Analyze qubit coordinates, amplitudes, and measurement chances easily. Convert states, inspect observables, and export reports. Learn Bloch sphere relations with practical physics steps.
| State | Theta (°) | Phi (°) | Bloch Coordinates (x, y, z) | P(|0⟩) | P(|1⟩) |
|---|---|---|---|---|---|
| |0⟩ | 0 | 0 | (0, 0, 1) | 1 | 0 |
| |1⟩ | 180 | 0 | (0, 0, -1) | 0 | 1 |
| |+⟩ | 90 | 0 | (1, 0, 0) | 0.5 | 0.5 |
| |i+⟩ | 90 | 90 | (0, 1, 0) | 0.5 | 0.5 |
A pure qubit state can be written as |ψ⟩ = α|0⟩ + β|1⟩, where |α|² + |β|² = 1.
Using Bloch angles, α = cos(θ/2) and β = eiφ sin(θ/2).
The Bloch coordinates are x = sin(θ)cos(φ), y = sin(θ)sin(φ), and z = cos(θ).
From amplitudes, the same coordinates become x = 2Re(α*β), y = 2Im(α*β), and z = |α|² - |β|².
Measurement probabilities follow P(|0⟩) = |α|² and P(|1⟩) = |β|². The density matrix is ρ = |ψ⟩⟨ψ|.
The Bloch sphere is a geometric model for one qubit. It maps every pure qubit state to a point on a unit sphere. The north pole is |0⟩. The south pole is |1⟩. Points on the equator represent balanced superpositions. This picture helps students, engineers, and researchers interpret quantum states quickly.
Two common descriptions exist. The first uses angles θ and φ. The second uses complex amplitudes α and β. Both describe the same state. This calculator accepts either form. It then normalizes the state and converts the data into Bloch coordinates, measurement probabilities, and matrix values. That makes validation easier during learning and analysis.
This Bloch sphere calculator computes θ, φ, α, β, P(|0⟩), P(|1⟩), and the expectation values for σx, σy, and σz. It also returns the density matrix. These outputs are useful in quantum mechanics lessons, quantum computing exercises, state preparation checks, and introductory simulation workflows.
A pure qubit always lies on the sphere surface. Its radius equals one after normalization. The z coordinate shows bias toward |0⟩ or |1⟩. The x and y coordinates show phase-sensitive superposition behavior. Small input changes can rotate the state around the sphere and alter measurement predictions in different bases.
You can use this page to study gate outputs, compare basis states, verify state-vector homework, or build intuition for phase. Teachers can use the example data table in class. Students can export CSV or PDF summaries for reports. The layout also keeps the result section above the form after submission, so review stays quick and clear.
The calculator is simple to enter, but detailed in output. It supports learning, checking, and documentation. With direct formulas and clear sections, it helps transform abstract quantum notation into readable numerical results.
A Bloch sphere is a unit sphere that represents a single qubit state. Each point on the surface corresponds to one pure quantum state.
Theta controls vertical position on the sphere. Phi controls the azimuth angle. Together they define the qubit orientation.
Alpha and beta are complex amplitudes of |0⟩ and |1⟩. Their squared magnitudes give measurement probabilities in the computational basis.
Quantum state vectors must sum to one in total probability. Normalization fixes scaled inputs and keeps the result physically valid.
They are expectation values of the Pauli operators. These coordinates locate the qubit on the Bloch sphere.
The density matrix stores the full pure-state information in matrix form. It is useful for calculations, comparisons, and later quantum operations.
This version is designed for pure qubit states. It reports purity as one and uses a pure-state density matrix formula.
Export CSV for spreadsheets and repeated analysis. Export PDF for documentation, sharing, or printable summaries of one calculation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.