This chapter establishes the theoretical foundation essential for understanding quantum teleportation protocols. It begins with mathematical prerequisites, including linear algebra concepts such as Hilbert spaces, vector operations, tensor products, and operator properties (Hermitian, unitary, and normal operators). Key quantum logic gates—single-qubit (e.g., Hadamard, Pauli X, Y, Z) and multi-qubit (e.g., CNOT, SWAP) gates—are detailed, emphasizing their roles in quantum computation. Core principles of quantum mechanics, such as the uncertainty principle, superposition, and the no-cloning theorem, are systematically explained. The density matrix formalism is introduced to describe pure and mixed states, alongside reduced density matrices for composite systems. Quantum entanglement and its applications, including entanglement swapping, are rigorously defined for both pure and mixed multiparticle states. Finally, the chapter addresses noise in open quantum systems, covering Kraus operators, measurement methods (projection, POVM), and quantum fidelity to quantify state proximity. These concepts collectively underpin subsequent chapters on noise-immune channel frameworks, quantum information splitting, and multi-degree of freedom protocols.