Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin-Meshkov-Glick D-level atom models

Collective spin operators for symmetric multi-quDit (namely identical D-level atom) systems generate a U(D) symmetry. We explore generalizations to arbitrary D of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrodinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric N-quDit states, expressed in the last two cases in terms of collective U(D)-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin-Meshkov-Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU(D) and relate it to pairwise D-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D-level atom models.

Automated summary

This paper investigates the U(D) symmetry generated by collective spin operators for symmetric multi-quDit systems, and generalizations of SU(2)-spin coherent states to arbitrary D. It explores the adaptation of these states to parity and multi-mode extensions, as well as the evaluation of entanglement in these systems using level and particle entanglement and entropies. The study also delves into numerical and variational analyses of ground states in specific models, and proposes an extension of SU(2)-spin squeezing to SU(D) for pairwise D-level atom entanglement characterization.