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Optics Express

Optical states with higher stellar rank

Jan Provazník, Olga Solodovnikova, Radim Filip, Petr Marek

Published Oct 31, 2025• 2 citations
Publisher

Abstract

Quantum non-Gaussian states of traveling light fields are crucial components of quantum information processing protocols; however, their preparation is experimentally challenging. In this paper, we discuss the minimal requirements imposed on the quantum efficiency of photon number resolving detectors and the quality of the squeezing operation in an experimental realization of certifiable quantum non-Gaussian states of individual photonic states with three, four, and five photons.

DOI
10.1364/oe.569547

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PRX Quantum•Jun 3, 2026•1 citation
Quantum non-Gaussian states of traveling light fields are crucial components of quantum information processing protocols; however, their preparation is experimentally challenging. In this paper, we discuss the minimal requirements imposed on the quantum efficiency of photon number resolving detectors and the quality of the squeezing operation in an experimental realization of certifiable quantum non-Gaussian states of individual photonic states with three, four, and five photons.
Non-Gaussian quantum states and operations constitute essential resources for achieving quantum computational advantage and enabling quantum error correction in bosonic platforms. However, their generation in optical settings remains a challenging experimental task, often relying on probabilistic heralded protocols. Here, we present an in-depth analysis of the suitability of photon catalysis between low number Fock states and squeezed states for the generation of squeezed coherent state superpositions. We employ the stellar rank formalism to characterize the non-Gaussian complexity of input resources (including both states and measurements) and the generated states. This enables a systematic comparison of the fidelity between the catalyzed output and the target states to the maximum fidelity achievable by any protocol with the same non-Gaussian input resources. In this sense, we identify instances where the catalysis protocols considered here are provably optimal. We identify parameter regimes in which high-fidelity approximations of the target states can be achieved with minimal resources. Furthermore, we benchmark the performance of photon catalysis against Gaussian boson sampling-inspired protocols in terms of success probability and state quality, highlighting the advantages of deterministic Fock state sources. We also investigate the generation of related non-Gaussian resources including squeezed Fock states, relevant for quantum error correction. To account for experimental imperfections, we model losses across all optical modes using a Hilbert space truncation approach in the Fock basis and analyze the robustness of the generated states under realistic conditions. Our results quantify the trade-offs between non-Gaussian resource complexity, achievable fidelity, and losses in photon catalysis protocols, providing practical guidelines for near-term photonic implementations.
Non-Gaussian states of light are essential for numerous quantum information protocols; thus, certifying non-Gaussianity is crucial. Full quantum state tomography, commonly used for this purpose, is a complicated procedure and yields inconclusive results for strongly mixed states. Certifying non-Gaussianity through directly measurable parameters is a simpler alternative, typically achieved by measuring photon-number probabilities - either directly, using photon-number resolving detectors, or through Hanbury Brown--Twiss type measurements with single-photon detectors. Here, we demonstrate theoretically and experimentally that optical parametric amplification combined with conventional intensity detectors can effectively replace this approach without the need for photon-number resolution. In our method, we measure the mean photon number and the second-order correlation function for the amplified state. Using it, we successfully certify the non-Gaussianity of a heralded quasi-single-photon state. Since optical parametric amplification is a broadband and multimode process, our method provides a foundation for developing high-dimensional quantum technologies utilizing broadband multimode non-Gaussian states.
Stellar rank of quantum state of light quantifies the amount of non-Gaussian resources required for its generation. One popular and practical approach to certification of stellar rank is based on measurement of click statistics with an array of binary detectors that can only distinguish the presence and absence of photons. Specifically, it was shown that measurements with an array of m + 1 detectors allow one to certify stellar rank m of approximate Fock state | m 〉 , even when the state is subjected to losses or certain noise. Here we address the question as to how many click detectors are in principle required to certify stellar ranks higher than one. We show that two click detectors arranged in a Hanbury Brown-Twiss setup suffice. Interestingly, detection of stellar ranks higher than one is greatly facilitated by making the total detection efficiency of the detectors sufficiently low but well calibrated. Losses affect the response of the detection scheme in a way that can be exploited to certify stellar rank of higher Fock states. We explicitly construct the corresponding stellar rank witnesses and discuss dependence of the stellar rank thresholds on parameters of the considered setup. Our results reveal that it is possible to certify stellar ranks higher than 1 even with a minimalistic scheme that provides only very coarse-grained information about the photon number statistics of the characterized state.
Photon-number-resolving (PNR) detectors are essential components of photonic quantum technologies, yet thus far, no practical metric exists to certify how many photons they can genuinely resolve in a single measurement. Here we introduce an operational framework for quantifying the capability of a PNR detector to distinguish between different numbers of photons, i.e. its genuine resolution. In turn, we develop a practical and scalable protocol for certifying the genuine resolution of a detector, which is based on coherent state probes. We apply the method to a 28-pixel photon-number-resolving superconducting nanowire single-photon detector (PNR-SNSPD) and certify genuine four-outcome resolution. Our work highlights the critical requirements in terms of detector efficiency towards achieving high genuine resolution. This approach provides an operational benchmark for PNR detectors and fills a crucial gap in the characterization of photonic quantum devices.
Non-Gaussian states of light are essential for numerous quantum information protocols; thus, certifying non-Gaussianity (NG) is crucial. Full quantum state tomography, commonly used for this purpose, is complicated and yields inconclusive results for strongly mixed states. Certifying NG through directly measurable parameters is a simpler alternative, typically achieved by measuring photon-number probabilities—either directly, using photon-number resolving detectors, or through Hanbury Brown-Twiss type measurements with single-photon detectors. Here, we demonstrate, theoretically and experimentally, that phase-sensitive optical parametric amplification (OPA), followed by conventional intensity detection, can effectively replace this approach. Our proposed witness relies on the mean photon number (relative to that produced by the amplifier without any input) and the second-order correlation function after OPA. Both can be directly obtained from the measured amplified intensity and are invariant to detection losses. The method therefore requires neither photon-number resolution nor high detection efficiency. In a proof-of-principle experiment, we successfully certify the quantum NG of a heralded quasi-single-photon state. Since OPA is a broadband and multimode process, our method provides a foundation for developing high-dimensional quantum technologies utilizing broadband multimode non-Gaussian states.