Entanglement: how CNOT connects qubits

Single-qubit gates rotate one Bloch vector. Only a two-qubit gate can make two qubits share a state. Here is exactly how CNOT does it, why CZ and ECR are the same thing in disguise, and what 'native gate' means on a real chip.

Step 0 · A product state

∣ ψ ⟩ = ∣ q_{1} ⟩ \otimes ∣ q_{0} ⟩ = (a ∣0 ⟩ + b ∣1 ⟩) \otimes (c ∣0 ⟩ + d ∣1 ⟩)

Two independent qubits: the joint state factorizes. Each qubit has its own Bloch vector of length 1. Nothing you do to q0 alone can change q1.

Step 1 · What CNOT does to each basis state

Input \|q1 q0⟩	Control q0	Output	Why
\|00⟩	0	\|00⟩	control is 0 → target untouched
\|01⟩	1	\|11⟩	control is 1 → target flips 0→1
\|10⟩	0	\|10⟩	control is 0 → target untouched
\|11⟩	1	\|01⟩	control is 1 → target flips 1→0

On basis states CNOT is just a classical XOR: target ← target ⊕ control. It only becomes quantum when the control is in superposition, because then both rows of the table happen at once.

Step 2 · H then CNOT: the Bell state

∣00 ⟩ H on q_{0} \frac{1}{2} (∣00 ⟩ + ∣01 ⟩) CNOT \frac{1}{2} (∣00 ⟩ + ∣11 ⟩)

Apply the table to each term: |00⟩ stays, |01⟩ (control 1) becomes |11⟩. The result cannot be written as (…)⊗(…): try it, you would need ad = 0 and bc = 0 but ac ≠ 0 and bd ≠ 0. That impossibility is entanglement.

ρ_{q_{0}} = Tr_{q_{1}} ∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣ = \frac{1}{2} I, r = (0, 0, 0), purity = \frac{1}{2}

Each qubit alone is now maximally mixed: its Bloch vector has length 0 (watch it shrink in the Playground). All the information is in the correlation: measure both and you always get 00 or 11, never 01 or 10, even if the qubits are far apart.

Try it: Playground → Bell state → press › twice and watch the two Bloch arrows collapse to the centre when CNOT is applied.

Step 3 · CZ, CNOT and ECR are the same resource

CNOT = (I \otimes H); CZ; (I \otimes H)

CZ adds a −1 phase to |11⟩ only. Sandwich the target with Hadamards and the phase flip turns into a bit flip: that is a CNOT. Because CZ is symmetric, 'control' and 'target' are only labels.

CNOT = (R_{Z} (- \frac{π}{2}) \otimes R_{X} (\frac{π}{2})); ECR; (\dots) up to single-qubit gates

ECR (echoed cross-resonance) is what IBM Eagle chips physically do: drive the control at the target's frequency so the target rotates about X by ±π/2 depending on the control. Add a few single-qubit gates and it is a CNOT. Every entangler with 'maximal entangling power' (CNOT, CZ, ECR, iSWAP) can be converted to any other with single-qubit gates alone.

Step 4 · What 'native gate' means

Chip	Native 1-qubit	Native 2-qubit	So a CNOT costs
IBM Eagle (127q)	RZ (free), √X, X	ECR	1 ECR + a few √X/RZ (≈ 300–500 ns, error ~0.5–1 %)
IBM Heron (133–156q)	RZ, √X, X	CZ (+ fractional RZZ)	1 CZ + 2 H (as RZ·√X·RZ) (≈ 70–100 ns, error ~0.2–0.5 %)
Trapped ions	R(θ,φ)	RXX (Mølmer–Sørensen)	1 RXX(π/2) + single-qubit rotations
Generic (QubitLab 'generic' target)	RZ, √X, X	CX	1 CX

The transpiler's translation stage rewrites every gate you drew into the native set. That is why Compile on fake_sherbrooke (Eagle) shows ECR gates and on fake_torino (Heron) shows CZ: same circuit, different hardware dialect. The two-qubit count after translation is the number that matters for error, because each native entangler is 10–50× noisier than a single-qubit gate.

Step 5 · Connectivity: entangling qubits that are not neighbours

A native two-qubit gate only exists between physically coupled qubits (the coupling map, heavy-hex on IBM). To entangle q0 with q5 on a chain, the router inserts SWAPs to carry the state next to its partner: three CNOTs per hop. The Levels tab shows how many SWAPs each optimization level needed; cutting (next chapter) is the alternative when the chip is simply too small.

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Dynamic circuits: if/else, loops and conditions

Learning path: from one qubit to algorithms

Input |q1 q0⟩

Control q0

Output

Why

|00⟩

control is 0 → target untouched

|01⟩

|11⟩

control is 1 → target flips 0→1

|10⟩

control is 0 → target untouched

|11⟩

|01⟩

control is 1 → target flips 1→0

Chip

Native 1-qubit

Native 2-qubit

So a CNOT costs

IBM Eagle (127q)

RZ (free), √X, X

ECR

1 ECR + a few √X/RZ (≈ 300–500 ns, error ~0.5–1 %)

IBM Heron (133–156q)

RZ, √X, X

CZ (+ fractional RZZ)

1 CZ + 2 H (as RZ·√X·RZ) (≈ 70–100 ns, error ~0.2–0.5 %)

Trapped ions

R(θ,φ)

RXX (Mølmer–Sørensen)

1 RXX(π/2) + single-qubit rotations

Generic (QubitLab 'generic' target)

RZ, √X, X

1 CX