Quantum Information: A Brief Overview and Some Mathematical Aspects

• Main Author: Maurice R. Kibler
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Mathematics
• Subjects: linearity; superposition; entanglement; mutually unbiased bases; SU(2); Galois fields; Galois rings
• Online Access: https://www.mdpi.com/2227-7390/6/12/273

The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) <i>quantum bits</i> or qubits (and more generally <i>qudits</i>), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, <i>linearity</i>, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and <i>entanglement</i> of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called <i>mutually unbiased bases</i> used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings.