Q048 · How an environment changes a qubit

Decoherence Channels & Information Loss

Does losing phase also mean losing energy? No. Pure phase damping destroys coherence but leaves the populations, and so the energy, unchanged. Relaxation into a cold bath does give energy away and ends in the ground state. Measuring along all three axes tells the two apart.

Watch the level populations and the phase alignment as two separate things. Predict what phase damping changes, then compare it with relaxation into a cold bath. Check the final state with three independent sets of simulated counts, and move on to state reconstruction and decay fits when you are ready.

Read the comparison and its values

Ground population $50\%$

Phase alignment $100\%$

The starting state is ready. Move time forward to see what changes.

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 $\text{—}$

Change one thing and watch what happens

Phase loss or energy loss?

Preparation and model assumptions

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

Every loop starts a new preparation of the same state. Dashed marks show that preparation; the colored diagram shows the selected elapsed time. Replaying adds no detector events.

1 · Choose and predict

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

2 · Play or drag through time

Observation time

0\,\mathrm{ms}

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

3 · Test with simulated counts

How counts are acquired

These actions acquire fresh preparations at the selected endpoint. Raw simulated counts and uncertainty appear below; the continuous preview stays separate.

Go further · parameters and state sphere

The simple comparison above always resets to its own starting state. Here you can change the whole experiment. The state sphere is a mathematical picture; it does not show a spin direction in space.

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 end of the timeline to change how long the noise acts. Drag the analyzer direction to change which basis is displayed; tomography still prepares and measures each of the three bases separately.

Thermal relaxation carries its own separate pure-dephasing rate. Phase damping leaves the populations unchanged; depolarization shrinks every component of the state. Only the rates of the selected channel take effect, and temperature and energy gap matter only in the thermal channel.

The channels are Markov processes acting on a two-level system, with ideal preparation and readout. The north pole of the Bloch sphere is the ground state. The thermal bath sets a two-level Gibbs population. State reconstruction uses radial projection, not maximum likelihood. The propagated Wilson bounds are approximate and leave out preparation, detector and model errors. Phase damping can leave the populations unchanged, and relaxation at zero temperature 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: Relaxation into a cold bath ends in the pure ground state. Pure phase damping leaves the populations alone, and depolarization drives the state toward 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}

The north pole is the ground state. Rates are non-negative and measured in inverse milliseconds. The transverse decay rate includes half of 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 comes from the displayed gap in GHz and temperature in kelvin. At zero temperature the excited fraction is zero. This is an effective bath, not a specific device.

Reconstruction 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 its own separately prepared ensemble. Radial Euclidean projection pulls the estimate back inside the Bloch ball so that the state is physical. It is not maximum likelihood, and with finite samples it can introduce 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}

Approximate joint Wilson bounds on the components are propagated into conservative ranges for purity and entropy. These ranges leave out systematic errors in preparation and measurement.

Common difficulties

Noise does not always lower purity

Typical misconceptionEvery channel always lowers the purity.

Better mental modelAmplitude damping is non-unital, so it can leave the qubit in the pure ground state. Depolarization and pure dephasing are different channels.

One qubit is not measured three times

Typical misconceptionThe three records are three measurements made one after another on the same qubit.

Better mental modelEvery recorded result comes from a fresh preparation. Averages over three separate ensembles estimate complementary components of the state.

A physical estimate does not certify the device

Typical misconceptionOnce the projected estimate is physical, the detector and the channel must be right.

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

Run the experiment

01
Measure in all three bases
Choose Thermal relaxation and acquire all three bases. Open the reconstructed density matrix and the count table.
What to observe: The green reconstructed state comes only from the counts. The purple state and the dashed curves are separate model references.
02
Follow the energy into a cold bath
Choose Cold excited state and acquire a time scan. Move the exposure from zero, through the middle of the range, to four milliseconds.
What to observe: The longitudinal population changes, and the purity first falls, then recovers. When the input has no transverse coherence, the fit flags its result as only weakly identified.
03
Lose phase without energy
Choose Phase loss only and acquire a time scan.
What to observe: The longitudinal probability stays fixed while the transverse probabilities approach one half. The entropy rises even though the populations do not relax.
04
Test the reconstruction
Choose No noise, reduce the number of shots and acquire three bases, then increase the shots. Also try an unpolarized input.
What to observe: Sampling noise can put the raw Bloch vector outside the ball. Projection makes the state positive again, but the uncertainty stays visible. An unpolarized input cannot reveal the depolarization rate.

Decoherence Channels & Information Loss

Use the records to identify the noise.

Inference from acquired counts

Current-state reconstruction

Reconstruct information after a channel

Core mathematical model

Thermal relaxation with pure dephasing

Two-level Gibbs bath

Reconstruction from counts

Purity and entropy

Common difficulties

Noise does not always lower purity

One qubit is not measured three times

A physical estimate does not certify the device

Run the experiment

Measure in all three bases

Follow the energy into a cold bath

Lose phase without energy

Test the reconstruction

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