Foundations

What Is a Qubit? From Classical Bits to Quantum States

A ground-up introduction to qubits for developers who already know code. Bloch sphere, Dirac notation, the normalization constraint, and why a qubit is not just a probabilistic bit — with runnable Qiskit code.

beginner · ~18 min · prereq: High-school algebra, A little Python

Superposition, Measurement, and the Born Rule

Measurement turns amplitudes into probabilities and destroys superposition. This tutorial walks through the Born rule, measurement in different bases, the no-cloning theorem, and why you can't just peek at a qubit without breaking it — with runnable Qiskit code.

beginner · ~22 min · prereq: Tutorial 1: What Is a Qubit?

Multi-Qubit Systems and Entanglement

Tensor products, the 2ⁿ-dimensional state space, separable vs entangled states, the four Bell states, and why entanglement is the real secret ingredient of quantum computing. With runnable Qiskit code and a measurement-correlation experiment.

beginner · ~25 min · prereq: Tutorial 1: What Is a Qubit?, Tutorial 2: Superposition, Measurement, and the Born Rule

Bell's Theorem and the CHSH Inequality: How We Know the Universe Isn't Locally Realistic

Bell's theorem (1964) is the most-cited result in the foundations of quantum mechanics. It proves that no theory based on local hidden variables can reproduce all the predictions of quantum mechanics — and the CHSH form gives a quantitative inequality that experiments can test. Every loophole-free Bell test since 2015 has confirmed quantum mechanics. This tutorial derives the CHSH inequality, explains the Tsirelson bound, and covers what 'loophole-free' actually means.

intermediate · ~16 min · prereq: Tutorial 3: Multi-Qubit Entanglement

The No-Cloning Theorem: Why Quantum Information Cannot Be Copied

The no-cloning theorem (Wootters-Zurek 1982; Dieks 1982) is the structural reason quantum information behaves differently from classical bits. There is no unitary operation that takes an unknown qubit and outputs two identical copies. This single result underlies quantum error correction's complexity, quantum cryptography's security, and the impossibility of quantum signal repeaters that simply amplify their input. This tutorial proves the theorem in three lines and walks through its consequences.

beginner · ~13 min · prereq: Tutorial 1: What Is a Qubit

Density Matrices and Mixed States: The Formalism for Real Quantum Systems

Pure-state quantum mechanics ($|\psi\rangle$ vectors) is enough for textbook quantum computing but not for real hardware. Real qubits are noisy, partially-known, or part of larger entangled systems whose other parts you've ignored. The density matrix is the formalism that handles all three cases. This tutorial defines density matrices, derives their properties, covers the partial trace and the purification theorem, and shows why density matrices are the natural language of quantum-information theory.

intermediate · ~17 min · prereq: Tutorial 1: What Is a Qubit, Tutorial 18: Noise and Decoherence

The Bloch Sphere: Every Single-Qubit State Is a Point in Three Dimensions

The Bloch sphere is the geometric picture that makes single-qubit quantum mechanics intuitive. Every pure single-qubit state is a point on the surface of a unit sphere; every mixed state is a point inside. Single-qubit gates are rotations of this sphere. This tutorial constructs the Bloch sphere from the density-matrix formulation, derives the geometry of common gates, and shows how the picture extends (and fails to extend) to multi-qubit systems.

beginner · ~14 min · prereq: Tutorial 1: What Is a Qubit, Tutorial 47: Density Matrices and Mixed States