From discrete to continuous variable systems via Jordan–Schwinger tomographic transformation

Annals of Physics, vol. 493, pp. 170593

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 sector wise bridge between DV spin systems and restricted CV bosonic 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 direct sector wise correspondences between the classical measurement statistical descriptions of DV systems and the corresponding restricted CV bosonic representations. Our tomographic mapping enables a direct transfer of measurement data between different quantum architectures by acting as an intrinsic data compression kernel that selects the Hilbert space sector relevant to the target representation. It allows one to obtain the tomogram of a target sector directly from experimentally acquired data in another representation, without reconstructing the density matrix. This provides a unified framework for transferring and comparing quantum information across heterogeneous quantum hardware platforms within the corresponding restricted sectors, facilitating hybrid protocols, device benchmarking, and the validation of error correction schemes that rely on mappings between finite-dimensional systems and sector restricted bosonic representations.

Authors 3

1. Liubov A. Markovich corresponding Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Applied Quantum Algorithms Leiden, The Netherlands

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

2. Alexey N. Rubtsov

   Lomonosov Moscow State University · Russian Quantum Center

   Affiliation as printed

   Department of Physics, Lomonosov Moscow State University, Moscow 119991, Russia

   Russian Quantum Center, Skolkovo, Moscow 121205, Russia

3. Vladimir I. Man’ko

   Russian Academy of Sciences · P.N. Lebedev Physical Institute of the Russian Academy of Sciences · Russian Quantum Center

   Affiliation as printed

   Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, Moscow 119991, Russia

   Russian Quantum Center, Skolkovo, Moscow 121205, Russia