A permutation of n particles is either even or odd according to the parity of the number of pairwise swaps required to realize it. Classically, deterministic identification of this parity requires n distinct local labels, one for each particle. A recently proposed quantum protocol [A. Diebra et al., Phys. Rev. Lett. 135, 260603 (2025)] reduces this requirement to a local dimension dmin = ⌈√n⌉ by exploiting multipartite entanglement, while retaining deterministic parity identification after a single application of the permutation. Here, we experimentally realize the minimal nontrivial case, n = 3 and d = 2, by mapping the parity-sensitive state structure onto a coherent superposition of Hermite–Gaussian spatial modes. A programmable multi-plane light converter implements all six elements of the permutation group S3, and a single projection measurement separates the even and odd permutation classes with a visibility of V ≃ 0.78. Our experiment reproduces, with structured classical light, the operational mechanism underlying this multipartite quantum-advantage protocol and establishes programmable spatial-mode processing as a versatile platform for exploring quantum protocols based on symmetry, interference, and reduced dimensional resources.
A permutation of n particles is either even or odd according to the parity of the number of pairwise swaps required to realize it. Classically, deterministic identification of this parity requires n distinct local labels, one for each particle. A recently proposed quantum protocol [A. Diebra et al., Phys. Rev. Lett. 135, 260603 (2025)] reduces this requirement to a local dimension dmin = ⌈√n⌉ by exploiting multipartite entanglement, while retaining deterministic parity identification after a single application of the permutation. Here, we experimentally realize the minimal nontrivial case, n = 3 and d = 2, by mapping the parity-sensitive state structure onto a coherent superposition of Hermite–Gaussian spatial modes. A programmable multi-plane light converter implements all six elements of the permutation group S3, and a single projection measurement separates the even and odd permutation classes with a visibility of V ≃ 0.78. Our experiment reproduces, with structured classical light, the operational mechanism underlying this multipartite quantum-advantage protocol and establishes programmable spatial-mode processing as a versatile platform for exploring quantum protocols based on symmetry, interference, and reduced dimensional resources.
A permutation of n particles is either even or odd according to the parity of the number of pairwise swaps required to realize it. Classically, deterministic identification of this parity requires n distinct local labels, one for each particle. A recently proposed quantum protocol [A. Diebra et al., Phys. Rev. Lett. 135, 260603 (2025)] reduces this requirement to a local dimension dmin = ⌈√n⌉ by exploiting multipartite entanglement, while retaining deterministic parity identification after a single application of the permutation. Here, we experimentally realize the minimal nontrivial case, n = 3 and d = 2, by mapping the parity-sensitive state structure onto a coherent superposition of Hermite–Gaussian spatial modes. A programmable multi-plane light converter implements all six elements of the permutation group S3, and a single projection measurement separates the even and odd permutation classes with a visibility of V ≃ 0.78. Our experiment reproduces, with structured classical light, the operational mechanism underlying this multipartite quantum-advantage protocol and establishes programmable spatial-mode processing as a versatile platform for exploring quantum protocols based on symmetry, interference, and reduced dimensional resources.
We investigate the task of identifying the parity (odd vs even) of an unknown permutation applied to n particles. Classically, using fewer than n distinct labels per particle limits the success probability to random guessing, whereas quantum mechanics, exploiting entanglement in both preparation and measurement, accomplishes the task perfectly with as few as ⌈n⌉ levels per particle [\href{https://doi.org/10.1103/yhyv-xnwq}{PRL {\bf 135}, 260603 (2025)}]. We show that even without entangled preparation, quantum theory still offers a probabilistic advantage over classical strategies. Moreover, such product preparations yield perfect success in locally quantum theories, where elementary systems are quantum but their composition follows the minimal tensor product structure of generalized probabilistic theories (GPTs). We further identify GPT models that accomplish the task with certainty without requiring entanglement either at the preparation stage or at the measurement stage. Our central result establishes that the linear dimension of the elementary systems, rather than entanglement, is the fundamental resource governing the existence of probabilistic advantage in the permutation parity problem. In particular, below the required dimension threshold, no amount of entanglement can improve upon the random-guessing limit.
We establish a sharp quantum advantage in determining the parity (even-odd) of an unknown permutation applied to any number n≥3 of particles. Classically, this is impossible with fewer than n labels, being that the success is limited to random guessing. Quantum mechanics does it with certainty with as few as ⌈sqrt[n] ⌉ distinguishable states per particle, thanks to entanglement. Below this threshold, not even quantum mechanics helps: both classical and quantum success are limited to random guessing. For small n, we provide explicit expressions for states that ensure perfect parity identification. We also assess the minimum entanglement these states need to carry, finding it to be close to maximal, and even maximal in some cases. The task requires no oracles or contrived setups and provides a simple, rigorous example of genuine quantum advantage.
The information-carrying capacity of a single photon can be vastly expanded by exploiting its multiple degrees of freedom: spatial, temporal, and polarization. Although multiple qubits can be encoded per photon, to date only two-qubit single-photon quantum operations have been realized. Here, we report an experimental demonstration of three-qubit single-photon, linear, deterministic quantum gates that exploit photon polarization and the two-dimensional spatial-parity-symmetry of the transverse single-photon field. These gates are implemented using a polarization-sensitive spatial light modulator that provides a robust, non-interferometric, versatile platform for implementing controlled unitary gates. Polarization here represents the control qubit for either separable or entangling unitary operations on the two spatial-parity target qubits. Such gates help generate maximally entangled three-qubit Greenberger-Horne-Zeilinger and W states, which is confirmed by tomographical reconstruction of single-photon density matrices. This strategy provides access to a wide range of three-qubit states and operations for use in few-qubit quantum information processing protocols.Photons are essential for quantum information processing, but to date only two-qubit single-photon operations have been realized. Here the authors demonstrate experimentally a three-qubit single-photon linear deterministic quantum gate by exploiting polarization along with spatial-parity symmetry.
We describe a mode sorter for two-dimensional parity of transverse spatial states of light based on an out-of-plane Sagnac interferometer. Both Hermite-Gauss (HG) and Laguerre-Gauss (LG) modes can be guided into one of two output ports according to the two-dimensional parity of the mode in question. Our interferometer sorts HG(nm) input modes depending upon whether they have even or odd order n+m; it equivalently sorts LG(l)(p) modes depending upon whether they have an even or odd value of their orbital angular momentum l. It functions efficiently at the single-photon level, and therefore can be used to sort single-photon states. Due to the inherent phase stability of this type of interferometer as compared to those of the Mach-Zehnder type, it provides a promising tool for the manipulation and filtering of higher order transverse spatial modes for the purposes of quantum information processing. For example, several similar Sagnacs cascaded together may allow, for the first time, a stable measurement of the orbital angular momentum of a true single-photon state. Furthermore, as an alternative to well-known holographic techniques, one can use the Sagnac in conjunction with a multi-mode fiber as a spatial mode filter, which can be used to produce spatial-mode entangled Bell states and heralded single photons in arbitrary first-order (n+m = 1) spatial states, covering the entire Poincar e sphere of first-order transverse modes.
The emergence and development of quantum optics as a discipline in the latter half of the twentieth century has established photons as one of the most promising candidates for future technologies in computing, metrology, communication, and imaging. In imaging and sensing, a series of recent theoretical works and proof-of-concept experiments have shown that by examining the mode structure of photons, one can build optical systems with resolutions below the Rayleigh–Abbe limit. Building upon these developments, in one of our works, we investigate the use of higher-order Hermite–Gauss (HG) modes for sensing optically induced transverse displacements. We show that projective measurements onto neighboring modes are optimal and sufficient in the small-displacement regime, and that the sensitivity—quantified by the Fisher information—scales linearly with the order of the mode used. We generalize the calculations to arbitrary displacements and varying degrees of coherence between the displaced modes, and perform a proof-of-concept experiment with higher-order HG modes, obtaining an order-of-magnitude enhancement in sensitivity compared to the fundamental mode. This approach enables enhanced displacement sensitivity with minimal measurements and highlights the potential of structured light for ultrasensitive displacement sensing, particularly for birefringence measurements with broadband or low-coherence light sources. We also consider displacement sensing in the light field originating from a biphoton source, which possesses rich high-dimensional correlations in its biphoton mode structure. We show that the Fisher information is proportional to the degree of entanglement, or equivalently, its Schmidt rank. In addition to sensing, photons structured in their transverse degrees of freedom also provide a convenient toolkit for quantum computing and communication due to the high dimensionality of their Hilbert space. In one of the works presented, we introduce a technique for nonlocal transfer of high-dimensional computational results using spatial correlations in photon pairs. We expect these findings to be useful in future photonic quantum networks, where computational resources are distributed and users can access the results of complex computations via a centrally located quantum processor. We also experimentally implement the protocol in down-converted photon pairs and demonstrate the transfer of unitaries in one-dimensional and two-dimensional transverse momentum spaces. In another work, we address the noise inherent in quantum computing circuits and simulate the interaction between a qubit and its environment by manipulating transverse momentum modes. This study, performed by fabricating three liquid-crystal metasurfaces, provides a practical solution for quantum state purification, demonstrating a versatile approach for simulating open-qubit dynamics. Combining the transverse spatial structure with polarization yields the so-called vector modes of photons. Structuring two-qubit interference, or the Hong–Ou–Mandel (HOM) effect, offers significant potential to extend the applications of such modes beyond the classical regime. We demonstrate the simultaneous generation of all four Bell states in polarization by exploiting HOM interference of vector modes. These results represent a significant step toward manipulating HOM interference within structured photons, offering promising applications in high-dimensional quantum information processing, quantum communication, and sensing. On the algorithmic side of photonic information processing, we propose a quantum pattern recognition algorithm to identify images obtained from quantum imaging techniques with correlated photons—such as “interaction-free” or “ghost” imaging. Interfacing these imaging techniques with quantum pattern recognition promises a significantly improved image acquisition and identification system, with potential applications in medical diagnostics and biological imaging.