Let X be an operator acting on the composite Hilbert space. We define the product numerical range Λ Λ ⊗ ⊗ ( X ) of X , with respect to the tensor product structure of H N , as Λ Λ ⊗ ⊗ ( X ) = { ⟨ ⟨ ψ ψ A ⊗ ⊗ ψ ψ B | X | ψ ψ A ⊗ ⊗ ψ ψ B ⟩ ⟩ : | ψ ψ A ⟩ ⟩ ∈ ∈ H K , | ψ ψ B ⟩ ⟩ ∈ ∈ H M } , where | ψ ψ A ⟩ ⟩ ∈ ∈ H K and | ψ ψ B ⟩ ⟩ ∈ ∈ H M are normalized.