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Squashed entanglement, also called CMI entanglement , is an information theoretic measure of quantum entanglement for a bipartite quantum system. If ϱ A , B {\displaystyle \varrho _{A,B}} is the density matrix of a system {\displaystyle } composed of two subsystems A {\displaystyle A} and B {\displaystyle B} , then the CMI entanglement E C M I {\displaystyle E_{CMI}} of system {\displaystyle } is defined by
where K {\displaystyle K} is the set of all density matrices ϱ A , B , Λ {\displaystyle \varrho _{A,B,\Lambda }} for a tripartite system {\displaystyle } such that ϱ A , B = t r Λ {\displaystyle \varrho _{A,B}=tr_{\Lambda }}. Thus, CMI entanglement is defined as an extremum of a functional S {\displaystyle S} of ϱ A , B , Λ {\displaystyle \varrho _{A,B,\Lambda }}. We define S {\displaystyle S} , the quantum Conditional Mutual Information , below. A more general version of Eq. replaces the ``min" in Eq. by an ``inf". When ϱ A , B {\displaystyle \varrho _{A,B}} is a pure state, E C M I = S = S {\displaystyle E_{CMI}=S=S} , in agreement with the definition of entanglement of formation for pure states. Here S {\displaystyle S} is the Von Neumann entropy of density matrix ϱ {\displaystyle \varrho }.
E C M I = 1 2 min ϱ A , B , Λ ∈ K S {\displaystyle E_{CMI}={\frac {1}{2}}\min _{\varrho _{A,B,\Lambda }\in K}S} ,