Q048 · Open systems / state reconstruction

Decoherence Channels & Information Loss

Watch level populations and phase alignment separately. Predict what changes under phase damping, then compare relaxation into a cold bath. Test the endpoint with three independent sets of simulated counts; explore state reconstruction and decay fits when ready.

Ground population $50\%$

Phase alignment $100\%$

The prepared state is ready. Move time forward to see the change.

Bars are modeled populations. The phase hand shows transverse coherence; its direction is a relative phase, not a spatial orientation. Gray marks the preparation.

Endpoint model and acquisition readouts

Model excited population $\text{—}$

Model displayed-basis probability $\text{—}$

Model purity $\text{—}$

Model entropy $\text{—}$

Bath excited fraction $\text{—}$

Model transverse rate $\text{—}$

Current tomography shots $\text{—}$

ONE CHANGE · WATCH WHAT HAPPENS

Phase loss or energy loss?

Start with equal ground and excited populations and a coherent phase. Compare phase damping with relaxation into a cold bath.

1 · Choose and predict

At the end, will the level populations stay the same?

2 · Play or scrub the time

Observation time

0\,\mathrm{ms}

Make a prediction, then play to the end to check it.

3 · Test with simulated counts

The animation is a model preview. These actions acquire fresh preparations at the selected endpoint; raw counts and uncertainty appear below.

Go further · parameters and state sphere

The simple comparison above resets to its stated preparation. Here you can change the full experiment. The state sphere is a mathematical representation, not a spatial spin direction.

Time in the channel

t

2\,\mathrm{ms}

Channel: thermal / phase / depolarizingThermal longitudinal rate

\Gamma_1

1\,\mathrm{ms^{-1}}

Pure-dephasing rate

\Gamma_\varphi

1\,\mathrm{ms^{-1}}

Depolarization rate

\Gamma_d

1\,\mathrm{ms^{-1}}

Bath temperature

T_b

0\,\mathrm{K}

Level separation

f_{01}

5\,\mathrm{GHz}

Preparation polar angle

\theta

90\,\mathrm{deg}

Preparation azimuth

\phi

0\,\mathrm{deg}

Preparation polarization

v

1

Display analyzer polar angle

\alpha

90\,\mathrm{deg}

Shots per basis and setting

N

400

Acquisition seed

s

619

Drag the orange timeline endpoint to change exposure. Drag the analyzer direction to change the displayed basis; tomography always acquires three separate preparations.

Thermal relaxation includes independent pure dephasing. Phase damping preserves population; depolarization shrinks every state component. Only the selected channel rates act. Temperature and gap act in the thermal channel.

Markov two-level channels with ideal preparation and readout. North is the ground state. The thermal bath follows a declared two-level Gibbs population. State reconstruction uses radial projection, not maximum likelihood; propagated Wilson bounds are approximate and exclude preparation, detector and model errors. Phase damping can conserve population, while zero-temperature relaxation can restore purity.

Explore the evidence · counts, fits and raw data

Predict which components change. Acquire three bases for the current state, then a time scan. Colored points and 95% Wilson bars come from separate preparation ensembles; dashed curves are model references.

No acquired record yet.

Inference from acquired counts

\text{—}

Current-state reconstruction

\widehat\rho=\text{—}

\text{—}

Current counts and 95% intervals

Basis	Positive / total	95% Wilson interval

Model density matrix · not reconstructed data

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

Latest raw events · full record in CSV

Event / run	Protocol / basis	Duration (ms)	Outcome

Scientific basis: IBM Quantum · channels · IBM Quantum · noise models

Start with the essentials

Focus question: Can a noisy process make an initially pure state mixed, then pure again?
One-sentence intuition: Cold relaxation ends in a pure ground state. Pure phase damping keeps population, and depolarization approaches the maximally mixed state.

Core mathematical model

Thermal relaxation with pure dephasing

r_z(t)=r_{z,\mathrm{eq}}+[r_z(0)-r_{z,\mathrm{eq}}]e^{-\Gamma_1t},\quad r_{x,y}(t)=r_{x,y}(0)e^{-(\Gamma_1/2+\Gamma_\varphi)t}

North is ground. Rates are nonnegative inverse ms; the transverse decay rate includes half the longitudinal rate.

Two-level Gibbs bath

q=\frac{1}{1+e^{hf_{01}/(k_BT_b)}},\quad r_{z,\mathrm{eq}}=1-2q

The bath fraction uses the displayed GHz gap and Kelvin temperature. Zero temperature gives zero excited fraction; this is an effective bath, not a specific device.

Reconstruct from counts

\widetilde r_j=2\frac{n_{j,+}}{N_j}-1,\quad \widehat{\boldsymbol r}=\frac{\widetilde{\boldsymbol r}}{\max(1,|\widetilde{\boldsymbol r}|)},\quad \widehat\rho=\frac{I+\widehat{\boldsymbol r}\cdot\boldsymbol\sigma}{2}

Each basis uses a separately prepared ensemble. Radial Euclidean projection enforces a physical Bloch ball; it is not maximum likelihood and can introduce finite-sample bias.

Purity and entropy

\mathcal P=\frac{1+|\boldsymbol r|^2}{2},\quad S=-\sum_{\pm}\lambda_\pm\log_2\lambda_\pm,\quad\lambda_\pm=\frac{1\pm|\boldsymbol r|}{2}

Propagated approximate joint Wilson component bounds give conservative purity and entropy ranges. They exclude preparation and measurement systematics.

Common difficulties

Noise need not monotonically reduce purity

Typical misconceptionEvery channel always decreases purity.

Better mental modelAmplitude damping is non-unital and can prepare a pure ground state. Depolarization and pure dephasing are different channels.

One qubit is not measured three times

Typical misconceptionThe three records are successive measurements of one specimen.

Better mental modelEach recorded result comes from a fresh preparation. Three ensemble averages estimate complementary components.

A fitted state is not a certified device

Typical misconceptionA physical projected estimate proves the detector and channel are correct.

Better mental modelProjection is a statistical constraint. The model assumes ideal preparation and readout; real device calibration and process tomography are outside this experiment.

Run the experiment

01
Acquire complementary information
Choose Thermal relaxation and acquire three bases. Open the reconstructed density matrix and count table.
What to observe: The green reconstructed state comes only from counts. The purple state and dashed curves remain separate model references.
02
Follow energy to a cold bath
Choose Cold excited state and acquire a time scan. Move exposure from zero through the middle to four milliseconds.
What to observe: Longitudinal population changes; purity falls then recovers. The acquisition fit reports weak identification when the input lacks transverse coherence.
03
Lose phase without energy
Choose Phase loss only and acquire a time scan.
What to observe: The longitudinal probability stays fixed while transverse probabilities approach half. Entropy increases without population relaxation.
04
Challenge reconstruction
Choose No noise, reduce shots, acquire three bases, then increase shots. Try an unpolarized input.
What to observe: Sampling may put the raw Bloch vector outside the ball. Projection restores positivity; uncertainty stays visible. An unpolarized input cannot identify depolarization rate.