Superresolution

Lost photon enhances superresolution

A. Mikhalychev,1, ∗ P. Novik,1 I. Karuseichyk,1 D. A. Lyakhov,2 D. L. Michels,2 and D. Mogilevtsev1 1B.I.Stepanov Institute of Physics, NAS of Belarus, Nezavisimosti ave. 68, 220072 Minsk, Belarus 2Computer, Electrical and Mathematical Science and Engineering Division, 4700 King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia (Dated: January 18, 2021) Quantum imaging can beat classical resolution limits, imposed by 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 n-photon entangled state is used and the √nth-order correlation function is measured, the point-spread function (PSF) effectively becomes n times narrower relatively to classical coherent imaging. Quite surprisingly, measuring n-photon correlations is not the best choice if an n-photon entangled state is available. We show that for measuring (n − 1)-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.

Diffraction of light limits the spatial resolution of clas- only n − 1 remaining photons. According to our results, sical optical microscopes [1, 2] and hinders their applica- measurement of (n − 1)th-order correlations effectively bility to life sciences at very small scales. Quite recently, leads to p2(n − 1)/n times narrower PSF than for com- a number of superresolving techniques, suitable for over- monly considered n-photon detection. It is even more coming the classical limit, have been proposed. The ap- strange in view of the notorious entanglement fragility proaches include, for example, stimulated-emission deple- [20]: if even just one of the entangled photons is lost, the tion microscopy [3], superresolving imaging based on fluc- correlations tend to become classical. tuations [4] or antibunched light emission of fluorescence The insight for understanding that seeming paradox markers [5], structured illumination microscopy [6, 7], can be gained from a well-established ghost-imaging tech- and quantum imaging [8–10]. nique [14–16, 21–25] and from a more complicated ap- Quantum entanglement is known to be a powerful tool proach of quantum imaging with undetected photons [26– for resolution and visibility enhancement in quantum 29]. In our case, detecting n − 1 photons and ignoring imaging and metrology [8–16]. It has been shown that the remaining nth one effectively comprises two possi- using n entangled photons and measuring the n-th order bilities (see the imaging scheme depicted in Fig. 1): the correlations, one can effectively√ reduce the width of the nth photon can either fly relatively close to the optical point-spread function (PSF) n times [9, 12, 13, 17] and axis of the imaging system towards the detector or go far beat the classical diffraction limit. The increase of the from the optical axis and fail to pass through the aper- effect with the growth of n can naively be explained as ture of the imaging system. In the first case, the photon summing up the “pieces of information” carried by each can be successfully detected and provide us its piece of photon when measuring their correlations. Such logic information. In the second case, it does not bring us the suggests that, being given an n-photon entangled state, information itself, but effectively modifies the state of the the intuitively most winning measurement strategy is to remaining n − 1 photons (as in Refs. [26, 27, 29–31]). It maximally exploit quantumness of the illuminating field effectively produces position-dependent phase shift, thus and to measure the maximal available order of the photon performing wave-function shaping [31] and leading to an correlations (i.e. the nth one). effect similar to structured illumination [6, 7], PSF shap- Surprisingly, it is not always the case. First, it is worth ing [32], or linear interferometry measurement [33], and arXiv:2101.05849v1 [quant-ph] 14 Jan 2021 mentioning that effective narrowing of the PSF and reso- enhancing the resolution. We show that for n > 2 pho- lution enhancement can be achieved with classically cor- tons, the sensitivity-enhancement effect leads to higher related photons [9] or even in complete absence of correla- information gain than just detection of the nth photon, tions between fields emitted by different parts of the im- and measurement of (n−1)-photon correlations surpasses aged object (as it is for the stochastic optical microscopy n-photon detection. [4]). Moreover, the maximal order of correlations is not The discussed sensitivity-enhancement effect can be necessarily the best one [18, 19]. In this contribution, used for increase of resolution in practical imaging we show that, for an entangled n-photon illuminating schemes. One can devise a conditional measurement set- state, it is possible to surpass the measurement of all n- up by placing a bucket detector outside the normal path- photon correlations by loosing a photon and measuring way of the optical beam (e.g. near the lens outside of its aperture) and post-selecting the outcomes when one photon gets to the bucket detector and the remaining n − 1 ∗ [email protected] ones successfully reach the position-sensitive detector, 2 used for the coincidence measurements. We show that E (s) h(s,r) E(r) such post-selection scheme indeed leads to additional in- 0 crease of resolution relatively to (n−1)-photon detection. Resolution enhancement by post-selecting the more informative field configuration is closely related to the spatial s r mode demultiplexing technique [34, 35]. However, in our Source OP Optical system IP case the selection of more informative field part is performed by detection of a photon while all the remaining FIG. 1. General scheme of an imaging setup. See the text for photons are detected in the usual way rather than by fil- details. An image of an object placed at the object plane OP tering the beam itself. Also, our technique bears some is formed by the optical system at the image plane IP. resemblance to the multi-photon ghost-imaging [14]. For drawing quantitative conclusions about the resolution enhancement of the proposed technique relatively accomplished by simple coincidence photo-counting. The to traditional measurements of n and (n − 1)-photon detection rate of the n-photon coincidence at a point ~r coincidences, we employ the Fisher information, which is determined by the value of the nth-order correlation has already proved itself as a powerful tool for analysis function (see Methods for details): of quantum imaging problems and for meaningful quan- Z 2 (n) 2 n n tification of resolution [18, 19, 32, 34–38]. Our simu- G (~r) ∝ d ~sA (~s)h (~s,~r) . (2) lations show that for imaging a set of semi-transparent slits (i.e., for multi-parametric estimation problem), one indeed has a considerable increase in the information per The signal, described by√ Eq. (2), includes the nth power measurement, and the corresponding resolution enhance- of the PSF, which is n times narrower than the PSF ment. While genuine demonstration of the discussed ef- itself. At least for the object of just two√ transparent fects requires at least 3 entangled photons, which can be point-like pinholes, such narrowing yields n times bet- generated by a setup with complex nonlinear processes ter visual resolution of the object than for imaging with (e.g. cascaded spontaneous parametric down-conversion coherent light (see e.g. Refs. [8, 13]). (SPDC) [39], combination of SPDC with up-conversion Alternatively, one may try to ignore one of the photons [40], cascaded four-wave mixing [41], or the third-order and measure correlations of the remaining (n − 1) ones. SPDC [42–44]), a relatively simple biphoton case is still The rate of (n − 1)-photon coincidences is described by suitable for observing resolution enhancement for a spe- the (n − 1)th-order correlation function: cific choice of the region where the nth photon (here, the Z second one) is detected. G(n−1)(~r) ∝ d2~s |A(~s)|2(n−1) |h(~s,~r)|2(n−1) . (3)

Here, the 2(n−1)th power of the PSF is present. For n > I. RESULTS p 2, the resolution√ enhancement factor 2(n − 1) is larger than the factor n achievable for n-photon detection. A. Imaging with entangled photons The result obtained looks quite counter-intuitive: each photon carries some information about the illuminated We consider the following common model of a quan- object, while discarding one of the photons leads to ad- tum imaging setup (Fig. 1). An object is described by ditional information gain. This seeming contradiction is a transmission amplitude A(~s), where ~s is the vector of just a consequence of applying classical intuition to quan- transverse position in the object plane. It is illuminated tum dynamics of an entangled system. Due to quantum by linearly polarized light in an n-photon entangled quan- correlations, an entangled photon can affect the state tum state of the remaining ones and increase their “informativity” Z even when it is lost without being detected [26–29]. Here 2~ 2~ (2) ~ ~ |Ψni ∝ d k1 ··· d knδ (k1 + ··· + kn) we show that in our imaging scheme such an enhancement by loss is indeed taking place. Moreover, an addi- Z n tional resolution increase can be achieved through condi- × a+(~k ) ··· a+(~k )|0i ∝ d2~s a+(~s) |0i, (1) 1 n tioning by detecting the photon outside the aperture of the imaging system (see Fig. 1). where a+(~k) and a+(~s) are the operators of photon creation in the mode with the wavevector ~k and at position ~s respectively. An optical system with the PSF h(~s,~r) B. Effective state modification maps the object onto the image plane, where the field correlations are detected. Let us consider the change of the (n − 1) photons state Features of the field passing through the analyzed ob- depending on the “fate” of the nth photon in more de- ject (and, thus, the object parameters) can be inferred tails. We follow the approach discussed in Ref. [29], from the measurement of intensity correlation functions which consists in splitting the description of an n-photon 3 detection process into 1-photon detection, density opera- • The effective state of n − 1 photons (and the (n − tor modification, and subsequent (n−1)-photon detection 1)th order correlation function) is modified, even if for the modified density operator. If we detect (n − 1)- the nth photon is not detected, and depends on its photon coincidence, the nth photon can be: (1) transmit- “fate” (the way the photon actually passes). ted through the object into the imaging system aperture, (2) transmitted through the object outside the imaging • The effective state of (n − 1)-photons might be system aperture, or (3) absorbed by the imaged object. changed in a way, which provides the object res- For the first possibility, the nth photon can reach the olution enhancement. detector and potentially be registered at certain point ~r0. The effective state of the remaining photons is (see • When n-photon coincidences are measured, the nth Methods): photon detection actually leads to the postselection due to discard of possibilities leading to the photon Z (1) 0 2 0 + n−1 loss. |Ψn−1(~r )i ∝ d ~sA(~s)h(~s,~r )(a (~s)) |0i. (4)

If we are interested in detection of all the n photons • For n > 2, the advantage gained from registering (i.e. postselect the cases when the nth photon is success- more photon coincidences with the nth photon de- fully detected at the position ~r0 = ~r), the information tection does not compensate for the information gain due to the nth photon detection results from the loss caused by discarding outcomes corresponding factor h(~s,~r) introduced into the effective state (4) of the to the strongly modified (n−1)-photon state, which remaining photons. It forces the photons to pass through is more sensitive to the object features (see Eqs. (2) the particular part of the object, which is mapped onto and (3)). the vicinity of the detection point ~r, and effectively re- Further, we discuss how the advantageous outcomes duces the image blurring. can be postselected, instead of being discarded, for reso- If, according to the second possibility, the nth pho- lution enhancement. ton goes outside the aperture of the imaging system and has the transverse momentum component ~k, the effective state of the remaining photons is C. Model example Z (2) ~ 2 i~k·~s + n−1 |Ψn−1(k)i ∝ d ~sA(~s)e (a (~s)) |0i. (5) To gain better understanding of the processes of reso-

i~k·~s lution enhancement by a photon loss and postselection, An important feature of Eq. (5) is the factor e , let us consider a standard model object illuminated by an which effectively introduces the periodic phase modula- n-photon entangled state and consisting of two pinholes, tion of the field, illuminating the object, and leads to a which are separated by the distance 2d and positioned at similar effect as intensity modulation for the structured the points d~ and −d~ (Fig. 2a). If the pinholes are small illumination approach [6, 7]. enough, the light passing through the object can be de- To take into account possible absorption of the nth composed into just two field modes, corresponding to the photon by the imaged object, one can introduce an addi- spherical waves emerging from the two pinholes and fur- tional mode and model the object as a beamsplitter (see ther denoted by the indices “+” and “−” for the upper e.g. Ref. [45]). Similarly, to the two previously consid- and the lower pinhole respectively. For simplicity’s sake, ered cases, the following expression can be derived for we assume that the PSF is a real-valued function, and the effective (n − 1)-photon density operator: that the light state directly after the object has the form Z (3) 2 2 + n−1 n−1 of a NOON-state of the discussed modes “+” and “−”: ρn−1 = d ~s[1 − |A(~s)| ](a (~s)) |0ih0|(a(~s)) . (6) |Φni ∝ |ni+|0i− + |0i+|ni−. (8) By averaging over the three discussed possibilities (see Methods), one can obtain the following expression for the The nth-order correlation signal contains separate con- effective state of the remaining (n − 1) photons: tributions from the single pinholes and a cross-term, Z caused by constructive interference and leading to ad- 2 + n−1 n−1 ρn−1 = d ~s(a (~s)) |0ih0|(a(~s)) , (7) ditional blurring of the image (Fig. 2b): which indicates the well-known effect of turning an n- G(n)(~r) ∝ h2n(d,~ ~r) + h2n(−d,~ ~r) + 2hn(d,~ ~r)hn(−d,~ ~r). photon entangled state into an (n − 1)-photon classically (9) correlated state when one of the photons is excluded from The (n − 1)th-order correlations include only sepa- consideration. rate single-pinhole signals and produce a sharper image The detailed derivation of the result, while being quite (Fig. 2c): trivial from formal point of view, helps us to get to the following physical conclusions: G(n−1)(~r) ∝ h2(n−1)(d,~ ~r) + h2(n−1)(−d,~ ~r). (10) 4

G(n)(x) the remaining photons: (a) (b) 1.0

d 0.8 ρn−1 ∝ |n − 1i+hn − 1| ⊗ |0i−h0| 0.6 + |0i h0| ⊗ |n − 1i hn − 1|. (13) 0.4 + − -d 0.2 x/d I.e., the cross-terms with constructive and destructive -2 -1 1 2 interference cancel each other, and resulting mixed state

(n-1) (n-1,1) (с) G (x) (d) G (x,k) allows for some resolution gain over the pure nth photon 1.0 1.0 NOON state. 0.8 0.8 0.6 0.6 0.4 0.4 0.2 D. Application to quantum imaging 0.2 x/d -2 -1 1 2 -0.2 x/d -2 -1 1 2 To illustrate possible application of the ideas to practical quantum imaging, we consider an object represented FIG. 2. Model example: imaging two pinholes (a), separated by a set of semitransparent slits (Fig. 3a,c). The resolu- by the distance 2d. Constructive (b), absent (c) and destruc- tion of the modeled optical system is limited by diﬀrac- tive interference (d) of the light passing through the pinholes. tion at the lens aperture, which admits only the photons Dot-dashed lines represent separate contributions from the ~ with the transverse momentum k not exceeding kmax: pinholes; dashed line shows the interference signal; solid line ~ represents the sum of all the contributions. |k| ≤ kmax.

(a) (b) Signal(x) Let us interpret these results in terms of detecting n−1 1.0 photons conditioned by the nth photon detection. Ac- 0.8 cording to Eq. (4), if the photon is detected at the point 0.6 ~r of the image plane, it transforms the state of the re- d 0.4 maining photons into 0.2

x/d (1) ~ 2 4 6 8 10 |Φn−1(~r)i ∝ h(d, ~r)|n − 1i+|0i− (с) (d) Signal(x) ~ 1.0 + h(−d, ~r)|0i+|n − 1i−. (11) 0.8

The state coherence is preserved, while the blurring, 0.6 caused by constructive interference, is slightly reduced 0.4 due certain “which path” information provided by the 0.2 nth photon detection. x/d If the nth photon is characterized by the transverse 2 4 6 8 10 ~ ~ momentum k, |k| > kmax, and does not get into the imaging system aperture, the eﬀective modiﬁed state of (e) W the remaining photons is (see Eq. (5)): x

(2) ~ i~k·d~ |Φn−1(k)i ∝ e |n − 1i+|0i− −i~k·d~ Source OP IP + e |0i+|n − 1i−, (12)

Now, the phase shift between the two modes depends on ~k and can lead to destructive interference, which en- FIG. 3. Model objects (a, c), imaging scheme for (n − 1)- hances the contrast of the image. For example, when photon coincidence conditioned by detecting a photon in the ~ ~ region Ω (e), and simulation results (b, d). Sets of semi- k · d = π/2, one has maximally destructive interference transparent slits with transmission amplitudes 0.5÷1 (a) and (n−1) ~ and the detected signal is proportional to |h (d, ~r) − 0.9 ÷ 1 (c) were used as model objects. The signal (G(n) h(n−1)(−d,~ ~r)|2 with 100% visibility of the gap between — dotted lines, G(n−1) — dashed lines, G(n−1,1) — solid the two peaks (Fig. 2d). lines) was simulated for the object from panel (a) and n = 3 Discarding the information about the nth photon (b) and 4 (d). The detection region for the nth photon is ~ ~ (measuring G(n−1)) corresponds to averaging over the Ω = {k : kmax ≤ |k| ≤ 2kmax}. The axis x is directed across possibilities to have the photon passing to the detector the slits. The coordinate x is normalized by the slit size d and missing it, and yields the following mixed state of indicated in plot (a). 5

We compare the following three strategies: (i) mea- all the unknowns {θµ}) by the trace of the inverse of FIM: suring G(n)(~r) along the direction perpendicular to the slits (the signal is described by Eq. (2)); (ii) measuring X 1 ∆2 = ∆θ ≥ Tr F −1, (17) G(n−1)(~r) at the same points (Eq. 3); (iii) measuring co- µ N µ incidence signal G(n−1,1)(~r, Ω) of n − 1 photons detected at the point ~r of the detection plane and the nth pho- where N is the number of registered coincidence events. ton being anywhere in certain region Ω outside the lens When the size of the analyzed object features (e.g. the aperture. For the latter case the signal can be written as slit size d in Fig. 3a) tends to zero, the bound in Eq. (17) Z diverges (the effect is termed “Rayleigh’s curse”). The G(n−1,1)(~r, Ω) ∝ d2~sd2~s0An(~s)An(~s0) achievable resolution can, therefore, be determined by the feature size d, for which Tr F −1 starts growing rapidly × hn−1(~s,~r)hn−1(~s0, ~r))g(~s − ~s0), (14) with the decrease of d. A more rigorous definition can be given by specifying certain reasonable threshold N for where max the maximal required number of registered coincidence Z 5 0 2~ i~k·(~s−~s0) events N (e.g. we take Nmax = 10 for further exam- g(~s − ~s ) = d ke . (15) −1 ~k∈Ω ples) and imposing the restriction Tr F ≤ Nmax (see Methods). Integration in Eq. (15) corresponds to detection of the The dependence of the predicted reconstruction er- nth photon by a bucket detector, similarly to multipho- ror on the normalized object scale d/d (where d = ton ghost imaging [14–16]. However, all the n photons R R 3.83/kmax is the classical Rayleigh limit for the consid- (including the n − 1 ones, which get to the position- ered optical system) is shown in Fig. 4. The sampling resolving detector) do pass through the investigated ob- points for the signal are taken with the step d/2 along ject. Our scheme can also be considered as a generaliza- a line perpendicular to slits. As expected, ignoring nth tion of hybrid near- and far-field imaging, when the en- photon and measuring (n−1)-photon coincidences brings tangled photons are analyzed partially in position space about larger achievable information per measurement, and partially in momentum space. and correspondingly lower errors in object parameter es- Simulated images are shown in Fig. 3b,d. One can timation, yielding (10 ÷ 20)% better resolution for n = 3 clearly see that (n − 1)-photon detection yields better and 4. According to the theoretical predictions, for n = 2 visual resolution than measurement of the nth-order co- (n−1,1) no resolution increase is observed. incidences, while G provides additional contrast Also, our results confirm that the proposed hybrid enhancement. scheme is indeed capable of increasing the resolution for Of course, the effective narrowing of the PSF by mea- n = 3 and 4 by conditioned detection of the nth pho- suring correlation functions does not necessarily mean a ton. An additional information that can be gained rela- corresponding increase in precision of inferring of the an- tively to the measurement of (n − 1)-photon correlations alyzed parameters (positions of the object details, chan- is about (10 ÷ 15)%. However, that gain vanishes in the nel characteristics, etc.) [18, 19]. However, at least for regime of deep superresolution (d . 0.2dR): the black certain imaging tasks (such as, for example, a corner- solid and red dot-dashed lines intersect with the green stone problem of finding a distance between two point dashed one for small d/dR in Fig. 4. The reason for such sources in the far-field imaging), narrowing of the PSF behavior is that the effective phase shift, introduced by can indeed lead to increase of the informational content detection of the nth photon with the transverse momen- per measurement, and to the potentially unlimited reso- tum ~k outside of the aperture of the imaging system, lution with increasing of n [19]. becomes insufficient for the resolution enhancement for To describe the resolution enhancement in a quanti- |~k|d 1. tative and more consistent way, we employ Fisher infor- For n = 2, the scheme also can give certain advan- mation. Let the transmission amplitude of the object be tages (Fig. 4e,f), which, however, are not so prominent decomposed as A(~s) = P θ f (~s), where the basis func- µ µ µ because they do not stem from the fundamental require- tions f (~s) can represent e.g. slit-like pixels for the con- µ ment of having better resolution for G(n−1) than for G(n). sidered example [37]. Then the problem of finding A(~s) Still, taking into account the difficulties in generation of becomes equivalent to reconstruction of the unknown de- 3-photon entangled states [39–44], an experiment with composition coefficients θ . If one has certain signal S(~r), µ biphotons can be proposed for initial tests of the ap- sampled at the points {~r }, Fisher information matrix i proach. (FIM) [46, 47], normalized by a single detection event, The plots, shown in Fig. 4, represent information per can be introduced as single detection event. Therefore, certain concerns about X 1 ∂S(~ri) ∂S(~ri) X the rates of such events may arise: waiting for a highly Fµν = / S(~ri). (16) S(~ri) ∂θµ ∂θν informative, but very rare event can be impractical. Fig. i i 5 shows the ratio of the overall detection probabilities Cramér-Raoinequality [47, 48] bounds the total recon- pn−1,1/pn, where pn−1,1 corresponds to (n − 1)-photon struction error (the sum of variances of the estimators for coincidence, conditions by detection of a photon outside 6

-1 Tr F -1 Tr F pn-1,1/pn

107 (a) (b) 106 106

5 10 104 0.20 104 1000 100 0.10 100

d/dR d/dR 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 0.05 -1 Tr F -1 Tr F d/dR 0.2 0.3 0.4 0.5 107 (с) (d) 6 6 10 10 FIG. 5. Dependence of the overall detection probabilities ra- 105 104 104 tio on the normalized slit size d/dR (see details in the text). ~ 1000 100 The nth photon is detected in the region Ω = {k : kmax ≤ 100 |~k| ≤ 2kmax} (solid black and dashed green lines) or Ω = d/dR d/dR 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 {~k : kmax ≤ |~k| ≤ 1.5kmax} (dot-dashed red and dotted blue

Tr F -1 Tr F -1 lines) for n = 4. The objects are shown in Fig. 3a (for solid (e) (f) black and dot-dashed red lines) and Fig. 3c (for dashed green 108 108 and dotted blue lines). Horizontal dotted lines indicate the 106 106 threshold, used for quantiﬁcation of resolution.

4 104 10

100 100

d/dR d/dR 0.25 0.30 0.35 0.40 0.45 0.50 0.25 0.30 0.35 0.40 0.45 0.50 is taken into account. Notice, that all the mentioned val- FIG. 4. Dependence of the trace of inverse FIM on the nor- ues of the resolved feature size d are quite far beyond the malized slit size d/dR for detection of nth-order correlations classical resolution limit dR. (blue dotted lines), (n−1)th-order correlations (green dashed lines), and (n − 1)th-order correlations, conditioned by de- At the first glance, the reported percentage of the res- tection of the nth photon in the region Ω = {~k : kmax ≤ olution enhancement does not look very impressive or |~k| ≤ 2kmax} (black solid lines) and Ω = {~k : kmax ≤ |~k| ≤ encouraging. However, one should keep in mind that the 1.5kmax} (red dot-dashed lines) for n = 4 (a,b), 3 (c,d), and increase of the number of entangled photons from n = 2 2 (e,f). The objects are shown in Fig. 3a (for plots a, c, and to n = 3, while requiring significant experimental efforts, e) and Fig. 3c (for plots b, d, and f). Horizontal dotted lines leads to the effective PSF narrowing just by 22% for the −1 5 indicate the threshold Tr F ≤ Nmax = 10 , used for quan- measurement of G(n). The transition from a 3-photon tification of resolution. entangled state to a 4-photon one yields only 15% narrower PSF. Moreover, the actual resolution enhancement is typically smaller than the relative change of the PSF the aperture, and p describes traditional measurement n width [18, 19], especially for high orders of the correla- of n-photon coincidences. For a signal S(~r ), the overall i tions, where it may saturate completely. The proposed detection probability is defined as p = P S(~r ) and rep- i i approach provides a similar magnitude of the resolution resents the denominator of Eq. (16). When plotting Fig. increase on the cost of adding a bucket detector to the 5, we do not include the measurement of G(n−1) into the imaging scheme, which is much simpler than changing comparison, because the ratio of probabilities for (n − 1) the number of entangled photons. and n-photon detection events strongly depends on details of a particular experiment, such as the efficiency of Similar concern about soundness of the results may the detectors. be elicited by recalling a simple problem of resolving The rate of (n − 1)-photon coincidences, conditioned two point sources, commonly investigated theoretically by the detection of a photon outside of the aperture, [32, 34, 38]. For such a simple model situation, the error is indeed 3 ÷ 20 times smaller than the rate of n- of inferring the distance d between the sources scales as photon coincidences. However, for the considered multi- ∆d ∝ d−1N −1/2 [32], where N is the number of detected parametric problem, the “Rayleigh curse” leads to very events. Therefore, to resolve twice smaller separation fast decrease of information when the slit size d becomes of the two sources with the same error, one just needs smaller than the actual resolution limit, and the effect to perform a 4 times longer experiment and collect 4N of rate difference is almost negligible. For example, for events. The situation becomes completely different when n = 4, the object, shown in Fig. 3a, and the thresh- a more practical multiparametric problem is considered old Tr F −1 ≤ 105, the minimal slit width d for suc- [37]: the achievable resolution becomes practically insen- cessful resolution of transmittances equals 0.212dR for sitive to the data acquisition time (as soon as the number (n) the measurement of G , 0.170dR for the measurement of detected events becomes sufficiently large). For exam- of G(n−1,1) with the nth photon detected in the region ple, for the situation described by the solid black line in ~ ~ Ω = {k : kmax ≤ |k| ≤ 2kmax}, and 0.177dR for the same Fig. 4a, a 100-fold increase of the acquisition time leads measurement of G(n−1,1) when the reduced detection rate to 14% larger resolution. 7

II. DISCUSSIONS ing standard deﬁnition:

n n (n) h (−) i h (+) i We have demonstrated how to enhance resolution of G (~r) = hΨn| E (~r) E (~r) |Ψni, (19) imaging with an n-photon entangled state by loosing a photon and measuring the (n − 1)th-order correlation where E(−)(~r) = [E(+)(~r)]+ is the negative-frequency function instead of the nth-order coincidence signal. The field operator. By substitution of Eq. (18) into Eq. (19), resolution gain occurs despite breaking of entanglement one can obtain the expression (2) in the Results section. as a consequence of the photon loss. We have explained The (n − 1)th-order correlation function is calculated the effect in terms of the effective modification of the according to the expression remaining photons state when one of the entangled pho- h in−1 h in−1 tons is lost. Measurement of (n − 1)-photon coincidences G(n−1)(~r) = hΨ | E(−)(~r) E(+)(~r) |Ψ i, for an n-photon entangled state not only discards some n n information carried by the ignored nth photon, but also (20) makes the resulting signal more informative in the con- which yields Eq. (3) after substitution of Eq. (18). sidered imaging experiment. The latter effect prevails for n > 2 and leads to increase of the information per measurement and to decrease of the lower bounds for the B. Effective (n − 1)-photon state object inference errors. The (n − 1)-photon detection represents a mixture of The density operator, describing the effective (n − 1)- different possible outcomes for the discarded nth pho- photon state averaged over the possible “fates” of the nth ton, including its successful detection at the image plane photon, discussed in the main text, is

(resulting in the n-photon coincidence signal). The in- (1) (2) (3) formation per a single (n − 1)-photon detection event is ρn−1 = ρn−1 + ρn−1 + ρn−1, (21) averaged over the discussed possibilities and, for n > 2, (k) is larger than the information for a single n-photon co- where the operators ρn−1 are indexed according to the in- incidence event. It means that certain outcomes for the troduced possibilities and normalized in such a way that (k) nth photon provide more information per event than the Tr ρn−1 is the probability of the kth “fate”. average value, achieved for G(n−1). We prove that propo- According to the approach, discussed in Ref. [29], de- sition constructively by proposing a hybrid measurement tection of the nth photon at the position ~r0 of the detec- scheme, which provides resolution increase relatively to tor effectively modifies the states of the remaining (n−1) detection of (n − 1)-photon coincidences. Intentional de- photons in the following way: tection of a photon outside the optical system, used for (1) 0 (+) 0 imaging of the object, introduces an additional phase |Ψn−1(~r )i ∝ E (~r )|Ψni. (22) shift and increases the sensitivity of the measurement performed with the remaining photons. Our simulations Substitution of Eqs. (1) and (18) yields Eq. (4). show that the effect can be observed even for n = 2, thus If we ignore the information about the position ~r0 of making its practical implementation much more realis- the photon detection, the contribution to the averaged tic. We believe that our observation will pave a way for density operator (21) is practical exploitation of entangled states by devising a Z superresolving imaging scheme conditioned on detecting (1) 2 0 (1) 0 (1) 0 ρn−1 = d ~r |Ψn−1(~r )ihΨn−1(~r )|. (23) photons not only successfully passing through the imaging system, but also those missing it. 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 III. METHODS density operator (21) is Z A. Expressions for field correlation functions (2) 2~ (2) ~ (2) ~ ρn−1 = d k|Ψn−1(k)ihΨn−1(k)|, (24) |~k|>kmax For the imaging setup, shown in Fig. 1, the positive- frequency field operators E(~r) at the detection plane are where kmax is the maximal transverse momentum transferred by the optical system: k = kR/s ; k is the connected to the operators E0(~s) of the field illuminating max o the object as wavenumber of the light, R is the radius of the aperture, and so is the distance between the object and the lens Z (+) 2 (+) used for imaging. E (~r) = d ~sE0 (~s)A(~s)h(~s,~r). (18) Calculating integrals in Eqs. (23), (24), and (6) and taking into account the connection between the PSF The nth-order correlation function for the n-photon shape and kmax (see e.g. Ref. [8]), one can obtain Eq. entangled state (1) is calculated according to the follow- (7) for the effective (n − 1)-photon state. 8

C. Model of point-spread function Therefore, one can define the spatial resolution, achievable under the described experimental conditions, as the For the simulations, illustrated by Figs. 3, 4, and 5, minimal feature size d, for which the condition (26) is we assume for simplicity that the magnification of the satisfied. optical system is equal to 1, neglect the phase factor in PSF, and use the expression IV. ACKNOWLEDGEMENTS Z 2 i~k·(~s+~r) h(~s,~r) = d ~k e All the authors acknowledge financial support from ~ |k|≤kmax the King Abdullah University of Science and Technology 2 = 2πkmax somb (kmax |~s + ~r|) , (25) (grant 4264.01), A. M., I. K., and D.M. also acknowledge support from the EU Flagship on Quantum Technologies, where somb(x) = 2J1(x)/x, J1(x) is the first-order Bessel project PhoG (820365). function, and kmax is the maximal transverse momentum transferred by the optical system. V. AUTHOR CONTRIBUTIONS

D. Quantiﬁcation of resolution The theory was conceived by A.M. and D.M. Numeri- cal calculations were performed by A.M. The project was Let us assume that a reasonable number of detected supervised by D.M., D.L.M., and A.M. All the authors coincidence events N in a quantum imaging experiment participated in the manuscript preparation, discussions is limited by the value Nmax and the acceptable total and checks of the results. reconstruction error (see Eq. (17)) is ∆2 ≤ 1. Therefore, Eq. (17) implies the following threshold for the trace of the inverse of FIM: VI. COMPETING INTERESTS

−1 2 Tr F ≤ N∆ ≤ Nmax. (26) The authors declare no competing interests.

[1] E. Abbe. Beiträgezur theorie des mikroskops und der [9] V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro. mikroskopischen wahrnehmung. Archiv fürmikroskopis- Sub-rayleigh-diffraction-bound quantum imaging. Phys. che Anatomie, 9(1):413–468, 1873. Rev. A, 79(1):013827, 2009. [2] L. Rayleigh. Xxxi. investigations in optics, with special [10] D.-Q. Xu, X.-B. Song, H.-G. Li, D.-J. Zhang, H.-B. reference to the spectroscope. The London, Edinburgh, Wang, J. Xiong, and K. Wang. Experimental obser- and Dublin Philosophical Magazine and Journal of Sci- vation of sub-Rayleigh quantum imaging with a two- ence, 8(49):261–274, 1879. photon entangled source. Appl. Phys. Lett., 106(17): [3] S. W. Hell and J. Wichmann. Breaking the diffrac- 171104, 2015. doi:10.1063/1.4919131. URL http://aip. tion resolution limit by stimulated emission: stimulated- scitation.org/doi/10.1063/1.4919131. emission-depletion fluorescence microscopy. Optics Let- [11] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, ters, 19(11):780–782, 1994. C. P. Williams, and J. P. Dowling. Quantum interfero- [4] Thomas Dertinger, Ryan Colyer, Gopal Iyer, Shimon metric optical lithography: Exploiting entanglement to Weiss, and JörgEnderlein. Fast, background-free, 3D beat the diffraction limit. Phys. Rev. Lett., 85(13):2733– super-resolution optical fluctuation imaging (SOFI). Pro- 2736, 2000. doi:10.1103/PhysRevLett.85.2733. arXiv: ceedings of the National Academy of Sciences, 106(52): quant-ph/9912052. 22287–22292, 2009. [12] L. A. Rozema, J. D. Bateman, D. H. Mahler, [5] O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, R. Okamoto, A. Feizpour, A. Hayat, and A. M. Stein- Z. Deutsch, and D. Oron. Superresolution microscopy berg. Scalable Spatial Superresolution Using Entangled with quantum emitters. Nano Letters, 13(12):5832–5836, Photons. Phys. Rev. Lett., 112(22):223602, 2014. doi: 2013. 10.1103/PhysRevLett.112.223602. URL https://link. [6] A. Classen, M. O. von Zanthier, J.and Scully, and G. S. aps.org/doi/10.1103/PhysRevLett.112.223602. Agarwal. Superresolution via structured illumination [13] L. Maccone V. Giovannetti, S. Lloyd. Quantum- quantum correlation microscopy. Optica, 4(6):580–587, enhanced measurements: Beating the stan- 2017. dard quantum limit. Science, 306(5700):1330– [7] A. Classen, J. von Zanthier, and G. S. Agarwal. Analy- 1336, 2004. doi:10.1126/science.1104149. URL sis of super-resolution via 3d structured illumination in- https://www.sciencemag.org/lookup/doi/10.1126/ tensity correlation microscopy. Optics express, 26(21): science.1104149. 27492–27503, 2018. [14] I. N. Agafonov, M. V. Chekhova, T. Sh. Iskhakov, and [8] Y. Shih. An introduction to quantum optics: photon and L.-A. Wu. High-visibility intensity interference and ghost biphoton physics. CRC press, 2018. imaging with pseudo-thermal light. Journal of Modern 9

Optics, 56(2-3):422–431, 2009. [30] C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez, and [15] K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd. High- P. Zoller. Creation of entangled states of distant atoms order thermal ghost imaging. Opt. Lett., page 3, 2009. by interference. Phys. Rev. A, 59(2):1025, 1999. [16] X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, [31] E. Brainis. Quantum imaging with n-photon states in M. V. Chekhova, and L.-A. Wu. Arbitrary-order lensless position space. Optics express, 19(24):24228–24240, 2011. ghost imaging with thermal light. Opt. Lett., 35(8):1166, [32] M. Paúr,B. Stoklasa, J. Grover, A. Krzic, L. L. Sánchez- April 2010. doi:10.1364/OL.35.001166. arXiv: 0902.3713. Soto, Z. Hradil, and J. Reháˇcek.ˇ Tempering rayleigh’s [17] M. Tsang. Quantum imaging beyond the diffraction limit curse with psf shaping. Optica, 5(10):1177–1180, 2018. by optical centroid measurements. Phys. Rev. Lett., 102 [33] C. Lupo, Z. Huang, and P. Kok. Quantum limits to (25):253601, 2009. incoherent imaging are achieved by linear interferometry. [18] M. E. Pearce, T. Mehringer, J. von Zanthier, and P. Kok. Physical Review Letters, 124(8):080503, 2020. Precision estimation of source dimensions from higher- [34] M. Tsang, R. Nair, and X.-M. Lu. Quantum theory of order intensity correlations. Phys. Rev. A, 92(4):043831, superresolution for two incoherent optical point sources. 2015. doi:10.1103/PhysRevA.92.043831. URL https: Physical Review X, 6(3):031033, 2016. //link.aps.org/doi/10.1103/PhysRevA.92.043831. [35] M. Tsang. Subdiffraction incoherent optical imaging via [19] S. Vlasenko, A. B. Mikhalychev, I. L. Karuseichyk, D. A. spatial-mode demultiplexing. New Journal of Physics, 19 Lyakhov, D. L. Michels, and D. Mogilevtsev. Opti- (2):023054, 2017. mal correlation order in superresolution optical fluctu- [36] L. Motka, B. Stoklasa, M. D’Angelo, P. Facchi, A. Garuc- ation microscopy. Phys. Rev. A, 102:063507, Dec 2020. cio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, doi:10.1103/PhysRevA.102.063507. URL https://link. J. Reháˇcek,etˇ al. Optical resolution from fisher infor- aps.org/doi/10.1103/PhysRevA.102.063507. mation. The European Physical Journal Plus, 131(5): [20] Ryszard Horodecki, Pawe lHorodecki, MichalHorodecki, 130, 2016. and Karol Horodecki. Quantum entanglement. [37] A. B. Mikhalychev, B. Bessire, I. L. Karuseichyk, Rev. Mod. Phys., 81:865–942, Jun 2009. doi: A. A. Sakovich, M. Unternährer,D. A. Lyakhov, D. L. 10.1103/RevModPhys.81.865. URL https://link.aps. Michels, A. Stefanov, and D. Mogilevtsev. Efficiently org/doi/10.1103/RevModPhys.81.865. reconstructing compound objects by quantum imaging [21] T. B. Pittman, Y. H. Shih, D. V. Strekalov, and with higher-order correlation functions. Communication A. V. Sergienko. Optical imaging by means of two- Physics, 2(1):134, December 2019. doi:10.1038/s42005- photon quantum entanglement. Phys. Rev. A, 52(5): 019-0234-5. URL http://www.nature.com/articles/ R3429–R3432, 1995. doi:10.1103/PhysRevA.52.R3429. s42005-019-0234-5. URL https://link.aps.org/doi/10.1103/PhysRevA. [38] M. Paúr,B. Stoklasa, D. Koutn`y, J. Reháˇcek,Z.ˇ Hradil, 52.R3429. J. Grover, A. Krzic, and L. L. Sánchez-Soto. Reading [22] D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and out fisher information from the zeros of the point spread Y. H. Shih. Observation of two-photon “ghost” interfer- function. Optics Letters, 44(12):3114–3117, 2019. ence and diffraction. Phys. Rev. Lett, 74(18):3600, 1995. [39] Hannes Hübel, Deny R Hamel, Alessandro Fedrizzi, Sven [23] Yanfeng Bai and Shensheng Han. Ghost imaging with Ramelow, Kevin J Resch, and Thomas Jennewein. Direct thermal light by third-order correlation. Phys. Rev. A, generation of photon triplets using cascaded photon-pair 76(4):043828, 2007. doi:10.1103/PhysRevA.76.043828. sources. Nature, 466(7306):601–603, 2010. URL https://link.aps.org/doi/10.1103/PhysRevA. [40] T. E. Keller, M. H. Rubin, Y. Shih, and L.-An. Wu. 76.043828. Theory of the three-photon entangled state. Phys. Rev. [24] B. I. Erkmen and J. H. Shapiro. Unified theory of ghost A, 57(3):2076, 1998. imaging with gaussian-state light. Phys. Rev. A, 77(4): [41] J. Wen, E. Oh, and S. Du. Tripartite entanglement gen- 043809, 2008. eration via four-wave mixings: narrowband triphoton w [25] Paul-Antoine Moreau, Ermes Toninelli, Peter A. Mor- state. JOSA B, 27(6):A11–A20, 2010. ris, Reuben S. Aspden, Thomas Gregory, Gabriel Spald- [42] M. Corona, K. Garay-Palmett, and A. B. U’Ren. Exper- ing, Robert W. Boyd, and Miles J. Padgett. Resolu- imental proposal for the generation of entangled photon tion limits of quantum ghost imaging. Opt. Express, triplets by third-order spontaneous parametric downcon- 26(6):7528, March 2018. doi:10.1364/OE.26.007528. version in optical fibers. Optics Letters, 36(2):190–192, URL https://www.osapublishing.org/abstract.cfm? 2011. URI=oe-26-6-7528. [43] M. Corona, K. Garay-Palmett, and A. B. U’Ren. Third- [26] C. Skornia, J. von Zanthier, G. S. Agarwal, E. Werner, order spontaneous parametric down-conversion in thin and H. Walther. Nonclassical interference effects in optical fibers as a photon-triplet source. Physical Review the radiation from coherently driven uncorrelated atoms. A, 84(3):033823, 2011. Phys. Rev. A, 64(6):063801, 2001. [44] N. A. Borshchevskaya, K. G. Katamadze, S. P. Kulik, [27] C. Thiel, Th. Bastin, J. Martin, E. Solano, J. von Zan- and M. V. Fedorov. Three-photon generation by means thier, and G. S. Agarwal. Quantum imaging with inco- of third-order spontaneous parametric down-conversion herent photons. Phys. Rev. Lett., 99(13):133603, 2007. in bulk crystals. Laser Physics Letters, 12(11):115404, [28] S. Oppel, Th. Büttner,P. Kok, and J. von Zanthier. Su- 2015. perresolving multiphoton interferences with independent [45] G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lap- light sources. Phys. Rev. Lett., 109(23):233603, 2012. kiewicz, and A. Zeilinger. Quantum imaging with unde- [29] D. Bhatti, A. Classen, S. Oppel, R. Schneider, and J. von tected photons. Nature, 512(7515):409–412, 2014. Zanthier. Generation of N00N-like interferences with two [46] R. A. Fisher. Theory of statistical estimation. In Math- thermal light sources. The European Physical Journal D, ematical Proceedings of the Cambridge Philosophical So- 72(11):191, 2018. ciety, volume 22, pages 700–725. Cambridge University 10

Press, 1925. [48] H. Cramer. Mathematical methods of statistics. Princeton [47] C. R. Rao. Information and accuracy attainable in the University Press, Princeton, 1946. estimation of statistical parameters. Bull Calcutta. Math. Soc., 37:81–91, 1945.