About the Quantum Probability Calculator
What is the Quantum Probability Calculator?
A qubit can be in a superposition α|0⟩ + β|1⟩ of its two basis states. When measured, it gives 0 with probability |α|² and 1 with probability |β|² – the Born rule.
Enter the real and imaginary parts of both amplitudes. The calculator normalises the state and shows its position on the Bloch sphere.
How to use the Quantum Probability Calculator
- Amplitude α – real part (sample: 0.6)
- Amplitude α – imaginary part (sample: 0)
- Amplitude β – real part (sample: 0.8)
- Amplitude β – imaginary part (sample: 0)
- Press Calculate Measurement Probabilities (results also refresh as you type). The result card shows the Probability of Measuring |0⟩, along with Probability of measuring |1⟩, Normalisation factor applied, Bloch sphere polar angle θ and Relative phase φ.
Formula used
P(0) = |α|² ÷ (|α|² + |β|²); P(1) = |β|² ÷ (|α|² + |β|²)
- |α|² = real part² + imaginary part²
- Bloch angle θ = 2 × arccos(√P(0))
The relative phase does not change measurement probabilities in this basis, but matters for interference.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Amplitude α – real part | 0.6 |
| Amplitude α – imaginary part | 0 |
| Amplitude β – real part | 0.8 |
| Amplitude β – imaginary part | 0 |
| Result | Value |
|---|---|
| Probability of Measuring |0⟩ | 36% |
| Probability of measuring |1⟩ | 64% |
| Normalisation factor applied | 1 |
| Bloch sphere polar angle θ | 106.26° |
| Relative phase φ | 0° |
| State | ψ = α|0⟩ + β|1⟩, with |α|² + |β|² = 1 after normalisation |
Tips for using the Quantum Probability Calculator
- An equal superposition (α = β = 1/√2) gives a 50–50 outcome.
- Global phase has no physical effect; only the relative phase between α and β matters.