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

TL;DR

This paper introduces a novel tomographic mapping framework that bridges discrete-variable and continuous-variable quantum systems, enabling direct transfer and comparison of measurement data across different quantum hardware platforms.

Contribution

It provides the first explicit demonstration of how Jordan--Schwinger and Holstein--Primakoff maps act on tomographic probability distributions, linking classical measurement descriptions of DV and CV systems.

Findings

First explicit action of maps on tomograms and Wigner functions

Enables direct transfer of measurement data between quantum architectures

Facilitates hybrid quantum protocols and device benchmarking

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, information and states 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 and Holstein--Primakoff maps. While both maps are well known at the operator level, their action on the classical counterparts of quantum states, namely tomograms and other probability representations, has not been addressed in the literature. To the best of our knowledge, this work provides the first explicit demonstration of how the Jordan--Schwinger and Holstein--Primakoff maps act on tomographic probability distributions and Wigner functions, thereby establishing a direct correspondence between the classical measurement statistical descriptions of CV and DV quantum systems. Our tomographic mapping enables a direct transfer of measurement data between different quantum architectures by acting as an intrinsic data-compression kernel. It allows one to obtain the tomogram of a target representation directly from experimentally acquired data in another, without reconstructing the density matrix. This provides a unified framework for transferring and comparing quantum information across heterogeneous quantum hardware platforms, facilitating hybrid protocols, device benchmarking, and the validation of error-correction schemes that rely on mappings between finite- and infinite-dimensional systems.