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Quantum Measurement Calculator

Calculate quantum measurement probabilities, expectation values, variance, and wave function collapse.

Inputs

Results

Outcome 1 Probability

50%

Outcome 2 Probability

50%

Expectation Value

0

Variance1
Standard Deviation1
Measurement Fidelity0.5
How to Use This Calculator
  1. Enter State 1 Amplitude and State 2 Amplitude — they should satisfy |α₁|² + |α₂|² = 1 for a valid normalized state.
  2. Choose Measurement Basis: Computational Basis (|0⟩, |1⟩) for standard qubit readout, Hadamard Basis (+/−) for X-basis measurement.
  3. Read Outcome 1 Probability and Outcome 2 Probability — these are the Born rule probabilities for each measurement result.
  4. The Expectation Value is the average of many identical measurements; Variance shows how spread out results will be.
  5. Measurement Fidelity indicates how faithfully the measurement distinguishes the two outcomes.

How the result changes with State 1 Amplitude

State 1 AmplitudeOutcome 1 ProbabilityOutcome 2 ProbabilityExpectation Value
-0.856.15%43.85%0.123
-0.315.26%84.74%-0.695
0.315.26%84.74%-0.695
0.856.15%43.85%0.123

What each input means

State 1 Amplitude
Amplitude for first quantum state
State 2 Amplitude
Amplitude for second quantum state
Measurement Basis
Measurement basis

How this is calculated

Formula

P(outcome) = |⟨outcome|ψ⟩|²

Worked example, using the default values

  1. Identify Input Parameters
    State 1 Amplitude = 0.707, State 2 Amplitude = 0.707, Measurement Basis = 0 = 3 input(s) provided
  2. Calculate Outcome 1 Probability
    Outcome 1 Probability
    50 = 50%
  3. Calculate Outcome 2 Probability
    Outcome 2 Probability
    50 = 50%
  4. Calculate Expectation Value
    Expectation Value
    0 = 0
  5. Calculate Variance
    Variance
    1 = 1
  6. Calculate Standard Deviation
    Standard Deviation
    1 = 1

Engine last updated . Checked against 3 independently-derived tests — how we verify calculators.

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