Holevo Bound — Quantum Information

The Holevo bound χ = S(ρ̄) − Σpᵢ S(ρᵢ) limits how much classical information can be extracted from quantum states. It equals 1 qubit for orthogonal states, less for mixed/overlapping states.

State Angle θ (°) 90°

Prior p₁ 0.50

Noise ε 0.00

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Holevo χ (bits)

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Avg entropy Σpᵢ S(ρᵢ)

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Ensemble entropy S(ρ̄)

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|⟨ψ₁|ψ₂⟩|

χ = S(ρ̄) − Σpᵢ S(ρᵢ) ≤ 1 bit. For pure states, S(ρᵢ)=0, so χ = S(ρ̄). Orthogonal states (θ=180°) maximize χ=1. Identical states give χ=0. Noise reduces χ further.