Alice and Bob share an entangled pair. The joint state is perfectly known — a pure state. But if Alice only has her half, what does her qubit look like alone? The answer is the operation that quietly underlies decoherence, measurement, and noise.
Bob takes his qubit to the far side of the galaxy. Alice can never touch it again, so for any experiment she does, Bob’s qubit may as well not exist. Tracing it out is the bookkeeping for that: it sums over all of Bob’s possibilities, leaving the best possible description of Alice’s qubit on its own. The startling part: even though the pair is a sharp, pure state, Alice’s half alone can be a mixed state — genuine ignorance, conjured purely from entanglement.
Watch a perfectly rehearsed dance duet from the balcony — every move is graceful and predictable, the whole thing fully “known.” Now watch only one dancer through a narrow keyhole, with the partner hidden. That dancer’s steps look erratic, unpredictable — because all the meaning lived in the coordination you can no longer see. Tracing out Bob is exactly that keyhole: Alice’s qubit looks random not because the pair is, but because its information hides in the link between them.
Tune the shared state cosθ|00⟩ + sinθ|11⟩ from a plain product at θ=0 to a maximally entangled Bell pair at θ=45°. Trace out Bob and watch Alice’s Bloch arrow sink from the surface to the dead center.
The partial trace takes the joint density matrix and sums over Bob’s basis, sandwiching out his degrees of freedom and leaving an operator on Alice’s space alone:
The cross-terms (the coherences) survive only when the two halves are not entangled. The instant entanglement appears, summing over Bob erases them, and Alice is left holding a classical-looking mixture. Her loss of purity is exactly the entanglement entropy of the pair — the same von Neumann S from chapter 13.
“If the whole is perfectly known, each part must be too.” The opposite can hold. A Bell pair is a pure state — maximal knowledge — yet each qubit alone is maximally mixed, a 50/50 coin. The information isn’t in either qubit; it lives in the correlations between them. That is the deep strangeness of entanglement, now made quantitative.
And this is precisely decoherence (chapter 09): a qubit entangles with an environment you can’t see, you trace that environment out, and a pure state turns mixed. Same operation, different name. Next: the cleanest way to see how much two qubits are entangled — the Schmidt decomposition.