Distillation of continuous variable qudits from single photon sources: A cascaded approach

Creation of high fidelity photonic quantum states in the continuous variable regime is indispensable for the implementation of quantum technologies universally. However, this is a challenging task as it requires higher nonlinearity or larger Fock states. In this article, we surmount this necessity by using a linear optical setup with a cascaded arrangement of beam splitters that relies solely on single photon sources and single photon detectors to tailor desired single mode nonclassical states. To show the utility of this setup, we demonstrate the generation of displaced higher photon states with unit fidelity and the family of Schrodinger cat states above $98%$ fidelity. In addition, we manifest the generation of GKP resource states, such as ON states and weak cubic phase states with $99%$ fidelity. Creating such a variety of important states in this single setup is made feasible by stating the output in the form of displaced qudits. This figure of merit facilitates efficient identification and optimization of input parameters required to generate the target single mode quantum states. We also account for the experimental imperfections by incorporating detector inefficiencies and non-unit single photon sources. This cascaded setup will assist the experimentalists to explore the feasible creation of target states using currently available resources, such as single photon sources and single photon detectors.

💡 Research Summary

The paper presents a fully linear‑optical protocol for generating a broad class of high‑fidelity continuous‑variable (CV) quantum states using only single‑photon sources, single‑photon detectors, and a cascade of beam splitters (BS). The authors start from a coherent state |α⟩ and inject a single photon into each BS. Conditional detection of a photon on one output port implements quantum optical catalysis, producing a new state that feeds the next BS. After ℓ stages the output can be written compactly as a “displaced qudit” (DQ):

|Ψ⟩ℓ = D(α√X₀) ∑{p=0}^{ℓ} A_{ℓp}|p⟩,

where X₀ = ∏{i=1}^{ℓ}R_i (the product of the reflectivities) and the coefficients A{ℓp} are analytic functions of the reflectivities, Touchard polynomials, and Stirling numbers of the second kind. This representation makes the otherwise cumbersome multi‑parameter state transparent: the DQ is simply a finite‑dimensional superposition of Fock states displaced by a Gaussian operator.

Using this formalism the authors systematically design the cascade parameters (α and each R_i) to generate several important CV resources:

1. Higher‑order Fock states – By setting |A_{ℓℓ}|² = 1, the protocol yields the pure displaced Fock state D(α√∏R_i)|ℓ⟩ with unit fidelity. The success probability is low (≈1 % for ℓ=2) but scales as η_s η_d^ℓ when source and detector efficiencies (η_s, η_d) are included.

2. Schrödinger cat families – By engineering the relative signs and magnitudes of the A_{ℓp} coefficients, even‑cat, odd‑cat, three‑headed cat, and compass‑cat states are produced with fidelities > 98 %. The displacement is automatically incorporated via the D operator.

3. GKP‑type resource states – The same DQ framework allows the construction of ON states and weak cubic‑phase states, which are essential for CV error‑correction. Optimized parameters give > 99 % fidelity.

4. Arbitrary target states – The authors propose a numerical algorithm that, given a desired state’s Fock expansion, solves the nonlinear equations for (α, {R_i}) to match the target coefficients A_{ℓp}. A pre‑computed database of (α, {R_i}) versus A_{ℓp} enables rapid lookup, turning the protocol into a versatile state‑engineering toolbox.

The paper also incorporates realistic imperfections. Photon‑source inefficiency (η_s < 1) and detector inefficiency (η_d < 1) are modeled as loss channels preceding each catalysis step. Simulations show that even with η_s = 0.9 and η_d = 0.9 the fidelities remain above 95 %, while the overall success probability is reduced but still experimentally accessible with modern superconducting nanowire detectors (η_d ≈ 0.98) and high‑brightness quantum‑dot single‑photon emitters (η_s ≈ 0.95).

A critical insight is that the non‑Gaussianity required for all these states is supplied entirely by the conditional single‑photon detection, while the linear optics (the cascade) merely reshapes the amplitude distribution. The DQ description isolates the Gaussian displacement from the non‑Gaussian core, allowing independent optimization of each.

The authors acknowledge the primary limitation: the probabilistic nature of the cascade leads to exponentially decreasing success rates with increasing ℓ. They suggest parallelization, fast optical switching, or integration with quantum memories as possible routes to mitigate this bottleneck.

In conclusion, the work demonstrates that a modest experimental platform—coherent light, a handful of single‑photon sources, and standard beam splitters—can, through a carefully engineered cascade, produce a wide spectrum of CV quantum resources with fidelities rivaling those obtained with strong nonlinearities or large‑Fock‑state ancillae. This opens a practical pathway for near‑term experiments in CV quantum computing, error correction, and metrology using currently available photonic technology.