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Q001 · Amplitudes / measurement

Quantum State & Born Measurement

Prepare a pure state or a mixture, turn the analyzer, and predict a probability. Acquire individual outcomes and compare their frequency and uncertainty with the density-matrix prediction.

Interactive modelQuantum State & Born Measurement
Model positive probability—\text{—}
Positive / negative / rejected—\text{—}
Current-setting trials—\text{—}
Acquired positive fraction—\text{—}
95% Wilson interval—\text{—}
Output-state purity—\text{—}

SIMULATED ACQUISITION

Make a prediction. Collect the evidence.

No acquired record yet.

Inference from acquired counts

Acquire a scan to estimate the signal.

—\text{—}
Density matrix and count conventions
ρ=I+r⋅σ2\rho=\frac{I+\boldsymbol r\cdot\boldsymbol\sigma}{2}

Latest raw events · full record in CSV
EventSettingOutcome

Scientific basis: Feynman III.6 · MIT 8.05 · NIST Ramsey

Physics tutorial

Reading a quantum state through different bases

BackgroundOne state predicts outcomes for every measurement axis. A mixture carries less polarization than a pure preparation.

Why it mattersInterference experiments and state characterization require relative phase and preparation statistics, not just one population.

Start with the essentials

Focus question
Which analyzer can distinguish two states with equal Z probabilities?
One-sentence intuition
Relative phase changes transverse overlap. Global phase cancels in the density matrix.

Core mathematical model

Mixture with an unpolarized ensemble

ρ=v∣ψ⟩⟨ψ∣+(1−v)I/2\rho=v|\psi\rangle\langle\psi|+(1-v)I/2

Mixture with an unpolarized ensemble

Density-matrix Born rule

P(+n)=Tr⁡(ρΠ+)=(1+r⋅n)/2P(+\boldsymbol n)=\operatorname{Tr}(\rho\Pi_+)=(1+\boldsymbol r\cdot\boldsymbol n)/2

Density-matrix Born rule

Purity, not an individual particle orientation

Tr⁡(ρ2)=(1+∣r∣2)/2\operatorname{Tr}(\rho^2)=(1+|\boldsymbol r|^2)/2

Purity, not an individual particle orientation

Common difficulties

A short vector needs a density matrix

Typical misconceptionA mixture is just a pure wavefunction with a smaller norm.

Better mental modelPure wavefunctions stay normalized. The short Bloch vector represents an ensemble density matrix, with unit trace and nonnegative eigenvalues.

Intervals have a scope

Typical misconceptionA 95% interval contains the truth in exactly 95% of this one record.

Better mental modelCoverage concerns repeated experiments and is approximate for Wilson intervals. It excludes apparatus and model errors.

Run the experiment

  1. 01

    Predict in Z

    Choose the Z preset. Predict whether a relative-phase scan changes the positive fraction.

    What to observe: The reference stays flat. Fresh acquired counts fluctuate around the same probability.
  2. 02

    Reveal phase

    Choose Read phase and acquire a scan. Turn the analyzer azimuth.

    What to observe: The fringe shifts relative to the preparation phase; its amplitude comes from transverse coherence.
  3. 03

    Mix preparations

    Choose Partial mixture, then Maximally mixed. Acquire scans using the same number of shots.

    What to observe: The density matrix changes. At zero pure-state fraction every analyzer gives a half probability.
  4. 04

    Test sample size

    Collect 20 then 2000 shots at a fixed setting. Change global phase and repeat.

    What to observe: Wilson intervals narrow with more shots; global phase changes no theoretical probability.
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