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Multi-Qubit Registers & Separability
Read |q1 q0⟩ labels the Qiskit way, build joint states with ⊗, and tell separable from entangled at a glance.
- 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.
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.
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.
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 3.0;
include "stdgates.inc";
qubit[2] q;
h q[0];
h q[1];
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 3.0;
include "stdgates.inc";
qubit[2] q;
h q[0];
cx q[0], q[1];
2 questions · one try each counts
A two-qubit circuit returns the outcome 01. Which qubit read 1?
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Which of these states is entangled?
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