Quantum imaging of biological organisms through spatial and polarization entanglement

Quantum imaging holds potential benefits over classical imaging but has faced challenges such as poor signal-to-noise ratios, low resolvable pixel counts, difficulty in imaging biological organisms, and inability to quantify full birefringence properties. Here, we introduce quantum imaging by coincidence from entanglement (ICE), using spatially and polarization-entangled photon pairs to overcome these challenges. With spatial entanglement, ICE offers higher signal-to-noise ratios, greater resolvable pixel counts, and the ability to image biological organisms. With polarization entanglement, ICE provides quantitative quantum birefringence imaging capability, where both the phase retardation and the principal refractive index axis angle of an object can be remotely and instantly quantified without changing the polarization states of the photons incident on the object. Furthermore, ICE enables 25 times greater suppression of stray light than classical imaging. ICE has the potential to pave the way for quantum imaging in diverse fields, such as life sciences and remote sensing.


Introduction
Since van Leeuwenhoek's first microscope, optical imaging has been widely used to noninvasively investigate the structures and dynamics of various physical and biological systems 1,2 . The key advantage of optical imaging is that the interaction of non-ionizing light with molecules provides rich molecular information about biological samples. Aided by the convenience and compactness of optical systems, optical imaging has served as the workhorse for biological researchers and medical practitioners behind a wide variety of discoveries 3 . In the past two decades, advanced optical imaging techniques have been developed to allow super-resolution 1,4 and high-speed 5,6 bioimaging. However, to achieve high resolution and high imaging speed, most optical imaging techniques require intense illumination that can disrupt or damage the biological processes under investigation 2 . Low-intensity illumination may lead to a low signal-to-noise ratio (SNR) due to shot noise and stray light.
Recently, to overcome the limitations of existing optical imaging techniques that rely on classical light sources, quantum imaging approaches that use correlated, entangled, or squeezed photons have been developed [7][8][9][10][11] . Compared with classical optical imaging, quantum imaging has the following advantages 12 . First, the classical shot-noise limit can be broken, allowing for sub-shotnoise (SSN) imaging under low-intensity illumination 11,[13][14][15][16][17] . Second, stray light can be suppressed 10,18,19 . Third, super-resolution imaging beyond the diffraction limit can be enabled 8,[20][21][22][23][24][25] . Empowered by these advantages, quantum imaging has been employed to investigate biological specimens 8,11,26 , which have complex structures and may be susceptible to photobleaching and thermal damage. Despite the advantages, quantum images of biological specimens reported to date still suffer a low SNR because (1) the conditions required to achieve SSN are stringent 13,[15][16][17]27 and (2) the SNRs in most quantum imaging approaches are low 10,12,19 . Moreover, existing quantum imaging approaches usually have low resolvable pixel counts (i.e., the ratios of the field of view (FOV) to the spatial resolution) 7-11 and thus are not suited for practical biological studies, which often demand systematic investigation of multiple parts in a biological system with an FOV across a whole organism. Finally, quantum imaging techniques so far only measure transmittance (absorption) or phase contrast, whereas classical techniques support additional contrast such as birefringence 1,2,28 .
Here we present ICE, a higher-SNR, greater-resolvable-pixel-count, and birefringence-sensitive quantum imaging technique that generates high-quality images of biological specimens. Under low-intensity illumination, ICE employs a new SSN algorithm that utilizes the covariance of the raw images to achieve a higher SNR than the classical counterpart. Concurrently, ICE substantially increases the SNR over existing quantum imaging techniques by accommodating multiple spatial modes of the entangled photon pairs in each pixel, where a single spatial mode is constrained by the diffraction limit of the system 29,30 . The spatial resolution of ICE is determined by both the signal and idler photons through a quantum effect named "entanglement pinhole". In the entanglement pinhole effect, when an entangled photon pair is captured concurrently by two detectors, one detector functions non-classically as a pinhole on the object being imaged by the other detector. Further, ICE increases the resolvable pixel counts indefinitely through raster scanning and is 25 times more resilient to stray light than classical imaging. Consequently, ICE enables quantum imaging of whole organ (mouse brain) slices and organisms (zebrafish) with an FOV of up to 7 mm × 4 mm, and can be operated in the presence of ambient lighting, thus suitable for practical biological studies. Finally, ICE exploits the polarization entanglement of the photon pairs for ghost birefringence imaging, where the birefringence properties of an object can be remotely and instantly measured without changing the polarization states of the photons incident on the object. The quantum advantages of ICE, therefore, enable the observation of biological specimens under conditions that cannot be satisfied with classical imaging, as well as the remote sensing of birefringence.

Sub-shot-noise quantum imaging using multi-mode entangled photons
In ICE (Fig. 1, details in Methods), we use two β-barium borate (BBO) nonlinear crystals with perpendicularly aligned optical axes to produce hyperentangled photon pairs, which are simultaneously entangled in spatial mode, polarization, and energy 31,32 , through the type-I spontaneous parametric down-conversion (SPDC) process. Most quantum imaging techniques reported to date evenly distribute the spatial modes of entangled photons across multi-pixel cameras 10,13,19,33,34 , leading to a small number of spatial modes per pixel, a low coincidence rate, and, consequently, a low SNR in the image. In comparison, ICE increases the coincidence rate and SNR of quantum images by directly focusing the multi-mode SPDC beam onto the object, resulting in substantially more spatial modes in each pixel. We record the signal ( ), idler ( ), and coincidence ( ) counts from the two single-photon counting modules (SPCMs) while raster scanning the object through the focused SPDC beam to image the transmittance of the object.

Whereas
and provide classical and quantum (ICE) images of the object, respectively, can further improve the SNR of the images through SSN signal retrieval using our covariance-over-  8b), one can see that, within a 3D volume of 1000 × 1000 × 300 µm 3 , the carbon fibers in the ICE stack are clearly better resolved and have sharper edges than those in the classical stack.

Quantum imaging of biological organisms in the presence of stray light
By raster scanning the object, ICE provides an FOV that can be extended indefinitely. We imaged a slice of a whole organ (the cerebellum of a mouse brain) with a 7 mm × 4 mm FOV, whose In addition to the large FOV, ICE also demonstrates robust stray light resistance due to coincidence detection. To quantify ICE's resilience to ambient lighting, an LED was added to the system to introduce stray light ( Supplementary Fig. 10). We acquired classical and ICE images of a biological organism, i.e., an agarose-embedded zebrafish, in a 3.5 mm × 2.3 mm FOV while the LED was randomly turned on and off to simulate randomly fluctuating ambient light (Fig. 3). The zebrafish was positioned such that its torso was oblique to the imaging plane ( Supplementary Fig.   11). As shown in Fig. 3a, while the classical imaging is severely degraded by the stray light, ICE is almost unaffected. We further quantify the robustness of ICE to stray light by acquiring a series of images of carbon fibers under different stray light intensities (Fig. 3b). Using the images acquired without the stray light as the ground truth, we calculated the structural similarity index measure (SSIM) of each image to quantify the degradation of the image quality due to stray light 35 (Methods). The SSIM ranges from 0 to 1, where higher values indicate less degradation. The SSIM versus the stray light optical power is plotted in Fig. 3c. In accordance with the images in Fig. 3b, the classical images degrade quickly with an LED optical power above 0.1 mW, while ICE maintains a high SSIM even with an LED optical power above 1 mW. To simplify the comparison, we use an order-of-magnitude degradation (SSIM = 0.1) as a threshold to find the corresponding LED optical powers, found as 0.18 mW and 4.41 mW for the classical imaging and ICE, respectively. Therefore, ICE suppresses stray light 25 times more effectively than classical imaging. The advantage of ICE can also be seen in the difference between the two SSIM curves, i.e., ∆SSIM, shown in Fig. 3c. This advantage of ICE is attributed to coincidence detection, which is disturbed only by accidental coincidence counts. Despite its sufficient intensity to degrade a classical image, stray light acts as an uncorrelated source, causing negligible coincidence counts.  Fig. 13), the ICE images were substantially modulated by the birefringence properties of the zebrafish (Fig. 4a). Following the theory in Supplementary Note 5, the four ICE images could be used to calculate the transmittance, the angle of the principal refractive index (Fig. 4b), and the phase retardation between the two refractive index axes (Fig. 4c) of the zebrafish, providing additional biologically relevant information that has not been obtained with existing quantum imaging techniques. Furthermore, because of polarization entanglement, measuring the idler photon's polarization state instantly determines the incident signal photon's, thus allowing instant measurement of the object's birefringence properties, regardless of its distance. With the capability to remotely and instantly measure the birefringence properties of an object by changing the polarization states of the photons that do not probe the object, ICE can be used in remote sensing applications where the source is too far to be controlled in real time ( Supplementary Fig. 14).

Discussion
Although imaging by coincidence can be achieved with a classical source 39 , the SNR of the image will be substantially lower compared to that of ICE under the same illumination intensity and polarization entanglement, such as SSN performance and ghost birefringence imaging, will be unavailable. We also note that, despite the similarity in using spatially entangled photon pairs and detecting coincidence for imaging, ICE fundamentally differs from ghost imaging (GI) 40 or correlation plenoptic imaging (CPI) 41 for the following reasons: (1) ICE generates a direct image of the object through raster scanning over a theoretically unlimited FOV, whereas GI and CPI provide an indirect, ghost image of the object through triggering a multi-pixel camera with a limited FOV; (2) The signal arm of ICE contributes to spatial resolution, whereas the signal arms of GI and CPI do not; (3) ICE images substantially more spatial modes per pixel than GI and CPI; (4) ICE exploits polarization entanglement in addition to the spatial entanglement used in GI and CPI (see Supplementary Note 7 and Supplementary Fig. 17 for detailed comparison).
Despite the advantages, ICE has the following limitations. First, the pixel dwell time is currently 1 s, limited by the low SPDC efficiency of the BBO crystal 42 . Second, due to the utilization of multi-mode SPDC beams, ICE has a lower spatial resolution compared to the Abbe limit of resolution 1,2 . These problems could be solved in the future by using a more powerful quantum source 42 . A strong entangled photon source with high coincidence rates could substantially improve the imaging speed, and the SPDC beam could be filtered to a single spatial mode for diffraction-limited imaging while maintaining a sufficient SNR. Third, the entanglement pinhole is a virtual pinhole that filters SPDC modes in coincidence detection. In practice, all the SPDC photons in the signal arm still transmit through the object, which undergoes an illumination intensity higher than the two-photon coincidence used for quantum imaging. Nevertheless, the photon flux of all the SPDC photons on the object is less than 20 kHz (Supplementary Fig. 16), which equals 4.9 × 10 −15 W, an ultralow illumination intensity that is safe for photosensitive biological specimens.
To conclude, we have experimentally demonstrated ICE using hyperentangled photon pairs, achieving high-quality quantum bioimaging with higher SNR, greater resolvable pixel counts, and ghost birefringence quantification. As showcased using the thick biological organism (whole zebrafish) and the whole organ (mouse brain) slice with an FOV substantially larger than those of existing quantum images ( Supplementary Fig. 18

Experimental setup
In our system (Fig. 1), a paired set of BBO crystals ( As shown in Supplementary Fig. 12, our system shows a strong violation of the CHSH inequity with = 2.78 ± 0.01 > 2 estimated by calculating the mean and standard error of values measured from 10 rounds of Bell's tests.

Sample preparation
Four types of objects have been imaged. The wild-type zebrafish was fixed by 4% paraformaldehyde (PFA) solution five days post fertilization. After fixation, the zebrafish was washed 3-4 times using PBS in a fume hood prior to agarose embedding. The agarose-embedded zebrafish was mounted onto a glass slide and sealed with a coverslip to prevent dehydration during the experiment. To prepare the brain slice, a brain was obtained from a Swiss Webster mouse (Hsd: ND4, Harlan Laboratories) and fixed in 3.7% paraformaldehyde solution at room temperature for 24 h. After paraffin embedding, coronal sections (10 µm thick) of the brain were cut. Standard hematoxylin and eosin (H&E) staining was performed on the sections, which were examined using a bright-field microscope (NanoZoomer, Hamamatsu) with a 20 × 0.67 NA objective lens. All animal procedures were approved by the Institutional Animal Care and Use Committee of California Institute of Technology. We used a 2" × 2" positive 1951 USAF resolution target (58-198, Edmund Optics) to quantify the spatial resolution and DOF of our system. To prepare the thick object, carbon fibers with a diameter of 6 µm were randomly embedded in a 4% agarose block (A-204-25, GoldBio) in 3D. A 500 µm thick section was created from the agarose block using a vibratome (VT1200S, Leica). Next, the section was placed onto a standard microscope glass slide and fixed by applying cyanoacrylate glue around the edge. A coverglass was put on top of the sample and sealed using epoxy glue to prevent dehydration of the agarose.

Data acquisition and processing
A custom-written LabVIEW (National Instruments) program was used to synchronize the raster scanning of the 3-axis motor with the data acquisition of the time controller and acquire the raw singles and coincidence counts of the two SPCMs. When acquiring 2D imaging data, the LabVIEW program raster scanned the x-and y-axis motors and converted the raw singles counts of the signal channel and coincidence counts into classical and ICE images, respectively. The images were displayed on screen and saved to the computer in tag image file format (TIF). For imaging thick objects, multiple 2D images each captured at a z-position were combined to form a 3D stack. The TIF files were imported into MATLAB (MathWorks) and processed with customwritten scripts. Depending on the objects being imaged, the images were rotated, cropped, or inverted before being used to extract line profiles or edge spread functions for estimating resolution and DOF. Additionally, to compensate for the low contrast between the brain structure and the background, the brain slice images were denoised by block-matching and 3D filtering 43 followed by a variance-stabilizing transformation 44 .

Measurements of resolution and depth of field
To measure the spatial resolution of our system, the profile of a line along perpendicular to an edge in the USAF resolution target (e.g., the yellow dashed line in Fig. 2a)  ℛ = 2√ln 2 . The mean value of the resolution was estimated to be 2√ln 2 times the fitted , and the standard error was calculated to be √ln 2 1.96 ⁄ times the 95% confidence interval of the fitted . To measure the DOF of our system, resolution, ℛ, was estimated at each z position (e.g., Fig. 2d). The curves were fitted for to a hyperbolic function, i.e., ℛ( ) = where ℛ 0 is the focal resolution and is the Rayleigh length. The mean DOF was estimated to be 2 , and the standard error was estimated to be 1/1.96 times the 95% confidence interval of the fitted .

Imaging with stray light
A white LED (MNWHL4, Thorlabs) powered by an LED driver (DC2200, Thorlabs) was used to randomly generate stray light during imaging, as shown in Supplementary Fig. 10. The LED driver was externally triggered by an analog output device (PCI-6711, National Instruments) installed on the computer. While raster scanning the object prepared on the microscope slide, at each pixel, the LabVIEW program generated a random number uniformly distributed between 0 and 1 to determine whether to trigger the white LED to output stray light. If the random number was less than 0.2, the LED was triggered to generate stray light; otherwise, no stray light would be generated. Therefore, approximately 20% of the pixels would be disrupted by stray light. To evaluate how robust the classical imaging and ICE were against stray light, we acquired images under different stray light optical powers. We calculated the structural similarity index measure (SSIM) between each image and the ground truth at zero stray light by SSIM = , where and 2 ( = 1 or 2) are the average and variance of each image, respectively, and 12 is the covariance of the two images 35 .

Data availability
All data used in this study are available from the corresponding author upon reasonable request.

Code availability
All custom codes used in this study are available from the corresponding author upon reasonable request.

Supplementary Note 1 Sub-shot-noise signal retrieval in ICE
Each round of ICE acquisition generates three images: the signal image ( ), the idler image ( ), and the coincidence image ( ). ( ) and ( ) contain photon counts from both SPDC photon pairs (whose averaged value is denoted as SPDC ) and stray light (whose averaged value is denoted as stray ). For simplicity, we assume the signal and idler detectors have the same background light intensity and detection efficiency, denoted as .
The imaging of an object here measures its transmittance ( ). Although the following derivation applies to both and , we use as an example. Classically, the transmittance is estimated as where 0 denotes the signal image when the object is absent. In ICE, we estimate 0 using � b ( )� , where b denotes a background region of the image outside the target, and ⟨… ⟩ denotes averaging over spatial locations.
By using the correlation between the SPDC photon pairs, two types of sub-shot-noise (SSN) algorithms have been adopted to enhance the SNR of the transmittance measurements. The first type relies on the ratio of the two images, where the object's transmittance is estimated as 27,45 The second type of SSN algorithms, termed optimized subtraction, suppresses the noise in by subtracting the variation of 15,46 : where ( ) is the unknown spatially varying multiplier, and Δ ( ) = ( ) − ⟨ ( )⟩ . The ideal ( ) is proportional to the transmittance ( ), the ground truth of which is unknown. To estimate the ( ), one may use the approximated transmittance � ( ). For example, � ( ) can be acquired using 0 as in Eq. (S1) or as in Ref. 15 . The object's transmittance estimated using the second type of SSN algorithms is given by Both Eqs. (S2) and (S4) achieve higher SNR than Eq. (S1), demonstrating the quantum advantage.
However, these methods require either prior knowledge of ( ) or assumptions on the photon distribution and minimal stray light intensity. Here, inspired by the two algorithms, we introduce the covariance-over-variance (CoV) algorithm to further improve the SSN performance with fewer assumptions.
The workflow of the CoV algorithm is shown in Supplementary Fig. 1. Combining Eqs. (S1), (S3), and (S11) completes the s-CoV algorithm: To compare the performances of the three algorithms, we simulate the 1D case where the object is In all cases, the CoV algorithm outperforms the others consistently.

Supplementary Note 2 Entanglement pinhole in ICE
In the simplified schematic of the imaging system shown in Supplementary Fig. 5, 0, , 1, , and 2, represent the coordinates of the BBO, the object, and the two detectors for signal photons, respectively. 0, and 2, represent the transverse coordinates of the BBO and the detector for idler photons, respectively. 0, and 0, represent the wavevectors of the entangled signal and idler photons emitted from the BBO, respectively. 1, denotes the wavevector of the signal photon after the first objective. 1, ′ denotes the wavevector of the signal photon emitted from the object.
2, denotes the wavevector of the signal photon on the detector . All wavevectors have the same constant magnitude . Here, the subscripts s and i denote signal and idler, respectively.

Supplementary Note 3 ICE using accidental coincidences
If the two detectors only measure accidental coincidences, the results could be treated as coincidences of photons from different positions � 0, , 0, � with different wavevectors � 0, , 0, �.
The source state is replaced by which can be written as a product state | ⟩ acc = ∑ Because last term is a constant with a determined and , acc (2) ∝ CI (1) . Imaging with accidental coincidences therefore provides the same resolution and DOF as classical imaging.

Supplementary Note 4 Characterization of polarization entanglement through Bell's test
Bell-type inequalities provide a standard to The Bell-CHSH inequality is then given by where ′ and ′ denote the second choices of the analyzer angles.
In contrast, quantum theory predicts that CHSH > 2 is possible with specific combinations of observation angles. According to quantum mechanics, we can model coincidence counts � ( , | , ) for a maximally entangled Bell state (i.e., the EPR state) as 37 where 0 is the maximum true coincidence count, and 1 represents the contribution of accidental coincidences. For such states, quantum theory predicts a maximum violation of Eq. (S37) at . (S39) The CHSH value was then evaluated based on the value of according to Eq. (S37). The results are shown in Supplementary Fig. 12.
By performing Bell's test, our system shows a strong violation of the CHSH inequality with = 2.78 ± 0.01 > 2 estimated by calculating the mean and standard error of values measured from 10 rounds of Bell's tests. This result, which violates the Bell-CHSH inequality by more than 57 standard errors of the mean, indicates significant deviations of our result from the LHVT prediction.

Supplementary Note 5 Polarization entanglement-enabled ghost birefringence imaging
The polarization entanglement of the SPDC photons in the ICE system enables ghost birefringence imaging. By preparing the EPR state as described in Eq. (S38) and recording the coincidence counts, ICE can be used to measure the transmittance and the birefringence properties and Δ of the object, where is the angle of the principal refractive index axis and Δ is the phase retardation between the two refractive index axes. Here, we use Stokes vectors ( , , , ) and Mueller matrices to describe the state of polarization. The birefringence properties of the object can be denoted using a Mueller matrix 49 : Under the EPR state, we kept = 0° in the signal arm while rotating in the idler arm from 0° to 135° with a step size of 45°. The corresponding polarization states of the coincidence measurements can be represented by the Stokes vectors The coincidence counts for = 0°, 45°, 90°, 135°, therefore, can be represented as 0°= 1 2 ( 0°+ 0°) = 2 (1 + cos 2 2 + sin 2 2 cos Δ), (S45) Consequently, based on Eqs. (S45)-(S48), the transmittance and birefringence properties of the object can be extracted from the coincidence counts: In classical imaging, the birefringence properties of the object need to be measured using incident photons with different polarization states 50 . In ICE, however, birefringence imaging can be performed without changing the polarization states of the photons incident on the object. When the polarization of the signal photons was kept constant ( = 0°) while the polarization states of the idler photons were varied ( = 0°, 45°, 90°, 135°), the classical images acquired with the raw signal counts showed no differences, unable to extract the birefringence properties of the object ( Supplementary Fig. 13a). In comparison, the ICE images acquired with the coincidence counts exhibited substantial differences following Eqs. (S45)-(S48) (Supplementary Fig. 13b), which could be used to extract the transmittance and birefringence properties of the object ( Supplementary Fig. 13c). One may regard this approach as quantum "ghost birefringence imaging". Enabled by polarization entanglement, the ghost birefringence quantification of ICE demonstrates a true quantum advantage over classical imaging.

Supplementary Note 6 Imaging by coincidence with a classical light source.
Classical two-photon coincidence imaging can be achieved when the spatially entangled source is replaced with a classical pulse source 39 . We implemented such imaging as shown in Supplementary Fig. 15, where a 635-nm CW laser (MLL-III-635-100mW, CNI Laser) was modulated at 4 kHz by a mechanical chopper (MC1F60, Thorlabs). The modulated beam was split by a beam splitter (BS013, Thorlabs) and sent to the signal and idler arms of the ICE system in 16a and b, respectively. For a fair comparison, we used a neutral density filter to attenuate the classical beam such that it provided the same photon flux to the object as that of the SPDC signal beam (~19 kHz at maximum transmittance). Because the SPDC beam contained more spatial modes than the classical beam, the raw signal image (i.e., the classical image) generated with the quantum source exhibited a lower spatial resolution than the one generated with the classical source.
Using the quantum source, the ICE image in Supplementary Fig. 16c showed a maximum coincidence count rate at 485 Hz with an SNR of 22. In comparison, using the classical source, the ICE image in Supplementary Fig. 16d exhibited a maximum coincidence count rate at 16 Hz with an SNR of 4, which was 5.5 times lower than the SNR from the quantum source. Therefore, whereas it is possible to use classical correlation to generate ICE images, the SNR of the image is substantially lower than that generated with a spatially entangled quantum source. Consequently, to generate ICE images with the same SNR, the object needs to be illuminated with a classical source that is 30 times stronger than that of a quantum source, which could cause damage to photosensitive biological samples. Moreover, classical correlation is incompatible with either the sub-shot-noise algorithms (Supplementary Note 1), which require a spatially entangled quantum source, or the ghost birefringence quantification that is enabled by polarization entanglement (Supplementary Note 5). Therefore, the quantum correlation of hyperentangled SPDC photons in ICE is advantageous over classical correlation, especially for imaging photosensitive biological samples.

Supplementary Note 7 Comparison of ICE, GI, and CPI.
Here, we compare ICE ( Supplementary Fig. 17a) with two existing quantum imaging methods.
Quantum ghost imaging (GI) utilizes spatially entangled photons to record an image of an object using photons that have not interacted with the object 40 . A typical ghost imaging setup uses entangled SPDC photons generated by a nonlinear medium, e.g., a BBO crystal, and exploits the spatial correlations between the positions of the photon pairs in the signal and idler arms ( Supplementary Fig. 17b). The signal photons interact with the object and is detected by a nonspatially resolving bucket detector. The idler photons are detected by a multi-pixel camera, which, upon coincidence detection of signal and idler photons, provides a ghost image of the object. It is noted that neither signal nor idler beams alone contain enough information to reconstruct an image of the object. However, the spatial entanglement between the signal and idler photons can be utilized to extract the image.
As an extension of GI, correlation plenoptic imaging (CPI) is another quantum imaging method that utilizes spatial correlations of photon pairs 41 . Beyond the position correlation utilized in GI, CPI also exploits the momentum correlation of the photon pairs, capturing the light field (position and direction of the light) emanating from the object, thus allowing refocusing, DOF extension, and 3D visualization. Unlike GI, which requires a bucket detector and a camera, CPI utilizes two well-aligned multi-pixel cameras to simultaneously record the position and momentum of the photon pairs. Thus far, CPI has only been demonstrated experimentally with chaotic light 51 .
Although achieving CPI with entangled SPDC photons has been proved theoretically 52 , it has not been demonstrated experimentally. A theoretical framework of CPI with entangled photons adopts a multi-pixel camera in the idler arm to record the positions of the object, which, upon coincidence detection with the camera in the signal arm, provides a ghost image of the object ( Supplementary   Fig. 17c). Additionally, through a lens that conjugates the BBO crystal and the camera in the signal arm, the momentum of the photons at each pixel of the object is recorded. The simultaneous recording of the position and momentum of the spatially entangled photons enables the reconstruction of a plenoptic image of the object.
Despite the similarity in using entangled photon pairs and coincidence detection for imaging, ICE is fundamentally different from GI or CPI (Supplementary Fig. 17). First, ICE directly images the object by focusing the SPDC beam onto the object and recording the coincidence of two singlepixel detectors (SPCMs) while raster scanning the object. The raster scanning extends the resolvable pixel counts indefinitely, enabling quantum imaging over a large FOV. In comparison, both GI and CPI provide indirect ghost images of the object by triggering a multi-pixel camera using either a bucket detector or another camera. Because the multi-pixel cameras have a limited number of resolvable pixel counts, the FOVs of GI and CPI are limited. Second, ICE provides spatial resolution of the object, so it is capable of classically imaging the object using the signal arm alone. GI and CPI, however, do not provide spatial resolution of the object, and hence cannot image the object by only using the signal arm. Third, owing to the focusing and single-pixel detection of the SPDC beam, ICE measures substantially more spatial modes per pixel than GI and CPI, where the modes of the SPDC beam are evenly distributed across the multi-pixel cameras.
The larger number of spatial modes per pixel leads to a higher SNR of ICE over GI or CPI under With a satellite emitting polarization-entangled photon pairs 53,54 , ICE can measure the birefringence properties of a remote object by changing the polarization states of the photons without interacting with the source and object. Through polarization entanglement, measuring the idler photon's polarization state instantly determines the incident signal photon's and, consequently, the remote object's birefringence properties, regardless of its distance.
detection, a ghost image of the object is retrieved from the camera and triggered by the bucket detector. c, Schematic of CPI. The BBO crystal is imaged to the camera in the signal arm through the lens with a focal length of , where the labeled distances satisfy the thin-lens equation 1/( + ′ ) + 1 ′′ ⁄ = 1⁄ . The ghost image of the object is imaged to the camera in the idler arm though the lens with a focal length of , satisfying the condition 1 ( + ) ⁄ + 1 ′ ⁄ = 1⁄ .
Through coincidence detection, a ghost image of the object is retrieved from the camera in the idler arm, triggered by the camera in the signal arm.