As noted by Lindblad and Uhlmann, if, in equation (1), one takes K = 1 and r = 1 − − p , A = ρ ρ and B = σ σ and differentiates in p at p = 0 , one obtains the joint convexity of relative entropy : i.e., if ρ ρ = ∑ ∑ k λ λ k ρ ρ k , and σ σ = ∑ ∑ k λ λ k σ σ k , then