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Q042 · Ask the black box

Quantum State & Process Tomography

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

Interactive modelQuantum State & Process Tomography
Recorded trials00
From the current records—\text{—}
REPEATED INDEPENDENT COPIES01

Prepare

Identical copies

Unknown object

?

State source

Choose a question

First direction

Recent binary outcomes

A measurement protocol, not a hardware layout. Each symbol is one recorded outcome; it is not a particle path.

Ready. No recorded evidence 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.

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03 / MAKE A PREDICTION BEFORE LOOKING

Keep a fresh direction out of the fit.

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n=(1,1,1)/3\mathbf n=(1,1,1)/\sqrt3 · A diagonal measurement; channel mode also uses this fresh input.

The first holdout acquisition freezes the estimate. Further fitting requires a new sample. These are point predictions, not state confidence regions or optional-stopping guarantees.

04 / KEEP THE EVIDENCE

Compare complete recorded samples.

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

Matrix inspection & 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.

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Direct inversion

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Physical projection

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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 · Surawy-Stepney et al. · Projected process tomography

Physics tutorial

Quantum State & Process Tomography

BackgroundChoose how to interrogate an unknown qubit or channel. Collect independent outcomes, reconstruct from the counts, then test your prediction on measurements kept out of the fit.

Why it mattersMake a prediction from the information you can actually observe, then perform a separate check.

Start with the essentials

Focus question
One direction cannot tell the whole story.
One-sentence intuition
Many identical preparations are needed. Measuring one component does not determine the other components, and finite records may produce an unphysical direct estimate.

Core mathematical model

One qubit, three components

ρ=I+rxX+ryY+rzZ2,r^j=Nj,0−Nj,1Nj,0+Nj,1\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 settings. The physical state projection shortens a vector outside the unit ball to the nearest boundary.

A channel changes the whole Bloch vector

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

Six known positive and negative axis inputs and three output directions determine the affine map. Unweighted cell means define the direct estimate.

Physical process constraints

J⪰0,Tr⁡outJ=Iin/2,Tr⁡J=1J\succeq0,\qquad \operatorname{Tr}_{\rm out}J=I_{\rm in}/2,\qquad\operatorname{Tr}J=1

The normalized Choi matrix uses input then output tensor order. A Dykstra projection enforces complete positivity and trace preservation, with residuals reported separately.

A withheld question

P(0∣n)=1+n⋅r2,n=(1,1,1)3P(0|\mathbf n)=\frac{1+\mathbf n\cdot\mathbf r}{2},\qquad\mathbf n=\frac{(1,1,1)}{\sqrt3}

Freeze the estimate before recording the diagonal measurement. In channel mode, the same diagonal also defines a previously unused input.

Common difficulties

What was measured?

Typical misconceptionThe revealed model helped the estimator.

Better mental modelReconstruction receives only recorded fitting counts and the declared mode. The model comparison is separate.

Scope of evidence

Typical misconceptionA successful simulated run certifies a device.

Better mental modelSingle-qubit simulation with ideal input preparations and independent readout flips. Direct inversion is compared with a physical matrix projection, not maximum likelihood. An uncalibrated detector remains part of the inferred response; no device certification is claimed.

Run the experiment

  1. 01

    Choose a contrast

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

    What to observe: The unknown components remain unidentified.
  2. 02

    Complete the protocol

    Collect all settings, inspect direct and physical estimates, then use fresh held-out measurements.

    What to observe: Finite sampling can make direct inversion negative; physical projection does not guarantee an exact prediction.
  3. 03

    Stress the assumption

    Increase readout flips and compare a fresh reconstruction against the separately revealed model.

    What to observe: Detector bias is not eliminated by positivity. More shots alone cannot calibrate an unknown detector.