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arXiv:2502.05047

Incoherent behavior of partially distinguishable photons

Emilio Annoni, Stephen C. Wein

Published Feb 7, 2025• 1 citation
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Abstract

Photon distinguishability is a key factor limiting quantum interference in photonic devices, directly impacting the performance of protocols such as Boson Sampling and photonic quantum computing. We present a basis-independent framework for analyzing multi-photon interference, identifying a necessary and sufficient condition under which distinguishability behaves as a stochastic error process. This condition enables any multi-photon state to be uniquely expressed as a classical mixture of partition states -- discrete configurations representing different patterns of photon distinguishability. We introduce an experimentally implementable operation, analogous to Pauli twirling, that enforces this condition without compromising computational hardness. The resulting probability distribution over partition states defines the system's incoherent distinguishability spectrum, which we show can be fully characterized through a specific set of experiments. Building on this structure, we also demonstrate the existence of an error mitigation strategy. This framework clarifies key challenges in defining genuine multi-photon indistinguishability, links previous perspectives on partial distinguishability, and provides a rigorous foundation for robust photonic protocols.

DOI
10.48550/arxiv.2502.05047

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Photon distinguishability is a key factor limiting quantum interference in photonic devices, directly impacting the performance of protocols such as Boson Sampling and photonic quantum computing. We present a basis-independent framework for analyzing multi-photon interference, identifying a necessary and sufficient condition under which distinguishability behaves as a stochastic error process. This condition enables any multi-photon state to be uniquely expressed as a classical mixture of partition states -- discrete configurations representing different patterns of photon distinguishability. We introduce an experimentally implementable operation, analogous to Pauli twirling, that enforces this condition without compromising computational hardness. The resulting probability distribution over partition states defines the system's incoherent distinguishability spectrum, which we show can be fully characterized through a specific set of experiments. Building on this structure, we also demonstrate the existence of an error mitigation strategy. This framework clarifies key challenges in defining genuine multi-photon indistinguishability, links previous perspectives on partial distinguishability, and provides a rigorous foundation for robust photonic protocols.
Photon distinguishability is a fundamental property manifested in multiphoton interference and one of the main sources of noise in any photonic quantum information processing. In this work, rather than relying on first-quantization methods, we build on a generating-function framework based on the phase-space formalism to characterize the effects of partial distinguishability on the interference of single-mode squeezed states. Our approach goes beyond commonly used models that represent distinguishability via additional noninterfering modes and captures genuine multiphoton interference effects induced by the overlap of the internal state of the photons. This description provides a clear physical account of how distinguishability gives rise to effective noise in Gaussian boson sampling protocols while enabling a systematic investigation of phase effects arising from the overlap of the internal states.
Photonic fusion operations are a central primitive in photonic quantum information processing and are commonly characterized by their success probability and resource cost. Here, we identify an additional way in which different implementations of fusion operations differ, namely in how they transform partial distinguishability. We consider distinguishability originating both from imperfect entangled resource states and from imperfect single photons. We find that different successful fusion events can induce markedly different distinguishability transformations depending on the fusion protocol, heralding outcome, and error model. Under a collective resource-state error model, we identify a modification to an existing fusion protocol that improves the fidelity of the remaining state on average. Generally, however, these transformations adversely affect distinguishability, and many successful fusion outcomes are accompanied by distinguishability degradation. Our results show that partial distinguishability is dynamically shaped by interference and measurements and should be considered alongside overall fusion success probability when evaluating photonic fusion protocols.
Several photonic quantum technologies rely on the ability to generate multiple indistinguishable photons. Benchmarking the level of indistinguishability of these photons is essential for scalability. The Hong-Ou-Mandel dip provides a benchmark for the indistinguishability between two photons, and extending this test to the multi-photon setting has so far resulted in a protocol that computes the genuine n-photon indistinguishability (GI). However, this protocol has a sample complexity that increases exponentially with the number of input photons for an estimation of GI up to a given additive error. To address this problem, we introduce new theorems that strengthen our understanding of the relationship between distinguishability and the suppression laws of the quantum Fourier transform interferometer (QFT). Building on this, we propose a protocol using the QFT for benchmarking GI that achieves constant sample complexity for the estimation of GI up to a given additive error for prime photon numbers, and sub-polynomial scaling otherwise, representing an exponential improvement over the state of the art. We prove the optimality of our protocol in many relevant scenarios and validate our approach experimentally on Quandela's reconfigurable photonic quantum processor, where we observe a clear advantage in runtime and precision over the state of the art. We therefore establish the first scalable method for computing multi-photon indistinguishability, which applies naturally to current and near-term photonic quantum hardware.
We demonstrate how boson sampling with photons of partial distinguishability can be expressed in terms of interference of fewer photons. We use this observation to propose a classical algorithm to simulate the output of a boson sampler fed with photons of partial distinguishability. We find conditions for which this algorithm is efficient, which gives a lower limit on the required indistinguishability to demonstrate a quantum advantage. Under these conditions, adding more photons only polynomially increases the computational cost to simulate a boson sampling experiment.
Multiphoton indistinguishability is a key resource for photonic quantum technologies, yet its characterization typically relies on resource-intensive methods. In this work, we develop two efficient and experimentally friendly protocols to estimate or bound the fidelity FindF_{\mathrm{ind}}Find​ of an NNN-photon state to the closest perfectly indistinguishable state. The first protocol applies to sources preparing separable states, uses a single Fourier interferometer together with photon-number-resolving detection, and yields tight two-sided bounds on FindF_{\mathrm{ind}}Find​. The second protocol combines randomized implementations of linear-optical interferometers with photon counting, enabling direct estimation of FindF_{\mathrm{ind}}Find​ for arbitrary NNN-photon states. Both protocols can certify Find=1−O(ε)F_{\mathrm{ind}} = 1 - \mathcal{O}(ε)Find​=1−O(ε) using provably optimal O(1/ε)\mathcal{O}(1/ε)O(1/ε) samples, in contrast to previous approaches which required prior assumptions on the model of partial distinguishability. Our methods, based on a multiphoton generalization of the Hong-Ou-Mandel test, bring the rigorous and operationally meaningful certification of multiphoton indistinguishability within reach of current photonic technologies.
Partial distinguishability affects multiphoton interference, which is central to large-scale photonic experiments such as Boson Sampling and to applications of multiphoton states across quantum technologies. Characterizing the internal degrees of freedom responsible for this behavior is challenging: in many scenarios they cannot be accessed directly and even their dimension or physical nature may be unknown \emph{a priori}. Here we introduce Gram-matrix tomography for independently prepared photons with pure or nearly pure internal states. For pure states, the Gram matrix contains all information required to predict the statistics of experiments that are insensitive to the internal degrees of freedom. We reconstruct this matrix, up to physically irrelevant phase choices, from a quadratic number of measurable Bargmann invariants (quantities encoding overlap magnitudes and collective phases). We access these quantities using two complementary approaches: targeted Fourier interferometers and two- and three-body photon-number correlations measured in a single random interferometer. We derive explicit error bounds that control the total-variation distance between predicted and actual outcome distributions for any subsequent experiment that is insensitive to the internal states. These guarantees account for finite sampling and deviations from purity. Crucially, the total sample complexity of the protocol is independent of the internal Hilbert-space dimension and polynomial under mild assumptions on overlaps of internal states. Our protocol uses standard interferometric measurements and turns them into a robust and operationally grounded characterization of partially distinguishable photons.