From Discrete to Continuous-Variable Systems via Jordan-Schwinger Tomographic Transformation
Hybrid quantum systems that combine discrete-variable (DV) and continuous-variable (CV) architectures represent a promising direction in quantum information science. However, transferring concepts, information, and states between these platforms poses practical and theoretical challenges, as their formalisms assign different properties and physical interpretations to many shared notions. We construct a bridge between DV spin systems and restricted CV bosonic systems within corresponding sectors using the tomographic probability representation of quantum states and the Jordan--Schwinger and Holstein--Primakoff maps. Although these maps are well known at the operator level, their action on tomograms and other probability representations has not been addressed in the literature. To the best of our knowledge, we provide the first explicit demonstration of their action on tomographic probability distributions and Wigner functions, establishing direct correspondences between the measurement statistics of DV systems and the corresponding restricted CV bosonic representations. Our tomographic mapping transfers measurement data between quantum architectures through an intrinsic data compression kernel that selects the Hilbert space sector relevant to the target representation. It yields the tomogram of a target sector directly from experimental data in another representation, without reconstructing the density matrix. This provides a unified framework for transferring and comparing quantum information across different hardware platforms within the corresponding restricted sectors, with applications to hybrid protocols, device benchmarking, and the validation of error correction schemes based on mappings between finite-dimensional systems and sector restricted bosonic representations.