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Journal of the Optical Society of America B

Visibility-dependent resource scaling analysis for verifying single-particle quantum superposition via XOR games

Ting-Hsun Chien, Chieh-Yu Yao, Chun-Ming Chiang

Published Sep 17, 2026• 0 citations
Publisher

Abstract

The verification of single-particle quantum superposition states conventionally relies on the physical recombination of distinct spatial paths within an interferometer. However, this approach is highly susceptible to phase drifts and environmental noise in long-distance distributed quantum systems. To explore the feasibility of quantum verification without path recombination, this study establishes a numerical analysis model for single-particle quantum superposition based on the XOR-game framework. By introducing an ancilla photon indistinguishable from the test photon, the system analyzes the quantum verification capability through statistical correlations derived from local measurements and two-photon interference. Using the interference visibility V as the core parameter, we systematically investigate the minimum number of valid detection events required to achieve a 99% verification confidence level under different interference-quality conditions. The verification significance is evaluated using a one-sided binomial test against the classical local hidden variable (LHV) bound, with an expected-count criterion adopted to provide a deterministic resource-scaling baseline. Our numerical results show that under the ideal condition ( V =1.00), approximately 25 valid detection events are required to reach the 99% confidence threshold, whereas approximately 89 valid detection events are required at V =0.50. As the visibility decreases further, the required statistical resources increase rapidly. Quantitative fitting of the numerical confidence boundary shows that a power-law model provides a substantially better description than an exponential model, with the fitted scaling behavior close to the analytically predicted inverse-square relation N ∝ V −2 . The proposed resource analysis framework provides a quantitative baseline for evaluating the operational viability of task-based verification protocols in distributed photonic quantum architectures.

DOI
10.1364/josab.607774

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One of the most striking quantum phenomena is superposition, where one particle simultaneously inhabits different states. Most methods to verify coherent superposition are indirect, in that they require the distinct states to be recombined. Here, we adapt an game, in which a “test” photon is placed in a superposition of two orthogonal spatial modes, and each mode is sent to separated parties who perform local measurements on their modes without reinterfering the original modes. We show that by using a second identical “measurement” photon the parties are nonetheless able to verify if the test photon was placed in coherent superposition of the two spatial modes. We then turn this game into a resource-efficient verification scheme, obtaining a confidence that the particle is superposed which approaches unity exponentially fast. We demonstrate our scheme using a single photon, obtaining a 99% confidence that the particle is superposed with only 37 copies. Our work shows the utility of games to verify quantum resources, allowing us to efficiently detect quantum superposition without reinterfering the superposed modes.
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Measuring the phase stability of single photons is essential for evaluating performance in various phase-based quantum applications, yet systematic methodologies for quantitative characterization remain underexplored. In this work, we numerically demonstrate that time-resolved two-photon interference can be effectively utilized to estimate the phase noise of single photons. Our simulations reveal that as phase noise increases, the interference visibility decreases, causing the joint detection probability to approach the non-interfering classical envelope. Furthermore, we verify that the power spectral density (PSD) of the phase noise can be reconstructed using a derived mathematical relationship. This approach establishes a direct quantitative link between measurable interference patterns and the underlying statistical phase fluctuations. This method provides a practical framework for characterizing single-photon phase stability, offering significant potential for enhancing quantum technologies such as quantum key distribution and sensing.
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