Quantum Measurement Collapse Calculator

Calculator Inputs

The page uses a single stacked layout, while the input grid shifts to three columns on large screens, two on medium screens, and one on mobile screens.

α real part

α imaginary part

β real part

β imaginary part

Measurement basis preset

θ axis angle in degrees

φ phase angle in degrees

Collapse strength s (0 to 1)

Positive eigenvalue λ+

Negative eigenvalue λ−

Expected trial count

Complex amplitudes Custom basis axis Partial collapse model Bloch vector tracking CSV and PDF export

Formula Used

1) Normalize the input qubit |ψ⟩ = α|0⟩ + β|1⟩ ‖ψ‖ = √(|α|² + |β|²) |ψₙ⟩ = |ψ⟩ / ‖ψ‖

3) Apply the Born rule c₊ = ⟨m+|ψₙ⟩ c₋ = ⟨m-|ψₙ⟩ p₊ = |c₊|² p₋ = |c₋|²

4) Compute projective collapse states |ψ₊⟩ = (P₊|ψₙ⟩) / √p₊ |ψ₋⟩ = (P₋|ψₙ⟩) / √p₋ where P₊ = |m+⟩⟨m+| and P₋ = |m-⟩⟨m-|

5) Model partial collapse on the ensemble density matrix ρbefore = |ψₙ⟩⟨ψₙ| ρproj = p₊|m+⟩⟨m+| + p₋|m-⟩⟨m-| ρafter = (1 − s)ρbefore + sρproj s ranges from 0 to 1.

6) Observable expectation and information ⟨A⟩ = λ₊p₊ + λ₋p₋ Purity = Tr(ρ²) Binary entropy: H = −p₊log₂(p₊) − p₋log₂(p₋)

How to Use This Calculator

Enter the real and imaginary parts of α and β for the qubit state.
Select a basis preset or choose a custom axis.
If you select a custom axis, enter θ and φ in degrees.
Set the collapse strength between 0 and 1.
Enter the two measurement eigenvalues for the observable.
Add a trial count to estimate expected positive and negative detections.
Submit the form to view probabilities, collapsed states, density matrices, and Bloch vector changes.
Use the CSV and PDF buttons to save the result section.

Example Data Table

This example uses a balanced superposition in the Pauli-Z basis with full collapse strength.

α	β	Basis	θ	φ	s	Trials	p+	p-	Purity After
0.707107 + 0.000000i	0.707107 + 0.000000i	Pauli-Z	0°	0°	1.0	1000	0.500000	0.500000	0.500000

FAQs

1) What does this calculator measure?

It evaluates a two-level quantum state under projective measurement. It returns Born probabilities, collapsed outcome states, density matrices, purity, Bloch components, expected trial counts, and an observable expectation value based on your chosen eigenvalues.

2) Why are my amplitudes normalized automatically?

Quantum state vectors must have total probability one. Automatic normalization keeps the physical interpretation valid and allows the calculator to compare states even when raw user inputs are scaled differently.

3) What is the collapse strength input?

It interpolates between the original pure state and the fully decohered postmeasurement ensemble. A value of 0 leaves the density matrix unchanged, while 1 applies the full projective ensemble collapse model.

4) Does this simulate a true weak measurement?

Not exactly. It offers a practical partial-collapse interpolation for analysis and visualization. That makes it useful for intuition, but it is not a full continuous weak-measurement or stochastic trajectory simulator.

5) What do θ and φ represent?

They define the measurement axis on the Bloch sphere. θ controls the polar angle from the positive z direction, while φ sets the azimuthal phase around the equator.

6) Why can a collapsed state be undefined?

If an outcome probability is exactly zero, that outcome never occurs for the chosen state and basis. Dividing by the square root of zero is not valid, so the corresponding collapsed state is undefined.

7) What does the purity value tell me?

Purity shows how mixed the density matrix is. A pure state has purity 1. Smaller values indicate more decoherence or classical uncertainty in the postmeasurement ensemble.

8) What do the expected counts mean?

They are probability-weighted estimates for repeated identical measurements. For example, if p+ is 0.30 and trials are 1000, the expected positive count is about 300.