Revisiting Boole Equation in the Quantum Context

TL;DR

This paper explores the fundamental relationship between classical bits and quantum qubits using Boole's equation, deriving new expressions for basis vectors and projection operators applicable to higher-dimensional quantum systems.

Contribution

It introduces a generic expression for qubit basis vectors and projection operators, extending the analysis to qutrits and qudits, clarifying foundational quantum information concepts.

Findings

Derived a compact basis vector expression for qubits

Formulated a generic projection operator in quantum space

Extended results to higher-dimensional quantum systems

Abstract

In this work we try to clarify the fundamental relationship between bits and qubits, starting from very simple George Boole equation. We derive a generic and compact expression for basis vectors of qubit which can be useful in further applications. We also derive a generic form for the projection operator in the quantum information space. The results are also extended to higher d-level cases of qutrits and qudits.