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

TL;DR

This work constructs a bridge between DV and CV systems by means of the tomographic probability representation of quantum states complemented by the Jordan-Schwinger map, and facilitates the design of hybrid protocols where, for example, spin-based quantum memories interface with photonic communication channels or CV bosonic codes.

Abstract

Hybrid quantum systems that combine discrete-variable (DV) and continuous-variable (CV) architectures represent a promising direction in quantum information science. However, transferring concepts and methods between such fundamentally different platforms entails both practical and theoretical challenges. The formalisms of these two "universes" differ significantly, and many notions, although sharing the same names, possess distinct properties and physical interpretations. In this work, we construct a bridge between DV and CV systems by means of the tomographic probability representation of quantum states complemented by the Jordan-Schwinger map. We connect observable random variables such as spin projections, photon numbers, or quadratures arising in DV and CV architectures through their probabilistic representations, namely spin, photon-number, and symplectic tomograms, as well as the Wigner function. This makes it possible to directly obtain the tomogram in one architecture from measurement data in another, thereby reconstructing the corresponding state across representations realizing similar statistics. The proposed formalism enables a unified framework for comparing and transferring quantum information across different hardware platforms. This facilitates the design of hybrid protocols where, for example, spin-based quantum memories interface with photonic communication channels or CV bosonic codes. It also provides a practical tool for benchmarking quantum devices, validating algorithms across heterogeneous architectures, and exploring error correction schemes that rely on mappings between finite- and infinite-dimensional systems.

Figures (4)

• Figure 1: Unified tomographic framework connecting DV and CV quantum systems. The spin tomogram $\omega_j(m, \Omega)$, the photon-number tomogram $\mathsf{w}_n(\alpha)$, and the symplectic tomogram $\mathcal{W}(x|\mu,\nu)$ are related through explicit quantizer-dequantizer kernel transformations, establishing one-to-one correspondences among all tomographic representations and the Wigner function $\mathbb{W}(q, p)$. Each tomogram corresponds to a measurable random variable: spin projection $m$, photon count $n$ (after displacement $\alpha$), and quadrature $x_{\mu,\nu}$, providing experimentally accessible probability distributions that jointly span both DV and CV regimes. The Jordan–Schwinger operator mapping serves as the unifying link between the density operators $\hat{\rho}^{(j)}$ and $\hat{\rho}_{(1)}$, which represent the same physical state in finite and infinite Hilbert spaces, respectively, enabling direct translation of tomographic data across representations and architectures.
• Figure 2: A schematic of the two-mode bosonic lattice with total particle number $\hat{n} =\hat{n}_1 + \hat{n}_2$, where $\hat{n}_1$ and $\hat{n}_2$ denote the number of particles in each mode. Bold lines connect states with fixed $n = 2j$, corresponding to spin-$j$ representations. Dashed lines connect states with constant $\hat{S}_z = (\hat{n}_1 - \hat{n}_2)/2$, where $m$ is the magnetic quantum number.
• Figure 3: Symplectic tomograms of the states \ref{['1608_1']}–\ref{['1608_3']}, reconstructed from spin tomograms for $j \in {\tfrac{1}{2}, 1, \tfrac{3}{2}}$.
• Figure 4: Photon-number tomograms of the states \ref{['1608_1']}–\ref{['1608_3']}, reconstructed from spin tomograms for $j \in \{\tfrac{1}{2}, 1, \tfrac{3}{2}\}$ for different field amplitudes $\alpha_1=\alpha_2$ ($\mathrm{Im},\alpha_1=\mathrm{Im},\alpha_2=0$).