Q001 · Amplitudes and measurement

Quantum State & Born Measurement

Can two states look identical in one basis and different in another? Yes. Two states can give each outcome with the same probability in one basis and still differ in relative phase. A measurement in a transverse basis, such as x instead of z, tells them apart. To separate a real phase difference from a mixed state or sampling noise, you need several bases and plenty of counts.

Prepare a pure state or a mixture, turn the analyzer and predict the outcome probability. Then record outcomes one at a time and compare how often each appears, with its uncertainty, against the density-matrix prediction.

Model positive probability $\text{—}$

Positive / negative / rejected $\text{—}$

Current-setting trials $\text{—}$

Acquired positive fraction $\text{—}$

95% Wilson interval $\text{—}$

Output-state purity $\text{—}$

The sphere is a picture of state space. A mixture blends a pure state with an unpolarized ensemble, and its density matrix gives the shorter vector. A global phase does not change any count.

No acquired record yet.

Inference from acquired counts

Acquire a scan to estimate the signal.

\text{—}

Density matrix and count conventions

\rho=\frac{I+\boldsymbol r\cdot\boldsymbol\sigma}{2}

Latest raw events · full record in CSV

Event	Setting	Outcome

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

Start with the essentials

Focus question: If two states give the same probabilities along Z, which analyzer can tell them apart?
One-sentence intuition: A relative phase changes the probabilities for analyzers perpendicular to Z; a global phase cancels out of the density matrix.

Core mathematical model

Mixture with an unpolarized ensemble

\rho=v|\psi\rangle\langle\psi|+(1-v)I/2

A fraction of the ensemble is in the pure state, and the rest is completely unpolarized.

Born rule for a density matrix

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

The chance of a positive result depends only on how far the Bloch vector points along the analyzer direction.

Purity of the state

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

Purity grows with the length of the Bloch vector. It describes the ensemble, not the orientation of any single particle.

Common difficulties

A shorter vector means a mixture

Typical misconceptionA mixed state is just a pure wavefunction with a smaller norm.

Better mental modelA pure wavefunction always stays normalized. A Bloch vector shorter than the sphere's radius describes an ensemble, which needs a density matrix with unit trace and no negative eigenvalues.

What a confidence interval covers

Typical misconceptionThis 95% interval contains the true value with exactly 95% probability in this one record.

Better mental modelCoverage describes many repeated experiments, and for Wilson intervals it holds only approximately. It also leaves out errors in the apparatus and the model.

Run the experiment

01
Predict in Z
Choose the Z preset. Predict whether scanning the relative phase changes the fraction of positive results.
What to observe: The model curve stays flat. New counts scatter around that same probability.
02
Make the phase visible
Choose Read phase and acquire a scan. Then turn the analyzer azimuth.
What to observe: The fringe moves as the analyzer turns relative to the prepared phase. Its height is set by the transverse coherence.
03
Mix the preparation
Choose Partial mixture, then Maximally mixed. Acquire each scan with the same number of shots.
What to observe: The density matrix changes. When no pure state is left, every analyzer gives a positive result half the time.
04
Change the sample size
Keep the settings fixed and collect 20 shots, then 2000. Change the global phase and repeat.
What to observe: The Wilson intervals narrow as shots accumulate. The global phase leaves every predicted probability unchanged.

Quantum State & Born Measurement

Make a prediction, then collect the evidence.

Inference from acquired counts

Reading a quantum state through different bases

Core mathematical model

Mixture with an unpolarized ensemble

Born rule for a density matrix

Purity of the state

Common difficulties

A shorter vector means a mixture

What a confidence interval covers

Run the experiment

Predict in Z

Make the phase visible

Mix the preparation

Change the sample size

Related experiments and articles

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