What does it mean for a system to "be" in a quantum state?
Four rules, and everything else in quantum computing is a consequence of them. Together they answer four questions: what is a state, how does it change on its own, what happens when you look at it, and what happens when you put several together.
State. An isolated system's state is a vector of unit length. For one qubit that vector lives in a two-dimensional space; the two perpendicular directions are the two classical outcomes 0 and 1, and superposition is just what it means for a vector to point somewhere between them.
Evolution. Left alone, a quantum state doesn't decay or blur. It rotates. Every closed-system operation a quantum computer performs is reversible: run it backwards and you land exactly where you started. This is the deep reason a quantum computer cannot simply "try every answer and keep the right one". There is no step in the model that throws information away.
Measurement is the one place the rules stop being reversible, and it's covered on its own two topics below because it deserves the space.
Composite systems. Put two qubits together and their combined state space isn't the sum of the two: it's the product. That single fact is why n qubits carry 2ⁿ complex amplitudes instead of 2n, and it's the entire reason simulating a quantum computer classically gets hard so fast.
1. State space. The state of an isolated quantum system is a unit vector |ψ⟩ in a complex Hilbert space. For a single qubit, that space is ℂ²:
|ψ⟩ = α|0⟩ + β|1⟩ , α, β ∈ ℂ , |α|² + |β|² = 1
2. Evolution. The state of a closed system evolves by a unitary transformation: |ψ'⟩ = U|ψ⟩, where U†U = I. Unitarity is what guarantees reversibility (U⁻¹ = U† always exists) and preserves the normalisation |α|²+|β|²=1: a quantum gate can never make total probability exceed or fall short of 1.
3. Measurement is described by a set of measurement operators {Mm} satisfying Σm Mm†Mm = I. Outcome m occurs with probability ⟨ψ|Mm†Mm|ψ⟩, and the state collapses to Mm|ψ⟩ divided by the square root of that probability. Topic 03 unpacks the two special cases (projective measurement and POVMs) that this general rule contains.
4. Composite systems. The state space of a system built from components 1..n is the tensor product of the component spaces: ℋ = ℋ₁ ⊗ ℋ₂ ⊗ ... ⊗ ℋₙ. A basis vector of an n-qubit system is written |b₁b₂...bₙ⟩, and a general state is a superposition over all 2ⁿ of them, which is exactly where the exponential state space that makes quantum simulation hard, and quantum computation potentially powerful, comes from.
Every other page on this site The Bloch sphere is postulate 1 drawn as a picture. Every gate on the golf game is postulate 2 in action. Syndrome measurement is postulate 3, deliberately designed to extract only the information needed without collapsing the data. And the reason a logical qubit needs so many physical ones is postulate 4: the state space you're trying to protect is exponentially large from the very first two qubits onward.
Check yourself
Which of the four postulates is the only one where the rules stop being reversible?
A closed system evolves by a unitary, so every such operation can be run backwards. Measurement is the one place the rules stop being reversible.