QUANTUM / EDUCATIONAL MODEL
One qubit. Two outcomes.
Explore the probabilities of measuring a single qubit in X, Y, and Z measurement bases across the Bloch sphere. This is an interactive classical simulation—not quantum hardware and not a two-qubit entanglement model.
What you can test
Change polar angle θ and relative phase φ. In the Z basis compare P(0) = cos²(θ/2) with P(1) = sin²(θ/2); switch to X and Y to see the phase affect their probabilities. Explore θ = 0°, 90° and 180° and repeat measurements to see sampling variability.
The model represents a general ideal pure qubit using θ and φ. It does not simulate a device-specific noise model. Use the separate entanglement visualization on the homepage for a different two-qubit concept.
AUTHORED STUDY COMPANION / 3.3
Track a noisy state with a density operator
Prerequisites. Complex vectors, outer products, trace and eigenvalues.
A density operator represents both pure states and probabilistic mixtures. It must be Hermitian, positive semidefinite and trace one. For |+⟩ the matrix has four entries equal to 1/2; its purity Tr(ρ²) is one. A classical mixture of |0⟩ and |1⟩ instead has zero off-diagonal coherence and purity 1/2.
Amplitude damping uses K₀=diag(1,√(1−γ)) and K₁=[[0,√γ],[0,0]]. The channel sums KᵢρKᵢ†. For the initial |+⟩, the resulting matrix is [[(1+γ)/2,√(1−γ)/2],[√(1−γ)/2,(1−γ)/2]]. This is a specific relaxation channel, not a universal hardware noise model. At γ=1 it resets the state to pure |0⟩, illustrating that noise does not always monotonically reduce purity.
The notebook checks trace, eigenvalues and purity. A negative eigenvalue beyond numerical tolerance is a physical-validity error, not merely a strange probability plot. Compare channel composition with replacing amplitudes by classical probabilities; discarding off-diagonal terms removes measurable phase information.
| Element | Model | Interpretation |
|---|---|---|
| State | ρ, trace one and nonnegative eigenvalues | A valid probability operator |
| Channel | ΣᵢKᵢρKᵢ† | Trace-preserving map |
| Checks | Trace, spectrum, purity | Distinct invariants and diagnostics |
Predict, run, compare
At γ=0.4, P(0)=0.7, coherence=√0.6/2, and purity=0.88. At γ=0 the initial state is unchanged; at γ=1, P(0)=1 and purity returns to one.
Download executable notebooks and source Inspect source ↗
Open density.ipynb in the package. Run cells in order, compare with the reference, then change the stated input. Restart the kernel to reset.
Computed reference result — local CPU fixture
{
"density": [
[
0.6999999999999998,
0.3872983346207416
],
[
0.3872983346207416,
0.29999999999999993
]
],
"trace": 0.9999999999999998,
"eigenvalues": [
0.06411010564593264,
0.9358898943540672
],
"purity": 0.8799999999999996,
"z_zero": 0.6999999999999998,
"coherence": 0.3872983346207416
}This is the recorded result of the included code under the package environment. It is not a live service or device measurement.
Extend the experiment
Derive the matrix above, then compare amplitude damping with dephasing at the same final coherence. Explain why identical coherence does not require identical populations.
Check your reasoning: Which property is required of every density operator?
- Trace zero — Trace is one for a normalized state.
- Nonnegative eigenvalues and trace one — Correct. These conditions, with Hermiticity, make probabilities valid.
- Only real entries — Valid quantum states can contain complex coherence.
This is an explained self-check, not a certification of mastery.
Claim-specific sources
- IBM Quantum Learning — Fundamentals of Quantum Algorithms
- Preskill — Quantum Computation lecture notes
Reviewed 2026-10-02: this worked example and its explicit assumptions. Source links and notebook results do not establish production or hardware validation.
Problem, implementation and evidence
Computational-basis probabilities alone hide relative phase.
The existing general pure-state lab varies both θ and φ and measures X, Y and Z. New companions check probabilities and amplitude-damping density operators.
What the evidence establishes
Executed ideal-state calculations and classical browser simulation; not a quantum device or a two-qubit hardware experiment.
Reproduce or extend
Use θ=90°, φ=90° to distinguish Y certainty from Z randomness, then compare a pure state with a mixed state.
Open the related experiment or review surface →