A qubit is a normalized vector $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ with $|\alpha|^2+|\beta|^2=1$. Universal computation follows from arbitrary single-qubit rotations plus any entangling two-qubit gate.

Quantum algorithms exploit interference and entanglement to outperform classical strategies on specific tasks. Canonical examples include Shor's factoring algorithm, Grover search, phase estimation, and Hamiltonian simulation.

Quantum error correction protects logical information using redundancy in entangled code spaces. Fault-tolerant thresholds show that scalable quantum computation is possible if physical error rates remain below a critical bound.

The von Neumann entropy $S(\rho)=-\mathrm{Tr}(\rho\log\rho)$ generalizes classical Shannon entropy. Mutual information and conditional entropy quantify correlations and are indispensable in quantum communication.

The no-cloning theorem forbids copying an unknown quantum state. Quantum teleportation uses shared entanglement and classical communication to transfer an unknown state without moving the physical carrier of the original state.