A Kronecker Delta Framework for Normalization and Orthogonality in Multi-Qubit Registers

Source abstract

In quantum information theory and quantum mechanics, mathematical formalism relies heavily on the orthonormality of Hilbert space basis vectors. The Kronecker delta function, denoted as 𝛿ij, serves as a fundamental operator for expressing state orthogonality and normalization across single- and multi-qubit systems. Normalization ensures unit length ( = 1), representing the conservation of total probability, while orthogonality guarantees that distinct basis states share zero geometric overlap ( = 0 for i ≠ j), ensuring mutual exclusivity. This paper presents a formal analysis of the Kronecker delta within qubit state representations, matrix inner products, and multi-qubit bitwise inner product decompositions. Through computational examples on single-qubit states (|0>, |1>) and 5-qubit binary registers (|11101|>, |11111>), this article demonstrates how the delta framework systematically enforces the structural properties essential for quantum information processing.