To see why this is the case, we start observing that there is a unique decomposition ρ ρ − − σ σ = P − − Q with P , Q ≥ ≥ 0 positive semidefinite matrices with orthogonal support. With these operators we can write concisely | ρ ρ − − σ σ | = P + Q . Furthermore Tr ⁡ ⁡ ( Π Π i P ) , Tr ⁡ ⁡ ( Π Π i Q ) ≥ ≥ 0 , and thus | Tr ⁡ ⁡ ( Π Π i P ) − − Tr ⁡ ⁡ ( Π Π i Q ) ) | ≤ ≤ Tr ⁡ ⁡ ( Π Π i P ) + Tr ⁡ ⁡ ( Π Π i Q ) ) . We thus have ∑ ∑ i | Tr ⁡ ⁡ ( Π Π i ( ρ ρ − − σ σ ) ) | = ∑ ∑ i | Tr ⁡ ⁡ ( Π Π i ( P − − Q ) ) | ≤ ≤ ∑ ∑ i Tr ⁡ ⁡ ( Π Π i ( P + Q ) ) = Tr ⁡ ⁡ | ρ ρ − − σ σ | . This shows that max Π Π δ δ ( P Π Π , ρ ρ , P Π Π , σ σ ) ≤ ≤ T ( ρ ρ , σ σ ) , where P Π Π , ρ ρ denotes the classical probability distribution resulting from measuring ρ ρ with the POVM Π Π , ( P Π Π , ρ ρ ) i ≡ ≡ Tr ⁡ ⁡ ( Π Π i ρ ρ ) , and the maximum is performed over all POVMs Π Π ≡ ≡ { Π Π i } i .