How do I convert alpha and beta to Bloch coordinates?

After normalization, x = 2 Re(α*β), y = 2 Im(α*β), and z = |α|² − |β|², where α* is the complex conjugate of α.

Why does phi not change Z-basis probabilities?

Phi changes the relative phase of the amplitudes but not their magnitudes. Z-basis probabilities are |α|² and |β|², so they depend only on theta.

What do points inside the Bloch sphere mean?

Interior points represent mixed states described by density matrices. Their Bloch-vector length is below one; the center is the maximally mixed state.

Why are quantum amplitudes complex?

Complex amplitudes carry phase as well as magnitude. Relative phases control interference and therefore affect measurements in bases such as X and Y.

Quantum computing · Visualization · Private, in-browser

Bloch Sphere Simulator: Visualize a Qubit State

Explore a single-qubit quantum superposition: drag the state vector, apply gates, simulate measurements, or enter amplitudes and Bloch angles directly.

Interactive single-qubit simulator

State mode: drag/tap the sphere to move the vector. Camera mode: drag to orbit. Mouse wheel or +/− zooms. On the keyboard, arrows move the state; hold Shift to orbit the camera.

Polar angle θ

Azimuthal phase φ

Angle display

θ controls amplitude balance; φ controls relative phase. Values update live.

Current state
|ψ⟩ = |0⟩

Z basis

|0⟩100%

|1⟩0%

⟨Z⟩ = 1.000

X basis

|+⟩50%

|−⟩50%

⟨X⟩ = 0.000

Y basis

|+i⟩50%

|−i⟩50%

⟨Y⟩ = 0.000

Bloch vector

(0, 0, 1)

Angles

0°, 0°

Purity / eigenvalues

1.000 / (1, 0)

Apply quantum gates

Each gate animates a rotation path. X and Y generally change amplitude balance; Z, S and T are Z-axis phase rotations.

Rotation angledegrees

Last operation

I = [[1, 0], [0, 1]]

Result: |0⟩

No gates applied

Simulate measurements

Sampling uses Born-rule probabilities. Relative phase becomes visible in X- and Y-basis outcomes.

BasisShots Collapse state (one shot)

|0⟩

|1⟩

Choose a basis and number of shots.

Copy and export

Include equation, angles and labels in image exports

Worked qubit state examples

Guided exercise: phase changes interference

Load |+⟩: Z is 50/50 and X is 100/0.
Keep θ = 90° and change φ to 90°: Z remains 50/50 while Y becomes 100/0.
Change φ to 180°: Z still remains 50/50 while X becomes 0/100.

Bloch sphere concepts

What the sphere represents

A surface point represents one pure qubit after global phase is removed. The arrow coordinates equal the Pauli expectation values.

Angles and amplitudes

θ sets the balance between |0⟩ and |1⟩. φ is their relative phase, which rotates the arrow around Z.

Phase and interference

Changing φ leaves Z probabilities unchanged but changes X and Y probabilities. Those bases reveal interference between amplitudes.

How gates rotate states

Unitary gates rotate the Bloch vector. Only rotations around Z are purely relative-phase changes; X and Y rotations usually also change amplitude magnitudes.

What it cannot represent

One sphere cannot encode correlations or entanglement between multiple qubits. Interior vectors represent mixed one-qubit states, not imperfectly normalized pure states.

Bloch sphere formulas

Canonical pure state

|ψ⟩ = cos(θ/2)|0⟩ + e^iφ sin(θ/2)|1⟩

θ is the polar angle and φ the relative azimuthal phase.

Normalization

|α|² + |β|² = 1

α and β are complex amplitudes; their squared magnitudes are probabilities, not amplitudes.

Bloch coordinates

x = 2 Re(α*β), y = 2 Im(α*β), z = |α|² − |β|²

α* is the complex conjugate. For a pure state, x²+y²+z²=1.

Inverse conversion

θ = arccos(z), φ = atan2(y,x)

For pure states this is equivalent to θ=2 arccos(|α|), φ=arg(β)−arg(α).

Born rule in Pauli bases

P_X±=(1±x)/2, P_Y±=(1±y)/2, P_Z±=(1±z)/2

The expectation values are ⟨X⟩=x, ⟨Y⟩=y and ⟨Z⟩=z.

Density matrix

ρ = (I + r·σ)/2 = ½[[1+z, x−iy], [x+iy, 1−z]]

I is identity, r=(x,y,z), and σ=(X,Y,Z). Physical states require |r|≤1.

Numerical example

For α=1/√2 and β=i/√2, the norm is 1. Then x=0, y=1, z=0, so θ=90° and φ=90°. Born’s rule gives Z: 50/50, X: 50/50 and Y: 100/0. This is |+i⟩.

Bloch sphere and superposition FAQ

What is a quantum superposition?

A normalized combination α|0⟩+β|1⟩. The complex amplitudes determine measurement probabilities and relative phase.

How do I convert α and β to Bloch coordinates?

Normalize first, then use x=2Re(α*β), y=2Im(α*β), z=|α|²−|β|².

Why does φ not change Z-basis probabilities?

φ changes phase, not |α| or |β|. Since Z probabilities are |α|² and |β|², they depend only on θ.

What do points inside the sphere mean?

They are mixed states. Their Bloch-vector length is less than one; the center is the maximally mixed state. Normalized nonzero state vectors are pure and stay on the surface.

Can a Bloch sphere represent two entangled qubits?

No. It completely represents one qubit. A reduced state of one member can appear as a mixed point, but the sphere omits the two-qubit correlations.

What is global phase?

A common factor e^iγ multiplying all amplitudes. It has no observable effect and is not represented geometrically.

Why are amplitudes complex?

The complex phase produces interference. It can change X- or Y-basis measurements even when Z-basis probabilities are unchanged.

Method, review and limitations

Content owner
Starlight Robotics

Last reviewed
17 July 2026

Method
Direct complex arithmetic, Pauli-basis Born probabilities and browser pseudorandom sampling.

All calculations run client-side. The default simulator represents one ideal qubit and removes global phase. The canvas is an educational mathematical model, not experimental quantum hardware. Sampling varies randomly; it does not model device noise unless you enter a mixed state.

References: IBM Quantum Learning: single systems; Qiskit Statevector documentation; Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press.

Bloch Sphere Simulator: Visualize a Qubit State

Interactive single-qubit simulator

Z basis

X basis

Y basis

Apply quantum gates

Simulate measurements

Copy and export

Worked qubit state examples

Guided exercise: phase changes interference

Bloch sphere concepts

What the sphere represents

Angles and amplitudes

Phase and interference

How gates rotate states

What it cannot represent

Bloch sphere formulas

Numerical example

Bloch sphere and superposition FAQ

Method, review and limitations

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