Linear optical networks driven by quantum states of light are important building blocks of photonic quantum technologies. They access large bosonic Hilbert spaces through multiphoton interference. At the same time, their dynamics are generated by single-particle mode transformations, thereby defining a highly structured subset of multiphoton unitaries and setting boundary on linear optics capability. To elucidate this boundary, we reveal an underlying fine-grained symmetry structure that partitions the multiphoton operator space into invariant subspaces and generates a hierarchy of invariants. We experimentally confirm the conservation of high-order invariants and demonstrate their operational utility in characterizing state reachability and the metrological capability of multiphoton probes. Our framework provides a symmetry-based perspective for understanding and harnessing structured multiphoton dynamics across photonic quantum technologies.
Linear optical networks driven by quantum states of light are important building blocks of photonic quantum technologies. They access large bosonic Hilbert spaces through multiphoton interference. At the same time, their dynamics are generated by single-particle mode transformations, thereby defining a highly structured subset of multiphoton unitaries and setting boundary on linear optics capability. To elucidate this boundary, we reveal an underlying fine-grained symmetry structure that partitions the multiphoton operator space into invariant subspaces and generates a hierarchy of invariants. We experimentally confirm the conservation of high-order invariants and demonstrate their operational utility in characterizing state reachability and the metrological capability of multiphoton probes. Our framework provides a symmetry-based perspective for understanding and harnessing structured multiphoton dynamics across photonic quantum technologies.
Multiphoton correlations in linear photonic quantum networks are governed by matrix permanents. Yet, surprisingly few systematic properties of these crucial algebraic objects are known. As such, predicting the overall multiphoton behavior of a network from its individual building blocks typically defies intuition. In this work, we identify sequences of concatenated two-mode linear optical transformations whose two-photon behavior is invariant under reversal of the order. We experimentally verify this systematic behavior in parity-time-symmetric complex interferometer arrangements of varying compositions. Our results underline new ways in which quantum correlations may be preserved in counterintuitive ways, even in small-scale non-Hermitian networks.
Multiphoton quantum interference underpins fundamental tests of quantum mechanics and quantum technologies. Consequently, the detrimental effect of photon distinguishability in multiphoton interference experiments can be catastrophic. Here we employ correlation measurements in the photonic inner modes, time or frequency, to restore quantum interference between photons differing in their colors or injection times in arbitrary linear optical networks, without the need for additional filtering or postselection. Interestingly, we demonstrate how harnessing the multiphoton inner-mode quantum information enables us to infer information about symmetries of multiphoton networks and states and to observe arbitrary degrees of W-state entanglement between a small number of photons with a fixed interferometer. These results are therefore of profound interest for future applications of universal inner-mode-resolved linear optics across fundamental science and quantum technologies with photons with experimentally different spectral properties.
Over the past few years, various methods have been developed to engineer and to exploit the dynamics of photonic quantum states as they evolve through linear optical networks. Recent theoretical works have shown that the underlying Lie algebraic structure plays a crucial role in the description of linear optical Hamiltonians, as such formalism identifies intrinsic symmetries within photonic systems subject to linear optical dynamics. Here, we experimentally investigate the role of Lie algebras applied to the context of Boson sampling, a pivotal model to the current understanding of computational complexity regimes in photonic quantum information. Performing experiments of increasing complexity, realized within a fully reconfigurable photonic circuit, we show that sampling experiments do indeed fulfill the constraints implied by a Lie algebraic structure. In addition, we provide a comprehensive picture about how the concept of Lie algebraic invariant can be interpreted from the point of view of n th -order correlation functions in quantum optics. Our work shows how Lie algebraic invariants can be used as a benchmark tool for the correctness of an underlying linear optical dynamics within typical interferometric architectures for photonic quantum information. This opens avenues for the use of algebraic-inspired methods as analysis tools for photon-based quantum computing protocols.
The precise manipulation of quantum states of light lies at the heart of advancing photonic technologies, yet scalable solutions for hybrid photonic systems—where disparate optical modes such as polarization, time-bin, and frequency must coherently interact—remain a critical challenge. While current literature demonstrates various energy-conserving devices that effectively map between Hilbert spaces of unequal dimensions, there is a growing need for frameworks that address variable-dimensional transformations in a more unified manner. In this paper, we introduce an operator-level framework that systematically integrates both unitary and rectangular isometric energy-conserving transformations in multi-dimensional photonic systems. Unlike prior mode-specific approaches, our formalism enables modular modeling of hybrid optical modes within effective signal spaces, offering practical tools for scalable photonic quantum processors. By classifying these transformations into fixed- and variable-dimensional regimes, we derive universal design criteria for energy-conserving photonic signal processors. This work not only advances the fundamental understanding of photonic quantum processing but also delivers actionable tools for engineering next-generation scalable optical technologies, including multi-purpose, programmable photonic circuits, energy-efficient optical signal processing in reconfigurable networks, and high-dimensional quantum key distribution with hybrid mode encoding.
We develop a structural ontology of quantum light in which the photon is not a particle, not a wave, and not an excitation of a field, but the boundary-projection of an underlying β-structure. The β-field encodes the structural capacity of existence: intrinsic configurations with 0 < β < 1 support coherent organization and allow a system to maintain internal tension, whereas the limit β → 1 eliminates all internal structure and produces a projection event. In this framework, the wavefunction ψ is not an element of physical reality but a linear statistical shadow of β-modes; measurement accesses only the projection frequencies |ψ|² because projection occurs solely at β = 1, where intrinsic structure no longer exists. A “photon” therefore corresponds not to a single β-configuration but to an equivalence class of β-modes that collapse to the same β = 1 boundary signature. The apparent indivisibility, lack of rest frame, invariant propagation speed, discrete polarization, interference capability, and entanglement of photonic phenomena emerge from this multi-existence ensemble rather than from particle attributes. Wave-like behavior arises entirely from the β-phase geometry in the intrinsic domain, while particle-like detections correspond to boundary collapses of β-ensembles. Interference, delayed-choice behavior, polarization transformations, and entanglement correlations follow naturally from the global coherence of β-structure prior to projection. The framework preserves all operational predictions of quantum mechanics while supplying an ontological layer clarifying why ψ is unobservable and why photonic behavior appears paradoxical under particle or wave descriptions. The model predicts boundary-sensitive and geometry-dependent deviations—nanometer-scale fringe shifts, polarization-mode distortions, structured entanglement modulation, and stable residual phases in balanced interferometers—that originate from the nonlinear structural response of β near projection. These effects are experimentally testable with current photonic technology. The β-projection ontology thus provides a unified, falsifiable, and structurally grounded account of quantum light, explaining photonic phenomena as manifestations of β-mode ensembles interacting with the nonlinear boundary at β = 1 rather than as properties of a classical or quantized particle.
Single- and two-mode multiphoton states are the cornerstone of many quantum technologies, e.g., metrology. In the optical regime, these states are generally obtained combining heralded single photons with linear optics tools and post-selection, leading to inherent low success probabilities. In a recent paper [A. Gonz\'alez-Tudela et al., Phys. Rev. Lett. 118, 213601 (2017)], we design several protocols that harness the long-range atomic interactions induced in waveguide QED to improve fidelities and protocols of single-mode multiphoton emission. Here, we give full details of these protocols, revisit them to simplify some of their requirements, and also extend them to generate two-mode multiphoton states, such as Yurke or NOON states.