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 SS be the von Neumann entropy and In the maximal mixed state. We show that for quantum states ρ,σρ,σ, the state ρρ is unitarily equivalent to σσ if and only if S(λρ+μIn)=S(λσ+μIn)for allλ,μ∈[0,1]withλ+μ=1.