Part III. Circuit Model

Bell States and GHZ States

One Hadamard and one CNOT turn $∣ 00 ⟩$ into the most reused object in quantum information. This chapter builds Bell and GHZ states gate by gate, predicts their measurement statistics, and draws the line between correlation and control.

Listen to this chapter

Bell and GHZ states are the cheapest entanglement you can manufacture and the easiest to reason about badly: their entire value is joint structure that no per-qubit description can express — and none of that structure is a messaging channel.

You will trace the preparation circuits from $∣ 00 ⟩$ and $∣ 000 ⟩$ , compute local marginals and joint probabilities separately, and leave with a builder's checklist for reading any entanglement demonstration.

Core concepts: Bell states, GHZ states, entanglement, joint statistics.

Notice the fan-out: one control wire, a new target for each added qubit. Two gates entangle two qubits; one more CNOT extends the same correlation to three.

Two gates, one non-factorable state

Bell states are the simplest entangled states, and the recipe fits on an index card: put one qubit into superposition with a Hadamard, then let it control a CNOT onto a second qubit. GHZ states repeat the trick, fanning the same control out to more targets; the three-qubit version is:

\frac{∣ 000 ⟩ + ∣ 111 ⟩}{2}

These states earn their keep as resources: their joint measurement statistics cannot be reproduced by qubits that are merely correlated the classical way. What they do not do, by themselves, is send controllable messages. The usable pattern is always joint state, chosen measurement bases, and a classical comparison afterwards.

The states, precisely

The canonical Bell state is:

∣ Φ^{+} ⟩ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2}

Ideal computational-basis measurements return 00 or 11 with equal probability, and never 01 or 10 — perfect correlation from a state neither party controls alone. The preparation follows directly from the CNOT basis action:

CNOT ∣ c, t ⟩ = ∣ c, t \oplus c ⟩

Traced from |00>:

∣ 00 ⟩ H on first qubit \frac{∣ 00 ⟩ + ∣ 10 ⟩}{2} CNOT \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2}

GHZ preparation continues the same pattern: from $∣ 000 ⟩$ , apply H to the first qubit, then CNOT from the first qubit to the second, and from the first to the third.

There are actually four Bell states, reached by flipping phases or bits on the pattern above, and together they form an orthonormal basis for two qubits — the Bell basis. That basis is the standard currency of quantum protocols: Chapters 24 and 25 spend exactly this currency, in opposite directions.

Worked example: from |000> to GHZ

Run the trace. After H on the first qubit:

\frac{∣ 000 ⟩ + ∣ 100 ⟩}{2}

After CNOT from qubit one to qubit two:

\frac{∣ 000 ⟩ + ∣ 110 ⟩}{2}

After CNOT from qubit one to qubit three:

\frac{∣ 000 ⟩ + ∣ 111 ⟩}{2}

Measure all three qubits in the computational basis and every run returns 000 or 111, half and half. But measure any one qubit alone and its outcome is a coin flip that tells you nothing — the information lives entirely in the correlation. That is why GHZ states show up both in tests of quantum mechanics itself and in error-correction constructions.

Correlation is not control

The seductive mistake is to think entanglement lets one party choose the other's result. It does not. In a Bell state, each local measurement is individually random; Alice cannot steer Bob's outcome toward 0 or 1 by anything she does to her half. The correlation becomes visible only when the two compare notes over an ordinary classical channel — Chapter 7 (Entanglement Without Faster-Than-Light Myths) spends a full chapter on why the no-signaling theorem closes every workaround.

The quieter mistake is filing Bell and GHZ states under big superpositions. Product states can have just as many terms after expansion, as $∣ + ⟩ \otimes ∣ - ⟩$ showed in Chapter 19. What makes these states special is that no per-wire description exists at all.

Bell states as test equipment

For builders, Bell and GHZ circuits are the hello-world of correctness. A simulator should produce exactly the ideal outcome distribution. A real device will leak into 01 and 10 through gate error, readout error, decoherence, and crosstalk — and the size of that leakage is a quick, honest health metric.

The discipline is to keep four things separate when you look at output:

the ideal state;
the measurement distribution it implies;
the finite-shot sample you actually collected;
the device's error pattern.

Conflating the sample with the state, or the diagram with the device, is how small experiments get oversold.

Reading an entanglement demo

When a team demonstrates entanglement, ask five questions. Which state were they aiming for? Which basis did they measure? What fidelity or correlation metric did they report? What controls ran alongside? And how did they separate device error from ideal behavior? Those answers tell you whether you are looking at physics or at a rendering of physics.

Keep the ceiling in view as well: one small entangled state proves entanglement. It does not prove scalable computation, network utility, or application advantage — each of those needs its own evidence.

Exercise

Bell trace, then GHZ. Derive the Bell state from $∣ 00 ⟩$ using H and the CNOT basis action, then extend the trace to three qubits. Close with one sentence on what a Bell-state demonstration does and does not establish.

Submit: the circuit diagram, the state after each gate, and the predicted measurement distribution.
Check: compute the single-qubit marginal probabilities and the joint probabilities separately, and confirm the marginals are flat while the joints are perfectly correlated.
Repair: if your write-up implies one party can choose the other's outcome, or treats the qubits as independent after CNOT, rework Chapter 19 (Multi-Qubit Circuits) and Chapter 7 (Entanglement Without Faster-Than-Light Myths).

Check your understanding

From a blank page: build the Bell circuit, derive the state at each step, predict measurement counts for 1000 shots, and explain both the perfect correlation and the flat marginals.

Oral defense: a colleague says measuring one half of a Bell pair collapses the other half into a chosen value. Correct the sentence precisely.

If you get stuck

If you cannot reproduce the preparation trace, or your no-signaling explanation leans on vibes, work back through Chapter 19 (Multi-Qubit Circuits) for the mechanics and Chapter 7 (Entanglement Without Faster-Than-Light Myths) for the interpretation. Bell states are the test case every later protocol chapter reuses.