Squashed entanglement

Squashed entanglement, also called CMI entanglement (CMI can be pronounced 'see me'), 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 ( A , B ) {displaystyle (A,B)} composed of two subsystems A {displaystyle A} and B {displaystyle B} , then the CMI entanglement E C M I {displaystyle E_{CMI}} of system ( A , B ) {displaystyle (A,B)} is defined by E C M I ( ϱ A , B ) = 1 2 min ϱ A , B , Λ ∈ K S ( A : B | Λ ) {displaystyle E_{CMI}(varrho _{A,B})={frac {1}{2}}min _{varrho _{A,B,Lambda }in K}S(A:B|Lambda )} ,    Eq.(1) H ( A : B ) = H ( A ) + H ( B ) − H ( A , B ) {displaystyle H(A:B)=H(A)+H(B)-H(A,B),} .    Eq.(2) H ( A : B | Λ ) = H ( A | Λ ) + H ( B | Λ ) − H ( A , B | Λ ) = H ( A , Λ ) + H ( B , Λ ) − H ( Λ ) − H ( A , B , Λ ) {displaystyle {egin{matrix}H(A:B|Lambda )&=&H(A|Lambda )+H(B|Lambda )-H(A,B|Lambda )\&=&H(A,Lambda )+H(B,Lambda )-H(Lambda )-H(A,B,Lambda )end{matrix}}} .    Eq.(3) S ( A : B ) = S ( ϱ A ) + S ( ϱ B ) − S ( ϱ A , B ) {displaystyle S(A:B)=S(varrho _{A})+S(varrho _{B})-S(varrho _{A,B}),} ,    Eq.(4) S ( A : B | Λ ) = S ( ϱ A , Λ ) + S ( ϱ B , Λ ) − S ( ϱ Λ ) − S ( ϱ A , B , Λ ) {displaystyle S(A:B|Lambda )=S(varrho _{A,Lambda })+S(varrho _{B,Lambda })-S(varrho _{Lambda })-S(varrho _{A,B,Lambda }),} .    Eq.(5) P ( a , b , λ ) = P ( a | λ ) P ( b | λ ) P ( λ ) {displaystyle P(a,b,lambda )=P(a|lambda )P(b|lambda )P(lambda ),} ,    Eq.(6) E C M I ( P A , B ) = min P A , B , Λ ∈ K H ( A : B | Λ ) {displaystyle E_{CMI}(P_{A,B})=min _{P_{A,B,Lambda }in K}H(A:B|Lambda )} ,    Eq.(7) ϱ A , B , Λ = ∑ λ ϱ A λ ϱ B λ w λ | λ ⟩ ⟨ λ | {displaystyle varrho _{A,B,Lambda }=sum _{lambda }varrho _{A}^{lambda }varrho _{B}^{lambda }w_{lambda }|lambda angle langle lambda |,}     Eq.(8) Squashed entanglement, also called CMI entanglement (CMI can be pronounced 'see me'), 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 ( A , B ) {displaystyle (A,B)} composed of two subsystems A {displaystyle A} and B {displaystyle B} , then the CMI entanglement E C M I {displaystyle E_{CMI}} of system ( A , B ) {displaystyle (A,B)} 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 ( A , B , Λ ) {displaystyle (A,B,Lambda )} such that ϱ A , B = t r Λ ( ϱ A , B , Λ ) {displaystyle varrho _{A,B}=tr_{Lambda }(varrho _{A,B,Lambda })} . Thus, CMI entanglement is defined as an extremum of a functional S ( A : B | Λ ) {displaystyle S(A:B|Lambda )} of ϱ A , B , Λ {displaystyle varrho _{A,B,Lambda }} . We define S ( A : B | Λ ) {displaystyle S(A:B|Lambda )} , the quantum Conditional Mutual Information (CMI), below. A more general version of Eq.(1) replaces the ``min' (minimum) in Eq.(1) by an ``inf' (infimum). When ϱ A , B {displaystyle varrho _{A,B}} is a pure state, E C M I ( ϱ A , B ) = S ( ϱ A ) = S ( ϱ B ) {displaystyle E_{CMI}(varrho _{A,B})=S(varrho _{A})=S(varrho _{B})} , in agreement with the definition of entanglement of formation for pure states. Here S ( ϱ ) {displaystyle S(varrho )} is the Von Neumann entropy of density matrix ϱ {displaystyle varrho } . CMI entanglement has its roots in classical (non-quantum) information theory, as we explain next. Given any two random variables A , B {displaystyle A,B} , classical information theory defines the mutual information, a measure of correlations, as For three random variables A , B , Λ {displaystyle A,B,Lambda } , it defines the CMI as It can be shown that H ( A : B | Λ ) ≥ 0 {displaystyle H(A:B|Lambda )geq 0} . Now suppose ϱ A , B , Λ {displaystyle varrho _{A,B,Lambda }} is the density matrix for a tripartite system ( A , B , Λ ) {displaystyle (A,B,Lambda )} . We will represent the partial trace of ϱ A , B , Λ {displaystyle varrho _{A,B,Lambda }} with respect to one or two of its subsystems by ϱ A , B , Λ {displaystyle varrho _{A,B,Lambda }} with the symbol for the traced system erased. For example, ϱ A , B = t r a c e Λ ( ϱ A , B , Λ ) {displaystyle varrho _{A,B}=trace_{Lambda }(varrho _{A,B,Lambda })} . One can define a quantum analogue of Eq.(2) by