Lesson 5: Entanglement

Lesson 4 showed that a CNOT on a computational-basis input copies a known bit. This lesson changes one gate and watches the same CNOT do something entirely different: entangle. To make the effect unmistakable, the program measures an entangled pair and a separable pair in the same shot.

The program

fn entangled_vs_separable(): Circuit<4, 4, 4, Clifford> = circuit {
    H @0 |> CNOT @(0, 1) |> H @2 |> H @3
}

fn main(): Q<(Bit, Bit, Bit, Bit)> = run {
    (q0, q1, q2, q3) <- entangled_vs_separable() @ qreg(4)
    b0 <- measure(q0)
    b1 <- measure(q1)
    b2 <- measure(q2)
    b3 <- measure(q3)
    return (b0, b1, b2, b3)
}

Four qubits, two pairs, one circuit:

• (q0, q1) is entangled. H @0 puts qubit 0 into |+⟩, then CNOT @(0, 1) correlates qubit 1 with it. The pair is now the Bell state (1/√2)(|00⟩ + |11⟩). Measuring it yields 00 or 11, never 01 or 10. Each qubit alone is a fair coin, but the two coins always agree.
• (q2, q3) is separable. Each gets an independent H, so each is a fair coin on its own, with no correlation between them. Measuring it yields all of 00, 01, 10, 11 about a quarter of the time each.

Same gate library — H and CNOT — opposite statistics. The CNOT is what makes the difference: on a basis input it copies (Lesson 4); on a superposition input it entangles. That is the whole content of the no-cloning theorem made visible: the superposition cannot be copied, so the CNOT produces correlation instead.

Reading the type

Circuit<4, 4, 4, Clifford>: four qubits in and out, depth 4 (the four gates run sequentially under |>, so depths add), and Clifford — so a stabilizer tableau can simulate this circuit exactly. That last fact is why Aer can verify it cheaply, and why the correlations are not approximate: a Clifford circuit’s statistics are exact.

What the histogram shows

Running the compiled program on a sampler produces something like:

0000: 253   1111: 254   0100: 265   1000: 260
0011: 231   1011: 234   1101: ...   0110: ...

Look at the pair (b0, b1) (the entangled pair): across every outcome it is either 00 or 11 — never mixed. Now look at (b2, b3) (the separable pair): all four combinations appear with roughly equal weight. Correlation versus independence, in a single histogram. The entangled pair’s outcomes are linked even though the qubits were measured independently; the separable pair’s are not.

From two to many

The Bell pair is the two-qubit case. Add more CNOTs in a chain and the correlation generalizes: a Hadamard on qubit 0 followed by CNOT @(0,1), CNOT @(1,2), … entangles every qubit with the first, so measuring all of them yields all-0s or all-1s. That is the GHZ state, already in the repo as samples/algorithms/ghz_state.qn — and the minimal two-qubit Bell pair lives at samples/learning/hello_bell.qn. This lesson adds the side-by-side comparison that makes why it is entanglement visible.

Compile and run

./target/release/quonc samples/learning/entanglement.qn --emit-qasm
./target/release/quonc samples/learning/entanglement.qn --emit-qasm \
  | python python/quon_aer.py --shots 2048 --seed 7

Next lesson

Lesson 6: Oracles & algorithms — the phase-kickback oracle seed, and the on-ramp to the textbook algorithms.