We show that it is possible for completely distinguishable particles to interfere postselectively without operating on, or indeed having any knowledge of, the distinguishing degree of freedom. In particular, we find a family of three-mode spatial interferometers that, upon inputting, say, a red photon in port 1 and a green photon in port 2, produce a state such that when vacuum is detected at one output, the two photons in the other two outputs will pass a HOM test, despite their frequency degree of freedom remaining untouched. In doing so we develop a general approach to distinguishability based on the Schmidt decomposition between particles' "System" and "Label" degrees of freedom, corresponding to what has been called unitary-unitary duality in many-body physics. This also gives insight into the relationship between the first and second quantized pictures, which is useful in bringing other quantum information concepts across from the former to the latter.
We show that it is possible for completely distinguishable particles to interfere postselectively without operating on, or indeed having any knowledge of, the distinguishing degree of freedom. In particular, we find a family of three-mode spatial interferometers that, upon inputting, say, a red photon in port 1 and a green photon in port 2, produce a state such that when vacuum is detected at one output, the two photons in the other two outputs will pass a HOM test, despite their frequency degree of freedom remaining untouched. In doing so we develop a general approach to distinguishability based on the Schmidt decomposition between particles' "System" and "Label" degrees of freedom, corresponding to what has been called unitary-unitary duality in many-body physics. This also gives insight into the relationship between the first and second quantized pictures, which is useful in bringing other quantum information concepts across from the former to the latter.
Particle distinguishability is a significant challenge for quantum technologies, in particular for photonics, where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantization, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum-information-inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes it clear that the HOM experiment is an instance of a (mixed) state discrimination protocol, which can be generalized to interferometers that discriminate unambiguously between ideal indistinguishable states and interesting distinguishable states, leading to bounds on the success probability of an arbitrary HOM generalization for multiple particles and modes. After setting out the first quantized formalism in detail, we consider several scenarios and provide a combination of analytical and numerical results for up to nine photons in nine modes. Although the quantum Fourier transform features prominently, we see that it is suboptimal for discriminating completely distinguishable states.
Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of n pairs of photons over the spectral Schmidt modes, whose normalization q2n is the probability that 2n photons interfere as indistinguishable. For two pairs of photons q4=(1+2P)/3, where P is the purity of the squeezed states (K=1/P is the Schmidt number). For a fixed purity P, the probability q2n decreases exponentially fast in n. For example, in the experimental Gaussian boson sampling of H.-S. Zhong et al., [Science 370, 1460 (2020)], the achieved purity $\mathbb{P}\ensuremath{\approx}0.938$ for the average number of photons $2n\ensuremath{\ge}43$ gives ${q}_{2n}\ensuremath{\lesssim}0.5$, i.e., close to the middle line between n indistinguishable and n distinguishable pairs of photons. In derivation of all the results, the first-order quantization representation based on the particle decomposition of the Hilbert space of identical bosons serves as an indispensable tool. The approach can be applied also to the generalized (non-Gaussian) squeezed states, such as those recently generated in the three-photon parametric down-conversion.
Multi-photon interference is the foundation of many applications regarding quantum technologies. Notably, the interference of multiple photons, registered using resolved measurements, is related to the permanent of a matrix, a problem known to be a #P-hard task. Traditionally, achieving high-visibility interference requires a high degree of indistinguishability between the interfering photons. However, this restriction can be circumvented by observing interference in the conjugate variables. This method has been successfully demonstrated using resolved measurements in the temporal and frequency domain. In this work, we extend these approaches to the spatial degrees of freedom of light. Theoretically, we demonstrate that the joint probability of detecting two photons at different positions in the output ports (3 and 4) of a beam-splitter is indeed determined by the permanent of a matrix related to the unitary transformation described by the interferometer. From an experimental perspective, we present experimental results for spatially-resolved interference in position for photons that are distinguishable in transverse momentum. We employ a setup to perform spatially-resolved measurements on photons with an adjustable distinguishability in both the central transverse momentum variable and the lateral displacement from the optical axis. This setup provides full simultaneous tunability over the distinguishability in the spatial variables. Additionally, by varying the lateral displacement, we observe how distinguishability in the position impacts the interference pattern. Overall, our results align well with the theoretical model, reinforcing the validity of our approach. Thus, this work offers new insights into the available degrees of freedom for studying of multi-photonic interference and its applications.
Photons in a two-path interferometer best embody wave-particle duality (WPD), which is a core concept of quantum theory. So far, the WPD relation is commonly written as V2+D2≤1, where V is the interference fringe visibility and D is path distinguishability, i.e., the distinguishability of which path a photon passed. This inequality is saturated only when the which-way marker (WWM), which embodies which-path information (WPI) via an internal degree of freedom of photons, such as polarization, is in a pure state. For mixed-state WWM, conventionally defined distinguishability underestimates the amount of WPI and thus does not saturate the WPD relation. Here, we introduce a generalized measure of distinguishability D that properly quantifies the WPI and saturates the WPD relation for all pure- and mixed-state WWM within a purification-based framework. To this end, mixed-state WWM is treated as a result of entanglement formation between the WWM and an external degree of freedom, e.g., environment, and D is defined so that it incorporates the total WPI shared between the WWM and the environment. We show that D thus defined is experimentally quantifiable, independently of V, without access to the environment. We experimentally evaluate V and D using true single photons generated in the completely mixed (unpolarized) state, and thus verify the saturated WPD relation.
Abstract Quantum interference is known to become extinct with distinguishing information, as illustrated by the ubiquitous double-slit experiment or the two-photon Hong–Ou–Mandel effect. In the former case single particle interference is destroyed with which-path information while in the latter bunching interference tails-off as photons become distinguishable. It has been observed that when more than two particles are involved, these interference patterns are in general a non-monotonic function of the distinguishability. Here we perform a comprehensive characterization, both theoretically and experimentally, of four-photon interference by analyzing the corresponding correlation functions, contemplating several degrees of distinguishability across different parameters. This study provides all the necessary tools to quantify the impact of multi-photon interference on precision measurements of parameters such as phase, frequency, and time difference. We apply these insights to quantify the precision in the estimation of an interferometric phase in a two-port interferometer using a four-photon state. Our results reveal that, for certain phase values, partially distinguishable multi-photon states can achieve higher Fisher information values compared to the two-photon experiment. These findings highlight the potential of distinguishable multi-photon states for enhanced precision in quantum metrology and related applications.
Two-photon interference effects arise because photons are indistinguishable particles. In the wellknown Hong-Ou-Mandel (HOM) effect, the transmission of two photons at a beam splitter interferes destructively with the reflection of both photons, requiring both photons to "bunch up" by leaving the beam splitter on the same side. Here, we show that the interference between locally propagating photons and photons exchanged by a mode swap can be implemented by post-selecting spatially separated photon outputs of a four-path interferometer. Even though the photons detected at spatially separated locations must have travelled along paths that never met up at the same beam splitter, the Hong-Ou-Mandel effect can be observed in correlations between the output ports that originate from the association of detection events with non-local output modes defined by the two single photon inputs. Local phase shifts can be used to map out non-classical correlations between the photons detected at different output locations, clarifying the role of linear optics in generating entanglement between spatially separated photons. Our work thus establishes a fundamental relation between multiphoton interference and entanglement, opening the door to new possibilities in optical quantum technologies.