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.
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.
Quantum non-Gaussian states and operations serve as fundamental resources for universal quantum computation, error correction, and high-precision metrology, extending beyond the Gaussian limits. While the stellar rank provides a rigorous hierarchical measure of non-Gaussianity, it remains challenging to determine experimentally. Conversely, witnesses of non-Gaussian features, based on the expectation values and variances of measurable observables, offer an accessible method for certifying non-Gaussian behavior but lack a direct connection to stellar rank. In this work, we establish a quantitative connection between these witnesses and stellar rank, demonstrating that the former can provide certifiable lower bounds on stellar rank. We introduce normalized expectation value and variance-based quantifiers and show that these witnesses form a consistent hierarchy of thresholds corresponding to stellar rank. Our results bridge the gap between abstract hierarchical measures and experimentally accessible quantifiers, enabling scalable certification of non-Gaussian states.
Quantifying non-Gaussianity is essential for assessing the computational power of bosonic quantum systems. We introduce a framework to witness and quantify non-Gaussianity from displaced parity measurements, and we directly apply it to experimental data obtained with superconducting cavities. Our method estimates fidelities with target pure states from the displaced parity measurements, enabling certification of non-Gaussianity via the stellar rank. We further introduce refined witnesses computed by measuring the value of the Wigner function at a few points, providing efficient certification tools outperforming existing ones. Applying these techniques to Fock, binomial, cat, and GKP experimental data, we rigorously certify non-Gaussian resources and establish a stellar rank of 8 for a GKP state, setting a new benchmark across all quantum platforms. Our methods enable a rigorous quantification of non-Gaussian quantum resources directly applicable to superconducting devices.
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.
Bosonic quantum error correction (QEC) relies on non-Gaussian bosonic encodings whose preparation cost is a central practical constraint. In this work, we use stellar rank as a resource measure to design and benchmark bosonic codes under finite non-Gaussian resources. For fixed cat and Gottesman--Kitaev--Preskill (GKP) code families, we show that finite stellar rank creates a trade-off among state approximability, energy, and logical protection under photon loss and photon-number dephasing, evaluated with optimal recovery. This trade-off implies that codewords with better ideal error-correction properties need not be optimal once finite-rank preparation constraints are imposed. Going beyond fixed-target codewords, we directly optimize bosonic encodings at fixed stellar rank, revealing noise-adapted code structures and concrete resource thresholds. Grid-like encodings emerge under photon loss, whereas approximately rotation-symmetric encodings arise under dephasing. In the optimized search, stellar rank k=2 suffices to surpass break-even for all dephasing strengths considered, while under photon loss the required rank increases with the loss rate. These results establish stellar rank as an operationally meaningful resource measure for bosonic QEC under practical state-preparation constraints.
The quantum theory of the electromagnetic field enables the description of multiphoton states exhibiting nonclassical statistical properties, often reflected in non-Gaussian phase-space distributions. While non-Gaussianity alone does not fully characterize quantum states, several classifications have been proposed to hierarchize non-Gaussian states according to physically or informationally relevant resources. Here, we provide a physical interpretation of non-Gaussianity and connect it to a computational perspective by showing how a prominent classification-the stellar rank-emerges as a limiting case of the roots of polynomials that univocally represent bosonic states defined with a quantized phase reference, namely the Majorana polynomials. A direct consequence of our results is a revised interpretation of both the stellar rank and non-Gaussianity itself: when superselection rules are properly taken into account, quadrature non-Gaussianity - and nonzero stellar rank - act as witnesses of particle entanglement, rather than being linked with photon addition to Gaussian states as previously assumed. In addition, we show that because the stellar rank depends on a specific choice of coherent states, its relation to computational resources and potential quantum advantage is inherently basis-dependent, being naturally tied to quadrature eigenstates as the computational basis. Motivated by this observation, we generalize the notion of stellar rank to arbitrary computational bases, thereby establishing it as a genuine witness of bosonic resources that may enable quantum advantage.