Holevo Bound 2D: Polar Quantum-Alphabet Information Diagram

Quantum information theory's central limit on communication is Holevo's theorem: however cleverly Alice encodes classical symbols into a quantum state, and however cleverly Bob measures it, the average information he can extract can never exceed the Holevo bound χ. This 2D simulator draws the Bloch sphere as a polar (top-down) disk: a small "quantum alphabet" of 2–4 equiprobable qubit states sits on a cone around the pole, χ is computed exactly from the mixed-state entropy of the ensemble, and Bob's real projective measurement axis — set by dragging directly on the disk — reveals the accessible information I(A:B) he actually recovers, always at or below χ. A live angle-sweep chart and a numerically-searched "best axis" button make visible a subtlety the original 3D version's math got backwards: a measurement through the pole recovers zero bits, not the most.