---
title: "Quantum State & Process Tomography Interactive Lab | Sandbox Physics"
description: "Can measuring along one direction reveal a whole quantum state? No."
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language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/labs/quantum-state-process-tomography/"
last_modified: "2026-10-28"
---

Q042 · Ask the black box

# Quantum State & Process Tomography

**Can measuring along one direction reveal a whole quantum state?** No. Measure many identical copies along several directions, or, to characterize a process, send in several input states. The counts then reconstruct the state or the channel. With finite data a direct reconstruction can come out unphysical, such as a Bloch vector longer than one. Measurements held back from the fit test its predictions.

Choose how to measure an unknown qubit or channel. Collect independent outcomes and reconstruct it from the counts, then test your prediction on measurements that were kept out of the fit.

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

The simulation has one qubit, ideal input preparations and readout flips that occur independently. Direct inversion is compared with a projection onto physical matrices; maximum likelihood is not used. An uncalibrated detector stays mixed into the inferred response, and nothing here certifies a device.

> Interactive mathematical state-space view of simulated tomography measurements.

Ready. Nothing recorded yet.

02 / What the records say

## Build a description, one question at a time.

Observed proportions, with pointwise 95% Wilson intervals. Missing settings stay blank.

Reconstruction checks

$0/3$

$N=0$

03 / Predict before you look

## Keep a fresh direction out of the fit.

$\text{—}$

$\mathbf n=(1,1,1)/\sqrt3$ · A diagonal measurement; channel mode also uses this fresh input.

How the independent test is interpreted

The first held-out acquisition freezes the estimate; more fitting needs a new sample. These are point predictions. They are not confidence regions for the state and do not cover optional stopping.

04 / Save the evidence

## Compare complete recorded samples.

Raw outcome table and export details

Most recent twelve trials shown. Export keeps every zero and every failed outcome. Save up to six samples, then restore their settings and records.

Inspect the matrix and reveal the model

The reconstruction uses recorded fitting outcomes only. Reveal is an after-the-fact model comparison; it never changes the estimate. State estimates are projected onto the Bloch ball. Channels are projected onto positive Choi matrices with trace-preserving partial trace.

$\text{—}$

### Direct inversion

$\text{—}$

### Physical projection

$\text{—}$

A single-qubit stationary channel with fresh, uncorrelated environments is assumed. Incomplete settings cannot identify the state or channel. Unknown preparation errors and system-environment correlations are outside this model. Readout flips are not calibrated away; the inferred map includes that response. Channel matrices are displayed to two decimal places; diagnostics retain full precision. The projection is not maximum likelihood and the displayed matrix is not a certified device model.

[IBM / Qiskit · State tomography](https://qiskit-community.github.io/qiskit-experiments/manuals/verification/state_tomography.html) · [Surawy-Stepney et al. · Projected process tomography](https://arxiv.org/abs/2107.01060)

Physics tutorial

## Quantum State & Process Tomography

**Background** Choose how to measure an unknown qubit or channel. Collect independent outcomes and reconstruct it from the counts, then test your prediction on measurements that were kept out of the fit.

**Why it matters** Predict from what you can actually observe, then check the prediction separately.

Start with the essentials

- **Focus question**: Can measuring along one direction reveal a whole quantum state?
- **One-sentence intuition**: You need many identical preparations. Measuring one component does not fix the others, and with finite data the direct estimate can come out unphysical.

### Core mathematical model

#### One qubit, three components

$$
\rho=\frac{I+r_xX+r_yY+r_zZ}{2},\qquad \widehat r_j=\frac{N_{j,0}-N_{j,1}}{N_{j,0}+N_{j,1}}
$$

The linear estimate needs all three measurement settings. If the estimated vector lands outside the unit ball, the physical projection shortens it to the nearest point on the boundary.

#### A channel changes the whole Bloch vector

$$
\mathbf r_{\rm out}=T\mathbf r_{\rm in}+\mathbf t
$$

Six known inputs along the positive and negative axes, each measured in three output directions, determine the affine map. The direct estimate uses the unweighted mean of each setting.

#### Conditions for a physical process

$$
J\succeq0,\qquad \operatorname{Tr}_{\rm out}J=I_{\rm in}/2,\qquad\operatorname{Tr}J=1
$$

The normalized Choi matrix orders its tensor factors as input, then output. A Dykstra projection makes the estimate completely positive and trace preserving, and the residuals are reported separately.

#### A measurement held back from the fit

$$
P(0|\mathbf n)=\frac{1+\mathbf n\cdot\mathbf r}{2},\qquad\mathbf n=\frac{(1,1,1)}{\sqrt3}
$$

Freeze the estimate before you record the diagonal measurement. In channel mode, the same diagonal also defines an input that has not been used before.

### Common difficulties

#### What was measured?

**Typical misconception** The revealed model was used in the estimate.

**Better mental model** The reconstruction receives only the recorded counts used for fitting and the chosen mode. The comparison with the model is done separately.

#### What a simulation can show

**Typical misconception** A successful simulated run certifies a device.

**Better mental model** The simulation has one qubit, ideal input preparations and readout flips that occur independently. Direct inversion is compared with a projection onto physical matrices; maximum likelihood is not used. An uncalibrated detector stays mixed into the inferred response, and nothing here certifies a device.

### Run the experiment

1. 01

   #### Choose a comparison

   Record only one direction. Increase the shot count without adding directions.

   **What to observe:** The components you did not measure stay unknown.

2. 02

   #### Run the full protocol

   Collect all settings and inspect the direct and physical estimates. Then test them on fresh held-out measurements.

   **What to observe:** With finite samples, direct inversion can give a negative eigenvalue. The physical projection still does not guarantee an exact prediction.

3. 03

   #### Push the assumptions

   Increase the readout flips and compare a fresh reconstruction with the model, which is revealed separately.

   **What to observe:** Enforcing positivity does not remove detector bias, and more shots alone cannot calibrate an unknown detector.

Physical conclusion

The simulation has one qubit, ideal input preparations and readout flips that occur independently. Direct inversion is compared with a projection onto physical matrices; maximum likelihood is not used. An uncalibrated detector stays mixed into the inferred response, and nothing here certifies a device.

## Related experiments and articles

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 **How Bell tests distinguish quantum correlations**](https://sandboxphysics.com/stories/quantum-evidence/bell-tests-correlations-loopholes/)
2. [Lab **Bell Correlations & CHSH** See how sampling and detector efficiency affect a Bell test](https://sandboxphysics.com/labs/bell-chsh/)
3. [Lab **Quantum Error Correction** See how error correction differs from a Bell test](https://sandboxphysics.com/labs/quantum-error-correction/)
4. [Article **Freedman and Clauser test Bell’s inequality with calcium light**](https://sandboxphysics.com/stories/quantum-evidence/freedman-clauser-coincidence-normalization/)
5. [Article **Three-photon GHZ correlations defeat a shared answer sheet**](https://sandboxphysics.com/stories/quantum-evidence/ghz-parity-three-photon-records/)
6. [Lab **Three-Photon GHZ Correlation Bench** Collect three-photon coincidences and test GHZ parity](https://sandboxphysics.com/labs/three-photon-ghz-correlations/)
7. [Article **How the 1997 photon experiment transferred a quantum state**](https://sandboxphysics.com/stories/quantum-evidence/quantum-teleportation-conditional-records/)
8. [Lab **Quantum Teleportation & Swapping** Compare complete and partially successful Bell protocols](https://sandboxphysics.com/labs/quantum-teleportation/)
9. [Article **A joint measurement entangles two photons that never met**](https://sandboxphysics.com/stories/quantum-evidence/entanglement-swapping-fourfold-records/)
10. [Article **Reconstructing a photon state from calibrated counts**](https://sandboxphysics.com/stories/quantum-evidence/photon-tomography-counts-physical-states/)
11. [Lab **Homodyne Quantum Tomography** Rebuild the quantum state of light from homodyne measurements](https://sandboxphysics.com/labs/homodyne-quantum-tomography/)

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 **Let a second ion read out a narrow transition**](https://sandboxphysics.com/stories/quantum-evidence/quantum-logic-spectroscopy-shared-motion/)
2. [Lab **Two-Ion Quantum Logic Spectroscopy** Read one ion through the simulated fluorescence of its partner](https://sandboxphysics.com/labs/two-ion-quantum-logic-spectroscopy/)
3. [Lab **Trapped-Ion Sideband Spectroscopy** See how a sideband pulse moves an excitation into the motion](https://sandboxphysics.com/labs/trapped-ion-sideband-spectroscopy/)
4. [Lab **Ramsey Frequency Measurement** Compare reading out the level with reading out the phase](https://sandboxphysics.com/labs/ramsey-frequency/)
5. [Article **A cavity cat loses coherence before it loses its energy**](https://sandboxphysics.com/stories/quantum-evidence/cavity-cat-coherence-energy-decay/)
6. [Lab **Kerr Cats & Wigner Interference** Compare photon loss, branch interference and parity measurements](https://sandboxphysics.com/labs/kerr-cat-interference/)
7. [Lab **Decoherence Channels & Information Loss** Compare coherence loss in a qubit channel](https://sandboxphysics.com/labs/decoherence-channels/)
8. [Article **Keeping a chosen number of photons in a cavity**](https://sandboxphysics.com/stories/quantum-evidence/quantum-feedback-photon-number-verification/)
9. [Lab **Cavity Photon Counting & Feedback** Compare a limited feedback model with fresh verification data](https://sandboxphysics.com/labs/cavity-photon-qnd-feedback/)
10. [Lab **Quantum Error Correction** Find errors while leaving an encoded message unread](https://sandboxphysics.com/labs/quantum-error-correction/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

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)