Quantum entanglement is one of the most prominent features of quantum mechanics and forms the basis of quantum information technologies. Here we present a novel method for the creation of quantum entanglement in multipartite and high-dimensional systems. The two ingredients are (i) superposition of photon pairs with different origins and (ii) aligning photons such that their paths are identical. We explain the experimentally feasible creation of various classes of multiphoton entanglement encoded in polarization as well as in high-dimensional Hilbert spaces-starting only from nonentangled photon pairs. For two photons, arbitrary high-dimensional entanglement can be created. The idea of generating entanglement by path identity could also apply to quantum entities other than photons. We discovered the technique by analyzing the output of a computer algorithm. This shows that computer designed quantum experiments can be inspirations for new techniques.

A novel quantum imaging technique has recently been demonstrated in an experiment, where the photon used for illuminating an object is not detected; the image is obtained by interfering two beams, none of which ever interacts with the object. Here we present a detailed theoretical analysis of the experiment. We show that the object information is present only in the interference term and not in the individual intensities of the interfering beams. We also theoretically establish that the magnification of the imaging system depends on two wavelengths: the average wavelength of the photon that illuminates the object and the average wavelength of the photon that is detected. Our analysis affirms that the imaging process is based on the principle that quantum interference occurs when interferometric path information is unavailable.

We observe spatial fringes in the interference of two beams, which are controlled by a third beam through the phenomenon of induced coherence without induced emission. We show that the interference pattern depends on the alignment of this beam in an analogous way as fringes created in a traditional division-of-amplitude interferometer depend on the relative alignment of the two interfering beams. We demonstrate that the pattern is characterized by an equivalent wavelength, which corresponds to a combination of the wavelengths of the involved physical light beams.The relationship between path-information and interference is fundamental to quantum physics [1] and has been studied in various contexts [2][3][4]. A particularly remarkable manifestation of this relationship is the phenomenon of induced coherence without induced emission [5,6]. This phenomenon was used for fundamental tests of complementarity [7-9] and led to applications in imaging [10,11], metrology [12], spectrum shaping [13], and spectroscopy [14,15]. If two spatially separated nonlinear crystals produce photon pairs (signal and idler) by spontaneous parametric down-conversion (SPDC) [16,17], the emitted signal beams in general do not interfere, even if the two pump beams are mutually coherent. This is due to the fact that the idler beams carry information which source a down-converted pair originated from. By aligning the idler beams emitted by the two crystals, this information can be suppressed. If the path lengths are chosen accordingly, then lowest-order interference is observed between the two signal beams. It has been shown that path distinguishability can be introduced through a time delay between the two idler beams [18] or by attenuating the idler beam from one source using a partially transmissive filter. In the latter case, the resulting visibility was quantitatively connected to the transmission coefficient [5,6].Here, we analyze a situation, in which distinguishability is introduced in a different way. This is done by marginally misaligning the idler beams, either by tilting or by defocusing one with respect to the other. One could conjecture that this has the analogous effect of merely a reduced visibility. However, our experiment shows that because the misalignment can be described by a transverse phase gradient, it results in the observation of spatial interference fringes. This interference pattern leads to a reduction of visibility when intensities are obtained by integrating over a transverse section of the beam.We demonstrate analogies and differences between these fringes created by induced coherence and those cre- * Electronic address: armin.hochrainer@univie.ac.at † Electronic address: anton.zeilinger@univie.ac.at ated in traditional interferometry by division of amplitude, i.e. by interfering two beams, which are derived from a single beam by splitting it on a semi-reflecting surface. In a standard division-of-amplitude interferometer (e.g. a Michelson interferometer), a fringe pattern is observed, when one of the two interf...

We report a measurement of the transverse momentum correlation between two photons by detecting only one of them. Our method uses two identical sources in an arrangement in which the phenomenon of induced coherence without induced emission is observed. In this way, we produce an interference pattern in the superposition of one beam from each source. We quantify the transverse momentum correlation by analyzing the visibility of this pattern. Our approach might be useful for the characterization of correlated photon pair sources and may lead to an experimental measure of continuous variable entanglement, which relies on the detection of only one of two entangled particles.quantum correlations | single-photon interference | complementarity principle | photon indistinguishability | photonic spatial modes S patial entanglement (1) of photon pairs plays an important role in fundamental quantum mechanics (2-4), quantum cryptography (5, 6), quantum teleportation (7), and quantum computation (8). A widely used strategy to test spatial entanglement is to directly measure intensity correlations in both near and far fields of the source plane, which are interpreted as correlations in the transverse positions and momenta of the two photons. Measurements of the transverse momentum correlation between two photons have been performed using a variety of experimental methods (3), including the scanning of two detectors (9, 10) or two slits (11), using detector arrays (12), spatial light modulators (13), and cameras (14). All of these methods rely on the detection of both of the correlated photons. This fact restricts their applicability to situations where the wavelength of both photons lies in a spectral range, for which sufficiently efficient detectors are available.If two spatially separated nonlinear crystals emit pairs of photons (signal and idler) by the process of spontaneous parametric down conversion (SPDC) (15, 16), the two resulting signal beams in general do not interfere in lowest order (17). The absence of interference can be understood by the fact that the measurement on an idler photon would provide which-path information about a signal photon. However, lowest-order interference between the signal beams occurs if the respective idler beams are indistinguishable. This phenomenon, known as induced coherence without induced emission, was first observed experimentally in ref. 18, following a suggestion by Z.-Y. Ou of aligning the two idler beams (18,19). The interferometric visibility is reduced if path-distinguishability is introduced via different transmissions (18,19), different temporal delays (20), or different transverse sizes of the two idler beams (21,22). Recently, the phenomenon led to applications in imaging (23), spectroscopy (24, 25), metrology (26), spectrum shaping (27), and fundamental tests of complementarity (e.g., refs. 28-30).In this paper, we introduce and experimentally demonstrate a method for measuring the transverse momentum correlation between two photons by detecting only one of them. Our me...

We show that it is possible to generate a novel single-photon fringe pattern by using two spatially separated identical bi-photon sources. The fringes are similar to the ones observed in a Michelson interferometer and possess certain remarkable properties with potential applications. A striking feature of the fringes is that although the pattern is obtained by detecting only one photon of each photon pair, the fringes shift due to a change in the optical path traversed by the undetected photon. The fringe shift is characterized by a combination of wavelengths of both photons, which implies that the wavelength of a photon can be measured without detecting it. Furthermore, the visibility of the fringes diminishes as the correlation between the transverse momenta of twin photons decreases: visibility is unity for maximum momentum correlation and zero for no momentum correlation. We also show that the momentum correlation between the two photons of a pair can be determined from the single-photon interference pattern. We thus for the first time propose a method of measuring a two-photon correlation without coincidence or heralded detection.

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Quantum imaging with undetected photons is a recently introduced technique that goes significantly beyond what was previously possible. In this technique, images are formed without detecting the light that interacted with the object that is imaged. Given this unique advantage over the existing imaging schemes, it is now of utmost importance to understand its resolution limits, in particular what governs the maximal achievable spatial resolution. We show both theoretically and experimentally that the momentum correlation between the detected and undetected photons governs the spatial resolution — a stronger correlation results in a higher resolution. In our experiment, the momentum correlation plays the dominating role in determining the resolution compared to the effect of diffraction. We find that the resolution is determined by the wavelength of the undetected light rather than the wavelength of the detected light. Our results thus show that it is in principle possible to obtain resolution characterized by a wavelength much shorter than the detected wavelength.

This corrects the article DOI: 10.1103/PhysRevLett.118.080401.