Arithmetic sequences as quantum states

TL;DR

This paper introduces a quantum-inspired method to analyze arithmetic sequences by mapping them onto quantum states and quantifying their randomness through von Neumann entropy, revealing sequences with maximal entanglement.

Contribution

It presents a novel approach to study arithmetic sequences using quantum state representations and entropy measures, providing analytical approximations for sequence properties.

Findings

Identifies sequences that maximize entanglement entropy

Derives analytical approximation of entropy as a function of sequence length

Offers a new perspective on randomness in arithmetic sequences

Abstract

We consider arithmetic sequences, here defined as ordered lists of positive integers. Any such a sequence can be cast onto a quantum state, enabling the quantification of its `surprise' through von Neumann entropy. We identify typical sequences that maximize entanglement entropy across all bipartitions and derive an analytical approximation as a function of the sequence length. This quantum-inspired approach offers a novel perspective for analyzing randomness in arithmetic sequences.