Operational Entanglement Families of Symmetric Mixed N-Qubit States

We recall that a set 𝒜 \mathcal{A} of states is said open if any of its elements can be surrounded by an open ball fully included in 𝒜 \mathcal{A} . A set is closed if its complement is open. An open ball of radius r r around a given state | ψ ⟩ |\psi\rangle is the set of states | ϕ ⟩ |\phi\rangle such that the distance d ⁡ ( ϕ , ψ ) d(\phi,\psi) is strictly smaller than r r . The distance between two states | ϕ ⟩ |\phi\rangle and | ψ ⟩ |\psi\rangle is defined from the standard metric in Hilbert spaces as d ⁡ ( ϕ , ψ ) = ‖ ϕ − ψ ‖ d(\phi,\psi)=\|\phi-\psi\| . A state | ψ ⟩ |\psi\rangle is said to lie at the boundary of a set 𝒜 \mathcal{A} if any open ball surrounding the state | ψ ⟩ |\psi\rangle contains at least a state in 𝒜 \mathcal{A} and a state out of 𝒜 \mathcal{A} . The closure 𝒜 ¯ \overline{\mathcal{A}} of a set 𝒜 \mathcal{A} is the union of the set with its boundary. This is always a closed set. In finite dimension, all these standard concepts of topology apply similarly in the real Hilbert space of hermitian operators acting on the state space. In this operator space containing the mixed state density operators the distance between two hermitian operators A ^ \hat{A} and B ^ \hat{B} is defined as d ⁡ ( A ^ , B ^ ) = ‖ A ^ − B ^ ‖ op d(\hat{A},\hat{B})=\|\hat{A}-\hat{B}\|_{\mathrm{op}} , with ‖ A ^ ‖ op = [ Tr ⁡ ( A ^ 2 ) ] 1 / 2 \|\hat{A}\|_{\mathrm{op}}=[\mathrm{Tr}(\hat{A}^{2})]^{1/2}

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In finite dimension, a compact set of states is a closed and bounded set. The closeness condition is actually sufficient since Tr ⁡ ( ρ ^ 2 ) ⩽ 1 \mathrm{Tr}(\hat{\rho}^{2})\leqslant 1 for all states ρ ^ \hat{\rho} . A set 𝒜 \mathcal{A} of mixed states is convex iff p ​ ρ ^ + ( 1 − p ) ​ ρ ^ ′ ∈ 𝒜 p\hat{\rho}+(1-p)\hat{\rho}^{\prime}\in\mathcal{A} for all ρ ^ , ρ ^ ′ ∈ 𝒜 \hat{\rho},\hat{\rho}^{\prime}\in\mathcal{A} and p ∈ [ 0 , 1 ] p\in[0,1]

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