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Holevo Bound: Classical Information from a Quantum Alphabet

Interactive Bloch-sphere simulator of Holevo's theorem: place N quantum states around a cone, watch the Holevo bound on extractable classical information shrink toward the mixed-state entropy, and see how much of it any single projective measurement actually recovers.

Quantum Physics3DAdvanced60 FPS📱 Mobile-adapted⇄ 2D version

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 simulator builds a small "quantum alphabet" of 2–4 equiprobable qubit states arranged around a cone on the Bloch sphere, computes χ exactly from the mixed-state entropy of the ensemble, and lets you choose Bob's real projective measurement axis to see the accessible information I(A:B) he actually recovers — always at or below χ, with the gap making visible exactly how much classical information a single measurement leaves on the table.

⚙ Under the hood

Interactive Bloch-sphere simulator of Holevo's theorem: arrange 2-4 quantum states on a cone, compute the exact Holevo bound on extractable classical information, and see how much of it a chosen projective measurement actually recovers.

quantum informationholevo boundqubitbloch sphereentropymeasurement

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install