Pure vs. Mixed States

Ch. 3's Bloch sphere only ever showed you the surface. That wasn't a simplification for beginners — it was the surface specifically because every state on it was pure. Density matrices unlock the rest of the ball.

The purity test

One number tells you everything: Tr(ρ²).

• Tr(ρ²) = 1 → pure state. Sits exactly on the Bloch sphere's surface.
• Tr(ρ²) < 1 → mixed state. Sits somewhere inside the ball — the smaller Tr(ρ²) gets, the closer to dead center.
• Tr(ρ²) = 1/2 (the minimum, for a qubit) → the maximally mixed state, I/2, sitting exactly at the center. No direction is preferred at all — it's the quantum equivalent of a fair, unbiased coin.

Superposition is not the same thing as a mixture

This is the single most common mix-up at this stage, so it's worth stating bluntly: |+⟩ = (|0⟩+|1⟩)/√2 is a pure state — it sits right on the equator, Tr(ρ²)=1. A 50/50 classical mixture of |0⟩ and |1⟩ is a genuinely different, mixed state — I/2, dead center, Tr(ρ²)=0.5. Both "give 50/50 outcomes when measured in the Z basis," which is exactly why it's tempting to conflate them. But measure in the X basis and the difference is obvious: |+⟩ gives a certain "+" outcome every time (Ch. 7), while I/2 gives 50/50 no matter which basis you measure in. Superposition is coherent and basis-dependent; a mixture is incoherent classical uncertainty, the same in every basis.

Why Tr(ρ²) is the right quantity (not just a convenient one)

Writing ρ in terms of the Bloch vector r = (rₓ,r_y,r_z), with |r| ≤ 1, gives the clean closed form Tr(ρ²) = (1+|r|²)/2 — which is exactly what Fig. 9.1 computes as you drag. This isn't a coincidence of notation: Tr(ρ²) is one specific case of a whole family of quantities called Rényi entropies that measure "how mixed" a state is, and it has a special practical virtue over alternatives like von Neumann entropy (Ch. 15): it's just a trace of a matrix product, with no logarithm and no eigendecomposition required, which is why it's usually the first thing people check.