A von Neumann entropy condition of unitary equivalence of quantum states

Two unitarily equivalent quantum states have the same von Neumann entropy, but not vice versa. In the note, we give a sufficient and necessary condition of unitary equivalence of quantum states associated with the von Neumann entropy. Let S be the von Neumann entropy and I n the maximal mixed state. We show that for quantum states ρ , σ , the state ρ is unitarily equivalent to σ if and only if S ( λ ρ + μ I n ) = S ( λ σ + μ I n ) for all λ , μ ∈ [ 0 , 1 ] with λ + μ = 1 .