A p -Adic Model of Quantum States and the p -Adic Qubit

• Published in: Entropy (Basel, Switzerland) Vol. 25; no. 1; p. 86
• Main Authors: Aniello, Paolo, Mancini, Stefano, Parisi, Vincenzo
• Format: Journal Article
• Language: English
• Published: Switzerland MDPI AG 31.12.2022; MDPI
• Subjects: Computer science; Cryptography; Hilbert space; Inequality; Information theory; Linear operators; Mathematical analysis; Numbers; Operators (mathematics); p-adic probability; p-adic quantum mechanics; Probability theory; Quantum field theory; Quantum mechanics; Quantum physics; quantum state; Quantum theory; Qubits (quantum computing); Statistical analysis; ultrametric field; Valuation; ultrametric field; quantum state; p-adic probability; p-adic quantum mechanics
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e25010086

We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of -adic numbers. As in the standard complex setting, states and observables of a -adic quantum system are implemented by suitable linear operators in a -adic Hilbert space. In particular, owing to the distinguishing features of -adic probability theory, the states of an N-dimensional -adic quantum system are implemented by -adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)-a suitable -adic counterpart of a POVM in a complex Hilbert space-as a convenient mathematical tool describing the physical observables of a -adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of -adic qubit states and 2-dimensional SOVMs. The analogies-but also the non-trivial differences-with respect to the qubit states of standard quantum mechanics are then analyzed.