# Kaleidoscope of Quantum Coherent States

## TL;DR

This paper generalizes Schr"odinger cat states to a kaleidoscope of coherent states with n-polygon symmetry, exploring their mathematical properties and potential for quantum information processing.

## Contribution

It introduces a new class of coherent states based on regular n-polygon symmetry, linking them to quantum Fourier transforms and quantum groups, expanding the framework of quantum state engineering.

## Key findings

- States can be generated by Quantum Fourier transform.
- States provide qubit, qutrit, ququat, and qudit units.
- Normalization involves symmetry-specific exponential functions.

## Abstract

The Schr\"odinger cat states, constructed from Glauber coherent states and applied for description of qubits, are generalized to the kaleidoscope of coherent states related with regular n-polygon symmetry and the roots of unity. The cases of the trinity states and the quartet states are described in details. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry. We show that for an arbitrary $n$, these states can be generated by the Quantum Fourier transform and can provide qutrits, ququats and in general, qudit units of quantum information. Relations with quantum groups and quantum calculus are discussed.