Interactive visualizations — drag sliders and edit matrix entries to explore
Bloch sphere — orthogonal projection
Measuring in the Z-basis projects the state vector onto the Z-axis. The squared projection
length gives the Born probability. φ encodes relative phase but is invisible to Z-basis measurement.
θ — polar angle from |0⟩0°
φ — azimuthal angle0°
Z-basis outcome probabilities
P(|0⟩) = cos²(θ/2)1.000
P(|1⟩) = sin²(θ/2)0.000
|ψ⟩ = |0⟩
The red dot marks the projection of |ψ⟩ onto the Z-axis. φ affects only the relative phase —
invisible to Z-basis measurement. Try θ = 90° for the equator: both outcomes become equally likely.
Projective (von Neumann) measurement
A projective measurement is a rotatable orthonormal basis {|m₀⟩, |m₁⟩}. The outcome
probability equals the squared projection of |ψ⟩ onto that axis. The two axes are always perpendicular — this is
the fundamental constraint projective measurement cannot break.
State angle α60°
Basis angle β — rotate measurement0°
Outcome probabilities
P(m₀) = cos²(α − β)0.750
P(m₁) = sin²(α − β)0.250
Post-measurement collapse → |m₀⟩
Dashed lines drop perpendiculars from |ψ⟩ to each axis. Rotating β to align with |ψ⟩ gives
P(m₀) → 1. The basis axes are always 90° apart — the defining constraint that the POVM tab relaxes.
POVM — trine measurement
A POVM replaces orthogonal projectors with positive operators {Eₖ} satisfying only ΣEₖ = I.
The trine has 3 non-orthogonal outcome directions at 120° apart — more outcomes than dimensions, impossible
projectively.
PROJECTIVE (2 outcomes)
TRINE POVM (3 outcomes)
State angle α45°
Projective outcomes
P(Π₀)—
P(Π₁)—
POVM outcomes
P(E₀)—
P(E₁)—
P(E₂)—
Three outcomes from a 2-level system: no clean post-measurement
eigenstate. P(E₀)+P(E₁)+P(E₂) = 1 always, but notice none ever saturates to 1 — the non-orthogonality is the
cost.
Unitary evolution on the Bloch sphere
Enter a 2×2 unitary U. The dashed arrow is the initial state |ψ⟩; the solid arrow is U|ψ⟩.
The arc traces the geodesic rotation. Select a preset or type custom complex entries.