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Every qubit state is a point on a sphere

A single qubit needs two complex amplitudes, which sounds like four numbers. After the rules of normalization and global phase, only two angles are left. Those two angles place the qubit at a point on the surface of the Bloch sphere, a picture physicists use every day.

Sinclair College - Dr. Mike Libassi

Bloch sphere — a unit sphere where the north pole is |0⟩, the south pole is |1⟩, and every other surface point is a superposition. Named after physicist Felix Bloch.

A qubit drifting around the sphere. Drag to turn it.

01 · From amplitudes to angles

Two angles describe any qubit

Any single-qubit state can be written using a polar angle θ (how far down from the north pole) and an azimuthal angle φ (how far around the equator):

θ (theta), from 0° to 180° — latitude measured from the top. θ = 0° is |0⟩, θ = 180° is |1⟩, θ = 90° is the equator, an even 50/50 blend.

φ (phi), from 0° to 360° — longitude around the vertical axis. It sets the relative phase between the |0⟩ and |1⟩ parts, which a plain 0-or-1 measurement can't see.

|ψ⟩=cos(θ/2)|0⟩ + e^iφ sin(θ/2)|1⟩

|ψ⟩=cos(θ/2)e^iφ sin(θ/2) point on sphere = (sinθ cosφ, sinθ sinφ, cosθ)

Notice the θ/2. The sphere uses the full angle θ, so |0⟩ (θ = 0°) and |1⟩ (θ = 180°) end up at opposite poles. On the Bloch sphere, opposite points are states that can be told apart perfectly.

02 · The core idea

Move the qubit around the sphere

Try it: set θ and φ

Use the sliders or presets. Drag the sphere (or focus it and use arrow keys) to change your viewing angle. The amber arrow is the qubit.

drag to rotate the view

θ (polar)60°

φ (azimuthal / phase)45°

|ψ⟩=

P(0)

P(1)

Try this: set θ = 90° and slide φ all the way around. The probability bars never move, yet the qubit travels through |+⟩, |+i⟩, |−⟩, and |−i⟩. Phase is real and physically meaningful, but invisible to a 0-or-1 measurement.

03 · Landmarks

Six states worth memorizing

Each pair sits on opposite ends of one axis. Click a state to see where it lives.

04 · Limits of the picture

What the sphere can't show

The Bloch sphere is a perfect map for one qubit. It breaks down for two or more qubits, because entangled states (page 05) don't belong to either qubit individually. There's no pair of arrows that captures them.

THE SPHERE SHOWS

Any single, isolated qubit state: its odds (latitude) and its phase (longitude).

IT DOESN'T SHOW

Global phase (it has no physical effect), and anything about entanglement between qubits.

Real qubits that are losing their state to noise drift inside the sphere, toward the center. These are called mixed states; only pure states sit on the surface. The very center is the maximally mixed state, a completely random qubit. Quantum error correction doesn't push a qubit back to the surface. It spreads one logical qubit across many physical qubits and repeatedly checks for errors without reading the data itself, so the logical information survives even though each physical qubit is noisy.

05 · Security connection

The two axes BB84 uses

Quantum key distribution encodes bits using two axes of this sphere. The Z axis (|0⟩ and |1⟩ at the poles) is the rectilinear basis. The X axis (|+⟩ and |−⟩ on the equator) is the diagonal basis. The two axes sit at 90° to each other, so a state that is certain along one is a 50/50 coin flip along the other. Bases related this way are called mutually unbiased. Don't confuse this with orthogonal states such as |0⟩ and |1⟩, which sit 180° apart at opposite ends of one axis.

Why that matters: an eavesdropper who measures along the wrong axis gets a random answer and knocks the qubit onto the other axis. Page 07 turns that into an intrusion detector.