From Discrete to Continuous-Variable Systems via Jordan-Schwinger Tomographic Transformation

1. Problem Statement

Quantum information science currently operates in two distinct paradigms:

• Discrete-Variable (DV): Finite-dimensional systems (e.g., qubits, qudits, spins).
• Continuous-Variable (CV): Infinite-dimensional systems (e.g., optical modes, microwave cavities, vibrational modes).

While hybrid architectures combining both are emerging, transferring information and states between them is theoretically and practically challenging. Existing methods typically require:

1. Reconstructing the full density matrix from experimental data (an ill-posed inverse problem prone to noise and artifacts).
2. Applying operator-level mappings (like Jordan–Schwinger or Holstein–Primakoff) to the reconstructed matrix.

This workflow is computationally expensive, sensitive to truncation errors (due to infinite dimensions), and introduces unnecessary noise accumulation. Furthermore, standard mappings are defined at the operator level, but there was no established framework for how these maps act directly on tomographic probability distributions (the raw statistical data obtained from experiments).

2. Methodology

The authors propose a unified framework that bypasses density matrix reconstruction entirely. They utilize the Quantizer–Dequantizer formalism to construct direct integral transforms between the tomographic representations of DV and CV systems.

Core Components:

• Tomographic Representations: The paper focuses on:
  • Spin Tomograms ($\omega_j$): Probability distributions of spin projections for DV systems.
  • Symplectic Tomograms ($W$): Probability distributions of quadratures for CV systems (measured via homodyne detection).
  • Photon-Number Tomograms ($w_r$): Probability distributions of photon counts after displacement.
• Operator Mappings:
  • Jordan–Schwinger (JS) Map: Maps two bosonic modes to a spin-$j$ system. It establishes an isomorphism between a fixed total excitation sector ($n = 2j$) of the bosonic Fock space and the spin-$j$ Hilbert space.
  • Holstein–Primakoff (HP) Map: Maps a single bosonic mode to a spin-$j$ system, valid within a fixed excitation sector.
• The "Restricted" Approach: The authors introduce the concept of restricted states. Instead of mapping the entire infinite-dimensional Fock space, they project the CV state onto a specific excitation sector ($n=2j$). Within this sector, the mapping is an exact isomorphism.
• Kernel Formulation: The central innovation is deriving analytical transformation kernels ($K$) that act directly on the tomograms. The transformation is expressed as:
  $$\omega_j(m, \Omega) = \int W(x|\mu, \nu) K(m, \Omega; x, \mu, \nu) \, dx \, d\mu \, d\nu$$
  This allows the calculation of a spin tomogram directly from homodyne data (symplectic tomogram) without intermediate reconstruction.

3. Key Contributions

1. First Explicit Tomographic Mappings: The paper provides the first derivation of how Jordan–Schwinger and Holstein–Primakoff maps act on classical probability distributions (tomograms and Wigner functions), extending these concepts from the operator level to the statistical measurement level.
2. Direct Data Transfer Kernels: The authors derive closed-form analytical expressions for the kernels connecting:
   • Symplectic $\leftrightarrow$ Spin Tomograms.
   • Photon-number $\leftrightarrow$ Spin Tomograms.
   • Wigner Functions $\leftrightarrow$ Spin Tomograms.
     These kernels involve special functions (Laguerre polynomials, Wigner D-matrices) and are computationally efficient.
3. Intrinsic Data Compression: The mapping acts as a filter, automatically extracting only the relevant excitation sector ($n=2j$) from the CV data and discarding redundant sectors. This avoids the need for arbitrary truncation of the Hilbert space.
4. Avoidance of Ill-Posed Reconstruction: By fusing measurement, reconstruction, and mapping into a single integral transform, the method eliminates the numerical instability and regularization artifacts associated with reconstructing infinite-dimensional density matrices.
5. Experimental Validation: The method is tested using real experimental homodyne data (from Ref. [23]) of a non-classical "Schrödinger kitten" state. The authors successfully reconstructed spin-1/2 and spin-1 tomograms directly from the raw quadrature data, demonstrating high fidelity with theoretical predictions.

4. Results

• Analytical Success: The derived kernels (detailed in Appendices B–H) successfully link the probability distributions of different architectures. For example, the transition from a symplectic tomogram to a spin tomogram is shown to be a linear integral transform with a kernel expressible via Laguerre polynomials.
• Numerical Stability: The use of analytical kernels replaces large-scale numerical linear algebra (required for density matrix inversion) with direct evaluation, significantly reducing computational cost and improving stability.
• Experimental Agreement: Using the Kernel-based Quantum State Estimation (KQSE) method to estimate the characteristic function from raw homodyne data, the authors generated spin tomograms.
  • The reconstructed spin-1/2 and spin-1 tomograms matched theoretical predictions closely.
  • Discrepancies were attributed to experimental noise and model imperfections (e.g., phase noise), which were successfully modeled by refining the mixed-state parameters ($p_{diag}$ and $p_{coh}$).
  • The method proved robust against the "negativity" issues often seen in other reconstruction techniques, as the kernel ensures the input tomogram remains a valid probability density.

5. Significance and Impact

• Unified Benchmarking: This work provides a unified framework for comparing heterogeneous quantum hardware (e.g., superconducting qubits vs. trapped ions vs. optical modes). Researchers can now benchmark a CV device against a DV standard (or vice versa) by converting measurement data directly, without needing to agree on a specific density matrix representation.
• Hybrid Protocol Development: It facilitates the design of hybrid quantum protocols where information is encoded in CV modes but processed or stored in DV qubits, enabling seamless interoperability.
• Error Correction and Validation: The ability to map between representations allows for the validation of error-correction schemes that rely on finite- to infinite-dimensional mappings. It enables direct verification of state transfer fidelity in hybrid devices.
• Theoretical Bridge: The paper clarifies the relationship between the "CV limit" and fixed-$n$ sectors, showing that the continuous variable description is an approximation emerging from a large, fixed excitation sector, thereby providing a controlled theoretical bridge between the two paradigms.

In summary, the paper establishes a rigorous, practical, and experimentally verified bridge between discrete and continuous quantum variables, transforming how quantum states are transferred, compared, and validated across different physical platforms.