Multi-Qubit Registers & Separability

Read |q1 q0⟩ labels the Qiskit way, build joint states with ⊗, and tell separable from entangled at a glance.

Beginner12 min read8 sections · 2 questions+45

By the end you will be able to

01Read a label such as |10⟩ as q1 = 1 and q0 = 0 (little-endian order).
02Build a two-qubit state from two one-qubit states with the tensor product, without heavy matrices.
03Decide whether a state is separable or entangled from the Bloch arrows, the purity and the Q-sphere.

QubitLab, like Qiskit, writes qubit 0 on the right. Think of the digits of a number: the rightmost digit is the ones place. A label is read |q1 q0⟩, and as a binary number it gives the index of the basis state. The usual slip: in the drawing q[0] is the top wire, but in the label it is the last character.

Label	q1	q0	Index (binary value)
\|00⟩	0	0	0
\|01⟩	0	1	1
\|10⟩	1	0	2
\|11⟩	1	1	3

01The tensor product in plain words

When two qubits are independent, the joint state just says 'q1 is in this state and q0 is in that state'. The tensor product ⊗ is the bookkeeping: multiply every amplitude of q1 with every amplitude of q0, and glue the labels together.

(a ∣0 ⟩ + b ∣1 ⟩)_{q 1} \otimes (c ∣0 ⟩ + d ∣1 ⟩)_{q 0} = a c ∣00 ⟩ + a d ∣01 ⟩ + b c ∣10 ⟩ + b d ∣11 ⟩

Example: |+⟩ ⊗ |+⟩ has a = b = c = d = 1/√2, so all four labels get amplitude 1/2 and probability 1/4.

02Separable or entangled?

A state that can be written as one factor times another is separable. A state that cannot is entangled. For two qubits there is a quick test on the four amplitudes: the state is separable exactly when the 'outer' product equals the 'inner' one.

a_{00} a_{11} = a_{01} a_{10} ⟺ separable Bell: \frac{1}{2} \cdot \frac{1}{2} \neq = 0 \cdot 0

The test works for pure two-qubit states. For the Bell state (|00⟩ + |11⟩)/√2 the left side is 1/2 and the right side is 0, so no choice of a, b, c, d can produce it.

03Seeing it in QubitLab

Bloch view: in a separable pure state every qubit has a full-length arrow (purity 1). When qubits are entangled the arrows shrink toward the centre; for a Bell pair they have length 0 and purity 1/2.
Q-sphere: one dot per basis state of the whole register, size for probability and colour for phase, |00⟩ at the north pole and |11⟩ at the south pole. It shows the joint state; combine it with the test above to judge separability.

Hands-on

Two independent qubits

State tab: four Q-sphere dots of equal size, and in the Bloch view both arrows have full length on the equator. Each qubit has a state of its own.

OpenQASM 3

OPENQASM 3.0;
include "stdgates.inc";
qubit[2] q;
h q[0];
h q[1];

Open in the IDE Open in the Playground

Hands-on

Now entangle them

Same two qubits, but a CNOT instead of the second H. The Q-sphere keeps only |00⟩ and |11⟩, and both Bloch arrows collapse to the centre with purity 0.5: neither qubit has a state of its own any more.

OpenQASM 3

OPENQASM 3.0;
include "stdgates.inc";
qubit[2] q;
h q[0];
cx q[0], q[1];

Open in the IDE Open in the Playground

Knowledge check

2 questions · one try each counts

Question 1

A two-qubit circuit returns the outcome 01. Which qubit read 1?

Question 2

Which of these states is entangled?

Label

Index (binary value)

|00⟩

|01⟩

|10⟩

|11⟩