Let X be an operator acting on an N -dimensional Hilbert space H N . Let Λ Λ ( X ) denote its numerical range, i.e. the set of all λ λ such that there exists a normalized state | ψ ψ ⟩ ⟩ ∈ ∈ H N , | | ψ ψ | | = 1 , which satisfies ⟨ ⟨ ψ ψ | X | ψ ψ ⟩ ⟩ = λ λ .