# A Review of Super-Resolution Imaging through Optical High-Order Interference [Invited]
## 6 Super-Resolution via Quantum Entanglement  
Position-entangled n-photon source can be used for realizing the coherent SQL super-resolution imaging, as initially proposed by [PERSON] et al. [31]. Currently, the most well-developed entangled source is entangled two-photon source [59]. By placing the object in proximity to the output plane of an entangled two-photon source, [PERSON] et al. have experimentally demonstrated the SQL super-resolution imaging [39], as shown in Figure 7I. The entangled source was generated through the type-II parametric down conversion process (SPDC) [59], which was implemented with a betabarium-borate crystal pumped by a pulse laser with a center wavelength at 400 nm. The two-photon coincidence was measured by two single-photon detectors connected to a coincident circuit. The results show a \(\sqrt{2}\) times improved resolution for direct imaging for the objects as shown in Figure 7 II. In this case, the realized SQL imaging is coherent, since the two-photon paths originating from different points of the object are coherent with respect to each other. More interestingly, the entangled two-photon source can be employed to realize Heisenberg-resolution imaging. Super-resolution imaging at HL was initially discussed by [PERSON] et al. [31], which is based on synchronous-position n-photon interference (dependent on the sum coordinates of observation positions), but relies on a conceptual nonlinear n-photon transmitting screen. The Heisenberg-resolution imaging based on synchronous-position multi-photon interference can expressed as  
\[H_{coh}^{(n)}(\mathbf{x^{\prime}})\,\propto\,\Big{|}\int[O(\mathbf{x})]^{n} \cdot\mathrm{PSF}_{n}(\mathbf{x^{\prime}},\mathbf{x})\,\mathrm{d}\mathbf{x} \Big{|}^{2} \tag{5}\]  
Figure 6: **(I)**: Schematic of super-resolution confocal microscope for fluorescence imaging. Multi-photon coincidence counts are measured by using single-photon detectors configured in the tree structure with fiber beam splitters. **(II)**: Three partially overlapped NV centers in the conventional intensity image (**a**) is selected, which cannot be clearly distinguished as seen in the zoom-in image (**b**). The antibunching effect of NV emitters is seen by employing second-order (**c**) and third-order (**d**) correlation measurement. After processing numerical calculation, the second-order (**e**) and third-order (**f**) cumulant function gives rise to super-resolution image, revealing three clearly distinguishable NV centers. The figure is reproduced with permissions from Ref. [38] of APS. for coherent imaging, and is expressed as  
\[H_{inc}^{(n)}(\mathbf{x^{\prime}})\propto\int|O(\mathbf{x})|^{2n}\cdot[\text{ PSF}_{n}(\mathbf{x^{\prime}},\mathbf{x})]^{2}\,\text{d}\mathbf{x} \tag{6}\]  
for the incoherent imaging. Here, \(O(\mathbf{x})\) is the object aperture function, \(\text{PSF}_{n}(\mathbf{x^{\prime}},\mathbf{x})\) is the nth-order effective point-spread function with an effective wavelength \(\lambda/n\), giving rise to an n-time reduced FWHM in comparison to that of the \(\text{PSF}(\mathbf{x^{\prime}},\mathbf{x})\) for the conventional single-photon imaging. However, the nonlinear n-photon transmitting screen does not exist currently, and is quite difficult to be realized. Therefore, more feasible method needs to be explored. Besides the synchronous-position multi-photon interference [28; 31], Hanbury Brown-Twiss type interference (dependent on the coordinate difference of the observation points) has recently been introduced for realizing Heisenberg-resolution imaging [40]. The proposed imaging scheme is shown in Figure 8I, which can be realized with current well-developed technology. In the scheme, the object is placed on the top half of the Fourier plane of an entangled two-photon source [57; 59], while the bottom half of the Fourier plane is kept transparent.