So far every qubit has been a sharp arrow on the Bloch surface. But what if you’re also just unsure which one you were handed? That ordinary ignorance, stacked on top of the quantum kind, pulls the state inside the ball — and needs a richer description than a single arrow: the density matrix.
A qubit in superposition |+⟩ is in a definite state — you know exactly what it is, you just can’t predict a single measurement. That’s the quantum kind of not-knowing, and it’s already on the surface of the ball.
Now imagine a bag of qubits: half were prepared as |0⟩, half as |1⟩, and someone hands you one without saying which. That’s plain classical ignorance — a coin flip about which pure state you hold. Pile it on top of the quantum uncertainty and the state is no longer one arrow; it’s a blur of arrows. The blur sinks toward the center of the ball.
Two blurry photos. One is a long-exposure of a spinning propeller — every blade really is there at once; the blur is true, complete information about the motion. The other is a stack of ordinary snapshots printed on top of each other — the blur is just you not knowing which single photo you hold. Superposition is the first kind of blur; a classical mixture is the second. The density matrix is the one ledger that keeps them apart.
Here’s a slice through the Bloch ball. Drag the point. Its direction is still which qubit; but now its distance from the center — the length r — is how pure it is. On the rim: a crisp pure state. At the murky center: a total coin flip, all certainty gone. Watch the density matrix ρ rewrite itself as you drag.
A pure state needs a vector; a mixed one needs a matrix. Build ρ by averaging the pure pieces you might have — each |ψ⟩⟨ψ| weighted by how likely it is. Its diagonal holds the outcome probabilities; its off-diagonal holds the surviving quantum phase.
Purity is just the squared length of the Bloch arrow, rescaled. Drag to the rim and Tr(ρ²) reads 1; drag to the center and it bottoms out at ½ — the most ignorant a qubit can be.
“A mixed state is just another name for a superposition.” They couldn’t be more different. Superposition |+⟩ and the 50/50 mixture of |0⟩ and |1⟩ give the same 50/50 odds in the up/down basis — yet one is on the surface and one is dead center.
The tell is in ρ. |+⟩ has fat off-diagonal entries — live phase, ready to interfere — while the mixture is purely diagonal, those terms zeroed out. Measure in the |+⟩/|−⟩ basis and the difference screams: the superposition is certain, the mixture is still a coin. Coherence is the thing a mixture has lost.