Entanglement and nonlocality are inequivalent for any number of particles

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For a linear map Λ : ℬ ⁡ ( ℋ ) → ℬ ⁡ ( 𝒦 ) \Lambda:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K}) we define the dual map as a linear map Λ † : ℬ ⁡ ( 𝒦 ) → ℬ ⁡ ( ℋ ) \Lambda^{\dagger}:\mathcal{B}(\mathcal{K})\to\mathcal{B}(\mathcal{H}) that satisfies Tr ⁡ [ X ​ Λ ​ ( Y ) ] = Tr ⁡ [ Λ † ​ ( X ) ​ Y ] \mathrm{Tr}[X\Lambda(Y)]=\mathrm{Tr}[\Lambda^{\dagger}(X)Y] for any X ∈ ℬ ⁡ ( 𝒦 ) X\in\mathcal{B}(\mathcal{K}) and Y ∈ ℬ ⁡ ( ℋ ) Y\in\mathcal{B}(\mathcal{H}) . Recall that if Λ \Lambda is positive (in particular completely positive), its dual Λ † \Lambda^{\dagger} is also positive. Moreover, if Λ \Lambda is trace-preserving, i.e., Tr ⁡ [ Λ ⁡ ( X ) ] = Tr ​ X \mathrm{Tr}[\Lambda(X)]=\mathrm{Tr}X for any X X , the dual map Λ † \Lambda^{\dagger} is unital, i.e., Λ † ​ ( 𝟙 𝒦 ) = 𝟙 ℋ \Lambda^{\dagger}(\mathbbm{1}_{\mathcal{K}})=\mathbbm{1}_{\mathcal{H}} with 𝟙 𝒳 \mathbbm{1}_{\mathcal{X}} being the identity operator acting on 𝒳 \mathcal{X}

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