---
title: "Quantum State & Born Measurement Interactive Lab | Sandbox Physics"
description: "Can two states look identical in one basis and different in another? Yes."
canonical_url: "https://sandboxphysics.com/labs/qubit-state/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/qubit-state/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Simulated acquisition

## Make a prediction, then collect the evidence.

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](https://www.feynmanlectures.caltech.edu/III_06.html) · [MIT 8.05](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/lecture-notes-7/) · [NIST Ramsey](https://www.nist.gov/publications/ramsey-spectroscopy-displaced-frequency-jumps)

Physics tutorial

## Reading a quantum state through different bases

**Background** A single quantum state predicts the outcome probabilities along every measurement axis. A mixture is less polarized than a pure state.

**Why it matters** Interference experiments and state characterization need the relative phase and the statistics of the preparation. The population in one basis is not enough.

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 misconception** A mixed state is just a pure wavefunction with a smaller norm.

**Better mental model** A 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 misconception** This 95% interval contains the true value with exactly 95% probability in this one record.

**Better mental model** Coverage 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Measure in more than one basis and keep the raw counts. That is how you tell relative phase, mixing and sampling noise apart.

## Related experiments and articles

Quantum evidence · Reading path

### Clocks, traps and single-quantum control

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Counting the quantum noise left by a perfect detector**](https://sandboxphysics.com/stories/quantum-evidence/quantum-projection-noise-population-counts/)
2. [Lab **Ramsey Frequency Measurement** Compare Ramsey fringes measured with a finite number of atoms](https://sandboxphysics.com/labs/ramsey-frequency/)
3. [Article **How helium atoms escaped the light and formed two narrow peaks**](https://sandboxphysics.com/stories/quantum-evidence/subrecoil-cooling-dark-state-momentum/)
4. [Lab **STIRAP Dark-State Transfer** Move an atom through a dark state with STIRAP](https://sandboxphysics.com/labs/stirap-dark-state-transfer/)
5. [Lab **Laser Cooling & MOT** Compare the central momentum peak with the camera’s full spread](https://sandboxphysics.com/labs/laser-cooling-magneto-optical-trap/)

Quantum evidence · Reading path

### Probability and exclusion

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Born: how a scattering direction became a probability**](https://sandboxphysics.com/stories/quantum-evidence/born-scattering-probability/)
2. [Lab **Partial-Wave Scattering** Compare angular patterns in partial-wave scattering](https://sandboxphysics.com/labs/partial-wave-scattering/)

Quantum evidence · Reading path

### Bell tests and photon information

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Dense coding sends three distinguishable messages with one photon**](https://sandboxphysics.com/stories/quantum-evidence/dense-coding-three-messages-resources/)
2. [Lab **Quantum Teleportation & Swapping** Send messages by dense coding and count errors, erasures and resources used](https://sandboxphysics.com/labs/quantum-teleportation/)
3. [Lab **Hong–Ou–Mandel Two-Photon Interference** See the two-photon interference the analyzer uses](https://sandboxphysics.com/labs/hong-ou-mandel/)

Starter path · Step 1 of 6

## Start from a single measurement

[Next step → **Stern–Gerlach Sequence** Turn the middle magnet to a crossed axis and compare it with the same-axis case. What does the middle measurement change? Compare · 15 min](https://sandboxphysics.com/labs/stern-gerlach-sequence/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)