Quantum Measurement Calculator
Calculate quantum measurement probabilities, expectation values, variance, and wave function collapse.
About this calculator
This calculator implements the Born rule for a two-level quantum system (qubit) described by two real amplitudes, State 1 Amplitude and State 2 Amplitude, which are renormalized internally so Outcome 1 Probability and Outcome 2 Probability always sum to 100% however the raw inputs are scaled. Measurement Basis genuinely changes what is measured: Computational Basis reads the amplitudes directly (probabilities are the squared normalized amplitudes), while Hadamard Basis first rotates the state onto the |+⟩ = (|0⟩+|1⟩)/√2 and |−⟩ = (|0⟩−|1⟩)/√2 axes before squaring -- the standard Hadamard transform used throughout quantum computing to read out superposition states. At the default equal-superposition amplitudes (both ≈0.707), the state IS the |+⟩ eigenstate, so switching to Hadamard Basis collapses Outcome 1 Probability to a certain 100% even though Computational Basis reads the same state as a 50/50 split -- a direct illustration that "probability" in quantum mechanics depends on which basis you measure in, not on the state alone.
Expectation Value, Variance, and Standard Deviation are the usual statistical moments of the ±1 measurement outcome, computed FROM whichever basis's probabilities apply, so they inherit the same basis-dependence. Measurement Fidelity (prob1² + prob2²) is a simple purity-style figure that peaks at 1.0 when the state is basis-aligned (a deterministic outcome) and bottoms out at 0.5 for an even 50/50 split, in either basis.
Inputs
Results
Outcome 1 Probability
50%
Outcome 2 Probability
50%
Expectation Value
0
How to Use This Calculator
- Enter State 1 Amplitude and State 2 Amplitude — they should satisfy |α₁|² + |α₂|² = 1 for a valid normalized state.
- Choose Measurement Basis: Computational Basis (|0⟩, |1⟩) for standard qubit readout, Hadamard Basis (+/−) for X-basis measurement.
- Read Outcome 1 Probability and Outcome 2 Probability — these are the Born rule probabilities for each measurement result.
- The Expectation Value is the average of many identical measurements; Variance shows how spread out results will be.
- Measurement Fidelity indicates how faithfully the measurement distinguishes the two outcomes.
How the result changes with State 1 Amplitude
| State 1 Amplitude | Outcome 1 Probability | Outcome 2 Probability | Expectation Value |
|---|---|---|---|
| 0.35 | 19.95% | 80.05% | -0.601 |
| 0.53 | 35.98% | 64.02% | -0.28 |
| 1 | 66.67% | 33.33% | 0.333 |
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⟩|²Worked example, using the default values
- Identify Input Parameters3 parametersState 1 Amplitude = 0.707, State 2 Amplitude = 0.707, Measurement Basis = 0 = 3 input(s) provided
- Calculate Outcome 1 ProbabilityOutcome 1 Probability50 = 50%
- Calculate Outcome 2 ProbabilityOutcome 2 Probability50 = 50%
- Calculate Expectation ValueExpectation Value0 = 0
- Calculate VarianceVariance1 = 1
- Calculate Standard DeviationStandard Deviation1 = 1
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why does switching Measurement Basis change the probabilities when I didn't touch the amplitudes?
Measurement Basis genuinely changes what is being measured. Computational Basis reads State 1 Amplitude and State 2 Amplitude directly; Hadamard Basis first rotates the same state onto the |+⟩ and |−⟩ axes -- (|0⟩+|1⟩)/√2 and (|0⟩−|1⟩)/√2 -- before computing probabilities. A state that looks like a 50/50 split in one basis can be perfectly certain in the other; that's a real feature of quantum measurement, not an inconsistency in this calculator.
Why do the default amplitudes (0.707, 0.707) give a 100% certain outcome in Hadamard Basis?
Because 0.707 is approximately 1/√2, the default state is already the |+⟩ eigenstate of the Hadamard basis -- exactly the state Hadamard-basis measurement is guaranteed to read with certainty. Change either amplitude away from that exact ratio, or switch Measurement Basis back to Computational, to see genuinely uncertain, probabilistic outcomes again.
What does Measurement Fidelity actually measure?
Measurement Fidelity is prob1² + prob2², a simple purity-style figure that peaks at 1.0 when one outcome is certain (either probability equals 1) and bottoms out at 0.5 for a perfectly even split. It answers "how deterministic is this measurement," not "how accurate is the measuring device" -- there is no hardware-error model in this calculator.
Do State 1 Amplitude and State 2 Amplitude need to already satisfy |α|² + |β|² = 1?
No -- the calculator renormalizes them internally, dividing each by the pair's own vector length, before computing any probability. Outcome 1 Probability and Outcome 2 Probability always sum to exactly 100% however you set the two amplitudes, so entering un-normalized values like 1 and 1 is equivalent to entering the already-normalized 0.707 and 0.707.
Why can Expectation Value and Variance change even though the ±1 eigenvalues never change?
Expectation Value and Variance are statistics of the OUTCOME distribution, not fixed properties of the eigenvalues themselves -- they're computed as prob1×(+1) + prob2×(−1) and the corresponding spread. Since Measurement Basis and the two amplitude inputs both change how probable each ±1 outcome is, they change these statistics even though the two possible outcomes stay fixed at +1 and −1.
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