Highly nonclassical states of light - such as the approximate Gottesman-Kitaev-Preskill states, states exhibiting cubic nonlinear squeezing, or cat-like states - can be generated from experimentally accessible Gaussian states via photon counting measurements on selected modes, conditioned on specific outcomes of these heralding events. A simplest yet important example of this approach involves performing photon number measurements on one mode of a two-mode entangled Gaussian state. The heralding probability of this scheme is a key figure of merit, as it determines the generation rate of the target nonclassical state. In this work we show that the maximum heralding probability for the two-mode setting can be calculated analytically, and we investigate its dependence on the number of detected photons n. Our results show that the number of required experimental trials scales only polynomially with n. Generation of highly complex optical quantum states with high stellar rank is thus in principle possible in this setting, given access to sufficiently strong squeezing.
Highly nonclassical states of light - such as the approximate Gottesman-Kitaev-Preskill states, states exhibiting cubic nonlinear squeezing, or cat-like states - can be generated from experimentally accessible Gaussian states via photon counting measurements on selected modes, conditioned on specific outcomes of these heralding events. A simplest yet important example of this approach involves performing photon number measurements on one mode of a two-mode entangled Gaussian state. The heralding probability of this scheme is a key figure of merit, as it determines the generation rate of the target nonclassical state. In this work we show that the maximum heralding probability for the two-mode setting can be calculated analytically, and we investigate its dependence on the number of detected photons n. Our results show that the number of required experimental trials scales only polynomially with n. Generation of highly complex optical quantum states with high stellar rank is thus in principle possible in this setting, given access to sufficiently strong squeezing.
Generation of highly non-classical quantum states of light is essential for optical quantum information processing and quantum metrology. Given the lack of sufficiently strong nonlinear interactions between optical fields, the commonly employed optical quantum-state preparation schemes are conditional, based on nonlinearity induced by heralding photon number measurement on a part of a multimode squeezed Gaussian state. Development and optimization of such probabilistic quantum-state engineering schemes represents one of the central challenges in current quantum optics. As technology advances and experiments progress to detection of higher numbers of photons, the maximization of the heralding probability becomes essential to ensure sufficiently high state-preparation rates. Here, we show that for the conditional quantum state preparation schemes based on Gaussian states and photon number measurements the maximization of the heralding probability can be formulated as finding solution to a system of polynomial equations, which offers an efficient way to find the optimal configuration and allows us to apply techniques dedicated specifically to solving such systems of equations. Our approach can seamlessly incorporate bounds on the available single-mode quadrature squeezing, which is highly experimentally relevant. We mainly consider generation of finite superpositions of Fock states but show that the approach can be straightforwardly extended to generation of squeezed superpositions of Fock states. We focus on Gaussian states with vanishing coherent displacements, hence the conditionally generated states have well defined photon number parity. We illustrate our general methodology through examples of generation of single-mode states with two and three heralding modes, and generation of two-mode states with two heralding modes.
We propose a heralded entanglement generation scheme based on Gaussian sources enhanced by photon addition and subtraction operations. By combining single-mode squeezing, linear interferometers, and conditional photon-number measurements on ancillary modes, our model can probabilistically generate dual-rail encoded Bell, GHZ, and W states. We systematically optimize the squeezing parameters and interferometer settings to maximize both the heralding success probability and the fidelity with the target states. Our results show that photon addition and subtraction significantly enhance the non-classicality of the output states and improve generation performance, while maintaining computational efficiency comparable to single-photon source models. We further analyze the robustness of the scheme under parameter perturbations and find that its performance remains stable under realistic experimental imperfections. This work provides a versatile and experimentally feasible framework for scalable heralded entanglement generation using Gaussian resources with non-Gaussian operations.
Non-Gaussian quantum states are crucial to fault-tolerant quantum computation with continuous-variable systems. Usually, generation of such states involves trade-offs between success probability and quality of the resultant state. For example, injecting squeezed light into a multimode interferometer and postselecting on certain patterns of photon-number outputs in all but one mode, a fundamentally probabilistic task, can herald the creation of cat states, Gottesman-Kitaev-Preskill (GKP) states, and more. We consider the addition of a non-Gaussian resource state, particularly single photons, to this configuration and show how it improves the qualities and generation probabilities of desired states. With only two modes, adding a single-photon source improves GKP-state fidelity from 0.68 to 0.95 and adding a second then increases the success probability eightfold; for cat states with a fixed target fidelity, the probability of success can be improved by factors of up to four by adding single-photon sources. These demonstrate the usefulness of additional commonplace non-Gaussian resources for generating desirable states of light.
Schrödinger cat states are important non-Gaussian resources for quantum information processing and quantum-enhanced metrology. Although photon subtraction from squeezed vacuum states has proven useful for generating such states in the optical domain, scaling to large amplitudes while maintaining high fidelity and heralding probability remains challenging. To address these obstacles, we analyze alternating squeezing and photon-number-subtraction operations. Rather than optimizing for a single heralding outcome, we formulate cat-state generation as a fidelity-constrained multibranch problem, jointly optimizing the total probability of all photon-number outcomes satisfying a prescribed fidelity threshold at one common experimental setting. Already for plain photon subtraction, this provides a more operational measure of the usable generation probability and can favor parameter configurations that differ from single-branch optima. Considering experimental constraints on available squeezing and photon-number-resolving capability, we demonstrate improvements in fidelity and success probability compared with plain photon subtraction, without necessarily increasing the required squeezing resources. Without restrictions on photon-number resolution, we find that the optimal architecture of the alternating scheme reduces to one of two single-subtraction endpoint configurations, depending on the available squeezing resources. When photon-number resolution is additionally limited, genuinely alternating branches can become advantageous. We derive analytical expressions for the conditional states, cat-state fidelities, and heralding probabilities, enabling efficient multibranch optimization and a systematic analysis of resource trade-offs, sensitivity to experimental parameter deviations, and heralding loss. Together, these results identify the architecture best suited to a given resource budget.
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.
Advanced quantum technologies rely on non-Gaussian states of light, essential for universal quantum computation, fault-tolerant error correction, and quantum sensing. Their practical realization, however, faces hurdles: Simulating large multimode generators is computationally demanding, and benchmarks such as the do not capture how effectively photon detections yield useful non-Gaussianity. We address these challenges by introducing the ( s 0 , δ 0 ) , a continuous and operational measure that goes beyond stellar rank. Leveraging these parameters, we develop a universal optimization method that reduces photon-number requirements and greatly enhances success probabilities while preserving state quality. Applied to the Gottesman-Kitaev-Preskill state generation, for example, our method cuts the required photon detections by a factor of 3 and raises the preparation probability by nearly 10 8 . Demonstrations across cat states, cubic phase states, Gottesman-Kitaev-Preskill states, and even random states confirm broad gains in experimental feasibility. Our results provide a unifying principle for resource-efficient non-Gaussian state generation, charting a practical route toward scalable optical quantum technologies and fault-tolerant quantum computation.