Heralded generation of entanglement with photons

TL;DR

Heralded generation of entanglement with photons surveys heralded entangled photonic states as a scalable alternative to postselected methods, detailing theory, experiments, and practical error sources. It spans Bell, NOON, GHZ, Werner, and W states, and discusses strategies to boost success probability (multiplexing, boosting, bleeding) and circuit-design methods (analytical Gröbner-basis and numerical optimization). The work highlights applications in measurement-based and fusion-based quantum computing, entanglement swapping for quantum networks, and quantum metrology, emphasizing how heralding enables on-demand, high-fidelity resources. It also identifies remaining challenges—loss, indistinguishability, detector imperfections, and the need for larger resource states and integrated, scalable hardware—pointing to a path toward fault-tolerant photonic quantum information processing.

Abstract

Entangled states of photons form the backbone of many quantum technologies. Due to the lack of effective photon-photon interactions, the generation of these states is typically probabilistic. In the prevailing but fundamentally limited generation technique, known as postselection, the target photons are measured destructively in the generation process. By contrast, in the alternative approach -- heralded state generation -- the successful creation of a desired state is verified by the detection of ancillary photons. Heralded state generation is superior to postselection in several critical ways: It enables free usage of the prepared states, allows for the success probability to be arbitrarily increased via multiplexing, and provides a scalable route to quantum information processing using photons. Here, we review theoretical proposals and experimental realizations of heralded entangled photonic state generation, as well as the impact of realistic experimental errors. We then discuss the wide-ranging applications of these states for quantum technologies, including resource states in linear optical quantum computing, entanglement swapping for repeater networks, fundamental physics, and quantum metrology.

Figures (11)

• Figure 1: Postselected and heralded entangled state generation circuits. (a): In postselected circuits, $n$ photons are injected into an $m$-mode linear optical circuit, $U$. Detectors are assigned to all modes such that all photons, notwithstanding loss, are measured. Typically, successful measurement patterns require only one photon being measured in certain groups of target modes (circled in purple). This is shown after the detectors where ticks correspond to photon detections and dashed ticks correspond to undetected photons, which can be due to the loss of a photon or lack of photon number resolution in the detector used. Green denotes a successful outcome, and red denotes an unsuccessful outcome (which might be mistakenly categorised as successful due to losses or detector inefficiencies, as would be the case for the far-right scenario shown). (b): In heralded entangled state generation, $n$ photons are injected into an $m$-mode circuit where a subset of modes, the heralding modes (circled in orange), are measured. Through the measurement of a subset of photons acting as ancillas in the heralding modes, the creation of an entangled state across the remaining, target modes is heralded. Since the photons encoding this state have not been measured, they can undergo further quantum information processing. A successful heralding pattern is shown verifying the creation of a state $\ket{\psi}$, whereas incorrect patterns denote a failed state generation attempt, denoted with a red cross in the target modes.
• Figure 2: Postselected Bell state generator circuit. The beam splitters in each pair of modes are used to create a uniform superposition, then one mode from each dual-rail is swapped. $U_1$ and $U_2$ can be configured to measure the photons in a desired Pauli basis to reveal quantum correlations or characterise the state. The postselection rule only keeps measurement patterns where each pair of modes contains only one photon.
• Figure 3: Single photon sources fall into two broad classes: pair sources (a) and single emitters (b). Pair sources emit photon pairs probabilistically through a nonlinear process such as spontaneous four-wave mixing, which uses a material's inherent optical nonlinearity in conjunction with an intense pump laser. Photons are separated using a degree of freedom of the photons, for example frequency using a spectral filter, and by measuring one photon in a pair, the presence of the other single photon can be heralded. Single emitters typically work by having an optical energy transition between some ground state, $\ket{g}$, and excited state, $\ket{e}$, whose energy difference corresponds to that of the desired single photon. As such, when the transition from the ground to excited state is pumped, for example optically with a laser, a single photon is emitted when the system relaxes from the excited to ground state.
• Figure 4: The simplest example of a pseudo-PNR detection set-up. (a) With a 50% probability, the two incident photons are split by the beam splitter and are resolved and registered as a two-photon event. (b) and (c): With 25% probability each, the two photons arrive together at one or the other detector and are thus not registered as a two-photon event.
• Figure 5: Schemes for heralded Bell state generation. Ticks suggest the (or, in the case of multiple options, one) detector "click" pattern that heralds the generation of a Bell state on the target modes. The labels $x$P$y$M indicate that the schemes are using $x$ photons in a linear optical circuit comprising $y$ modes, and DFT represents a linear optical circuit to carry out the discrete Fourier transform.