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 Incoming links: Weak monotonicity of quantum entropy

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 66 / 1 / vi / Solution Created 2026-10-03 Updated 2026-10-06

Take a purification of a density operator

∣Ψ ⟩_{A BC D}

of

ρ_{A BC}

. Apply Strong subadditivity of Von Neumann entropy to the reduced state on

A C D

, with

A

the conditioning system:

S (A C) + S (A D) \geq S (A) + S (A C D) .

Complementary subsystems of a pure state have the same Von Neumann entropy, by the Schmidt decomposition. Therefore

S (A D) = S (BC)

and

S (A C D) = S (B)

. Substitution gives

S (A) + S (B) \leq S (A C) + S (BC) .

This is weak monotonicity of quantum entropy. The purification of a density operator is only a proof device; no purity assumption is imposed on

ρ_{A BC}

.

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