Quantum Cookbook

Part II · Chapter 10

Partial Trace

Read ~18 min Prerequisites Ch. 5, Ch. 8–9 Tools entanglement/mixedness slider, code cell Live
Basic

Here's a question Ch. 8–9 left open: if two qubits are entangled, what's the state of just one of them, on its own? The answer turns out to be one of the most important and surprising facts in this whole book: even when the full 2-qubit state is perfectly pure, one qubit by itself is usually mixed.

What the partial trace does

Given a joint density matrix ρ_AB for two systems, the partial trace "sums away" everything about system B, leaving exactly the reduced density matrix that describes A alone:

ρ_A = Tr_B(ρ_AB)

Concretely, for two qubits using the index = 2q₁+q₀ convention from Ch. 5, tracing out q₀ means: (ρ_A)ᵢⱼ = Σₖ (ρ_AB)[2i+k][2j+k] for k=0,1 — pairing up entries that agree on q₁'s index and summing over q₀'s.

The showcase example: the Bell state

Fig. 10.1 — drag t from 0° (product) to 45° (max entangled)watch ρ_A's purity
|ψ⟩ itself is always pure — it's a single ket, by construction. But its subsystem ρ_A goes from pure (t=0°, an ordinary product state) all the way to maximally mixed (t=45°, the actual Bell state) as entanglement increases. This is the deepest reason mixed states matter at all in quantum computing.
Intermediate

Why this makes sense

At t=45°, |ψ⟩ = (|00⟩+|11⟩)/√2: knowing q₁'s value tells you q₀'s value with certainty, and vice versa — they're perfectly correlated. But q₁ by itself, with no access to q₀ at all, can't see any of that correlation. All that's left, looking at q₁ alone, is "50% chance of 0, 50% chance of 1, with absolutely no further structure" — which is precisely the maximally mixed state I/2. The information wasn't lost; it just isn't local to either qubit alone. It only exists in the correlation between them — which is, in fact, one workable definition of what entanglement even is.

Advanced

This is an "improper" mixture, and that distinction matters

Ch. 8's Advanced section mentioned proper vs. improper mixtures. Here's the improper case in the wild: ρ_A = I/2 from the Bell state isn't "secretly" |0⟩ or "secretly" |1⟩ waiting to be revealed — there's no hidden fact at all, because the global state is genuinely pure and the mixedness is purely an artifact of ignoring q₀. Contrast this with Ch. 8's lab-equipment example, where ρ = 0.95|0⟩⟨0|+0.05|1⟩⟨1| really does correspond to a definite (if unknown to you) state each time. Both give identical-looking density matrices and identical measurement statistics — but only the partial-trace case is inseparable from a fact about a larger system you don't have access to.

Fig. 10.2 — code, runs in your browser
Same partial-trace function, two very different outcomes — exactly the contrast Fig. 10.1's slider walks through continuously.
Check your understanding

What does Tr_B(ρ_AB) compute?

The reduced density matrix ρ_A — the correct description of subsystem A alone, with all information about B summed away.

Check your understanding

For the Bell state (|00⟩+|11⟩)/√2, what is ρ_A?

I/2, the maximally mixed state — even though the full 2-qubit state is perfectly pure.

Check your understanding

If ρ_AB describes an unentangled product state, what can you say about ρ_A?

It's pure — Fig. 10.1 at t=0° shows exactly this. No entanglement means no information is hidden in cross-qubit correlations, so tracing out the other qubit loses nothing.

Check your understanding

Why does tracing out an entangled partner make the remaining qubit's state mixed, if no information was actually destroyed?

The information isn't destroyed — it was never local to either qubit. It exists only in the correlation between them, so a description that looks at only one qubit necessarily can't see it, leaving an honestly mixed local description.

Check your understanding

What's the difference between the "improper" mixture from a partial trace and the "proper" mixture from imperfect state preparation (Ch. 8)?

In the proper mixture, there's a real, definite (if unknown to you) state each time. In the improper mixture from a partial trace, there's no such hidden fact — the global state really is pure, and the local mixedness comes entirely from ignoring an entangled partner. Both produce identical density matrices and statistics, but the underlying physical situations are conceptually different.

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