Bloch Sphere & Qubits

A light switch is either ON or OFF — that is an ordinary bit, the kind your laptop uses, always one of two values. A qubit is stranger and more generous. Picture instead a tiny arrow that can point anywhere on the surface of a globe. Straight up means "0", straight down means "1", and every direction in between is a genuine in-between state called a superposition. The globe is the Bloch sphere, and the arrow's direction is the qubit.

Where the angles come in

To name a direction on a globe you need two numbers, just like latitude and longitude. Here they are the polar angle $\theta$ (how far the arrow has tipped down from the north pole) and the azimuth $\varphi$ (how far it has swung around the equator). The north pole ($\theta=0$) is the state $|0\rangle$; the south pole ($\theta=180^\circ$) is $|1\rangle$; anywhere on the equator is a half-and-half mix. The compact rule that ties the geometry to the quantum state is

The reason for the half-angle is worth a worked number. If you tip the arrow to the equator, $\theta=90^\circ$, then the chance of reading "0" when you measure is $\cos^2(\theta/2)=\cos^2(45^\circ)=0.5$ — a perfect coin flip. Tip it only to $\theta=60^\circ$ and the odds shift to $\cos^2(30^\circ)\approx 0.75$, a 75 / 25 split. The arrow's height sets the probabilities; its swing around the equator ($\varphi$) sets the relative quantum phase, which you cannot see in a single measurement but which controls how amplitudes interfere later.

Gates just rotate the arrow

Everything a single qubit can "compute" is a rotation of this arrow. The Born rule $P(0)=\cos^2(\theta/2)$, $P(1)=\sin^2(\theta/2)$ reads off the probabilities; a gate like $X$ spins the arrow $180^\circ$ about the $x$-axis (it flips $|0\rangle\leftrightarrow|1\rangle$), while the Hadamard $H$ tips a pole onto the equator and builds a superposition. In the panel above, the $\theta$ and $\varphi$ sliders aim the arrow, the gate buttons rotate it along the glowing trail, and the readouts show $\langle X\rangle,\langle Y\rangle,\langle Z\rangle$ — the arrow's three shadows.

Try this in the sim above: (1) Drag $\theta$ to $90^\circ$ and watch $P(|0\rangle)$ settle at 50% — the equator is the fair-coin state. (2) Reset to $|0\rangle$, then click $H$ and see the arrow swing onto the $+x$ axis, the $|+\rangle$ state. (3) Switch to Rabi mode and raise $\Omega$: the arrow circles the $y$–$z$ plane and $\langle Z\rangle$ traces a $\cos(\Omega t)$ wave — a qubit being driven between 0 and 1.

A qubit is the quantum analogue of a classical bit. Where a classical bit is in one of two definite states (0 or 1), a qubit lives in any complex superposition of two basis states $|0\rangle$ and $|1\rangle$. The Bloch sphere is a geometric representation that maps every pure single-qubit state to a unique point on the surface of a unit sphere — a remarkably faithful translation of complex algebra into 3-D geometry.

1. The General Pure State

An arbitrary pure single-qubit state can be written:

Two complex numbers ($\alpha$, $\beta$) give 4 real parameters, but normalisation removes 1 and the global phase is unobservable, removing another. So the physical state space has 2 real parameters — exactly the dimensionality of a sphere.

2. Polar Form on the Bloch Sphere

Up to a global phase, every pure state can be written:

where $\theta \in [0,\pi]$ is the polar angle from the $+\hat z$ axis and $\varphi \in [0,2\pi)$ is the azimuth in the $xy$-plane. The point on the sphere is:

Special points:

• $+\hat z$: $|0\rangle$ (north pole)
• $-\hat z$: $|1\rangle$ (south pole)
• $+\hat x$: $|+\rangle = (|0\rangle+|1\rangle)/\sqrt 2$
• $-\hat x$: $|-\rangle = (|0\rangle-|1\rangle)/\sqrt 2$
• $+\hat y$: $|+i\rangle = (|0\rangle+i|1\rangle)/\sqrt 2$
• $-\hat y$: $|-i\rangle = (|0\rangle-i|1\rangle)/\sqrt 2$

3. Pauli Operators & Expectation Values

The three Pauli matrices are:

The Bloch vector components are precisely the expectation values:

For a pure state, $|\vec r| = 1$. For a mixed state (statistical mixture), $|\vec r| < 1$ and the state lives inside the Bloch ball, with the centre representing the maximally mixed state $\rho = I/2$.

4. Single-Qubit Gates as Rotations

Every unitary operation on a single qubit corresponds to a rotation of the Bloch sphere. The Pauli gates are 180° rotations:

5. Rabi Oscillations

A driven qubit (e.g., near-resonant microwave on a transmon) Hamiltonian is:

Solving Schrödinger's equation gives:

The qubit oscillates between $|0\rangle$ and $|1\rangle$ at frequency $\Omega/(2\pi)$. A pulse of duration $t = \pi/\Omega$ flips the qubit (a $\pi$-pulse, equivalent to an $X$ gate); a $\pi/2$-pulse creates an equal superposition.

6. Measurement Collapse

Measuring $\sigma_z$ on $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ gives:

After measurement, the state collapses to whichever eigenstate was observed. Repeated measurements on a Bloch state $\vec r$ along axis $\hat n$ give expectation $\langle \hat n\cdot\vec\sigma\rangle = \hat n\cdot\vec r$.

7. Two-Qubit Bell States

The Bell circuit applies Hadamard to the first qubit, then CNOT (control = first):

This is maximally entangled: tracing out either qubit leaves the other in the maximally mixed state $I/2$ (centre of the Bloch ball). Local single-qubit Bloch sphere descriptions cannot capture entanglement — that's why we need full density matrices for many-qubit systems.

Symbol Table

Mapping to the simulation

Sliders θ, φ set the initial point on the Bloch sphere. Click any gate (X, Y, Z, H, S, T, Rx, Ry, Rz) to apply it as a smooth rotation — the trail shows the geodesic path. Rabi mode applies a continuous $X$-axis drive at angular frequency Ω; you'll see ⟨Z⟩ oscillate as $\cos(\Omega t)$. Measurement mode samples 100 measurements and shows the empirical fraction converging on $|\alpha|^2$ and $|\beta|^2$. Bell mode shows that two qubits prepared as $|\Phi^+\rangle$ have a maximally mixed reduced state on each individual Bloch sphere.