Concavity of Von Neumann entropy

For density operators

ρ_{1}, ρ_{2}

and

0 \leq p \leq 1

, Von Neumann entropy is concave:

S (p ρ_{1} + (1 - p) ρ_{2}) \geq pS (ρ_{1}) + (1 - p) S (ρ_{2}) .

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- Entropy bound for a binary mixture Concavity of Von Neumann entropy

Entropy bound for a binary mixture

The entropy of a binary mixture also satisfies

S (p ρ_{1} + (1 - p) ρ_{2}) \leq pS (ρ_{1}) + (1 - p) S (ρ_{2}) + H (p),

where

H

is binary entropy. This follows by applying the operator monotonicity of logarithm to

p ρ_{1} \leq p ρ_{1} + (1 - p) ρ_{2}

and its counterpart for

ρ_{2}

.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 4 / a / Solution

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