Variations and measurements.
Change the parameters to observe how the representations vary and compare the resulting values with their defining relations; each experiment’s notes describe the limits of its interpretation.
State of a qubit
Changing θ varies the probabilities of a measurement in the computational basis, while φ describes the relative phase, which can be distinguished in other measurement bases; the vector represents a pure state in a projection of the sphere.
Angles are expressed in degrees; the vector represents the prepared state and the displayed probabilities refer to the {|0⟩, |1⟩} basis.
|ψ⟩ = cos(θ/2)|0⟩ + eⁱᵠ sin(θ/2)|1⟩; P(0) = cos²(θ/2)
Model, assumptions and references ↗Superposition and phase
Compare the two contributions with their sum while changing phase and amplitude, observing how peaks and troughs reinforce or cancel each other over the same spatial interval.
The diagram represents two harmonic waves of equal frequency at a fixed instant; A is relative to the first wave and x is measured in radians.
u₁(x) = cos x; u₂(x) = A cos(x + φ); u(x) = u₁(x) + u₂(x)
Model, assumptions and references ↗Probability and threshold
The surface retains the same function while you move the threshold, allowing you to compare a point’s changing assigned class with its unchanged probability value.
An analytic example on x,y ∈ [−2,2], independent of experimental data; the grid samples the function and the plane indicates threshold τ.
p(x,y) = 1 / (1 + exp[−(1.2x − 0.5y + 0.45xy)]); class 1 ⇔ p ≥ τ
Model, assumptions and references ↗