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.

Recorded trials $0$

From the current records $\text{—}$

01 / Change one thing yourself

Ask in three directions.

Drag the amber axis to explore the fit. The separate test uses a fixed diagonal.

Try a comparison

Preparation and acquisition directions

Object under testKnown input preparationMeasurement direction

Changing the box or readout error starts a new record. Save first.

Acquisition size, readout and seed

Shots per setting

N_s

100

Readout flip probability

p_r

0

Sampling seed

s_0

17421

The same seed repeats the same sample, so repeats are not independent evidence.

What the query and replay mean

The amber axis predicts a positive measurement from the count-derived physical estimate. Dragging, reversing or using its arrow keys changes only the query. Enter reverses the focused axis. A complete fit is required; one measured direction cannot identify a three-dimensional state.

The white pulse and highlighted outcome replay existing fitting records. The estimate always uses all fitting outcomes. Querying and replaying add no data, change no fitted matrix, and leave the independent held-out test untouched.

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.

Reconstruct the unseen state

BOX 01

Loaded simulated exampleUnknown stateIdentical copies

Drag the amber handle to query the fit.

P(+)=0.5

Direction 1

Replay of stored outcomes

X:+1

State space, not physical space. Replay uses stored outcomes; the full fit stays fixed.

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.

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$\mathbf n=(1,1,1)/\sqrt3$ · 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.

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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

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 misconceptionThe revealed model was used in the estimate.

Better mental modelThe 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 misconceptionA successful simulated run certifies a device.

Better mental modelThe 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