Bloch Sphere

qc/basics qc/math

Any single-qubit pure state can be written, up to an unobservable global phase, using two real angles instead of two complex amplitudes:

$|\psi ⟩=\cos \left(\frac{\theta }{2}\right)|0⟩+e^{i\varphi }\sin \left(\frac{\theta }{2}\right)|1⟩,\theta \in [0,\pi ],\varphi \in [0,2\pi )$

This is exactly a point on the surface of a unit sphere — the Bloch sphere. North pole ($\theta =0$) is $|0⟩$, south pole ($\theta =\pi$) is $|1⟩$, and every point on the equator is an equal superposition of $|0⟩$ and $|1⟩$ differing only in relative phase $\varphi$.

Key insight: single-qubit gates are rotations of this point around an axis.

Why only two degrees of freedom

A general 2-state complex system has 4 real parameters (2 complex amplitudes = 4 real numbers). Normalization ($|\alpha {|}^2+|\beta {|}^2=1$) removes one. Global phase — multiplying the whole state by $e^{i\gamma }$ — is physically unobservable (see Measurement and Collapse, the Born rule only ever depends on $|\cdot {|}^2$) and removes another. That leaves exactly 2 real degrees of freedom, $\theta$ and $\varphi$ — which is precisely enough to specify a point on the surface of a sphere.

from qiskit.visualization import plot_bloch_multivector
from qiskit.quantum_info import Statevector
from qiskit import QuantumCircuit
 
qc = QuantumCircuit(1)
qc.h(0)
plot_bloch_multivector(Statevector(qc))   # shows the point moved to the equator

Gates as rotations

Gate	Rotation
X Gate	180° about the X-axis
Z Gate and Relative Phase	180° about the Z-axis
H Gate	180° about the axis halfway between X and Z (swaps the poles with the equator)

Reading gates this way turns Pauli Operators from abstract matrices into concrete geometric moves — the same picture used later when circuits are described as sequences of rotations.

The limit of this picture

The Bloch sphere only exists for a single qubit’s pure state. There is no analogous single point for a two-qubit entangled state — that irreducibility is part of what “entangled” means. See Bell States.

Related

• What is a Qubit
• Superposition
• H Gate, X Gate, Z Gate and Relative Phase
• Pauli Operators
• S and T Gates — smaller Z-axis rotations than the full Z gate

Self-Check

• Could you sketch, or describe in words, where $|0⟩$, $|1⟩$, and $H|0⟩$ each sit on the Bloch sphere?
• Why does thinking of gates as rotations help build intuition that the raw matrices don’t?
• Why can’t a two-qubit entangled state be drawn as a point on a single Bloch sphere?
• Why does a 2-state complex system have exactly 2, not 3 or 4, real degrees of freedom?