Mathematical Approaches to Quantum Cryptography

Abstract

Quantum cryptography promises a paradigm shift in secure communications by leveraging the principles of quantum mechanics. This article explores mathematical frameworks central to quantum cryptographic protocols, such as linear algebra, Hilbert spaces, quantum probability, and number theory. It focuses on the theoretical underpinnings of key distribution, quantum security proofs, and error correction. The paper also highlights how mathematical tools like entropic uncertainty relations, complexity theory, and operator algebras underpin advancements in quantum cryptographic systems. These mathematical approaches ensure not only the security but also the efficiency and scalability of next-generation quantum communication networks.