Entropy of Entanglement

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Given a pure state, | \Psi \rangle , and a partition for the system, {A,B}, the entropy of entanglement is defined as,

E(\Psi) \equiv S(\rho_A)=S(\rho_B)

where the von Neumann entropy is S(ρ) = − Tr(ρlog2ρ) and \rho_A=\textrm{Tr}_{B}(|\Psi\rangle\langle\Psi|). Any entropy that results from performing a partial trace on the system must be a consequence of initial entanglement provided that the initial state is pure. For product states, |\Psi\rangle=|\Psi\rangle_A \otimes |\Psi\rangle_B, the entropy is zero since the single eigenvalue for each of the pure states ρA and ρB is one. The maximum entropy of entanglement given a partition with dimensions, \dim(A)=d_A and \dim(B)=d_B, with d_A\leq d_B, is log2(dA). A state that achieves this maximum is,

|\Psi\rangle = |0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B + \cdots + |d_A-1\rangle_A \otimes |d_A-1\rangle_B

The entropy of entanglement has the attractive feature that it is straightforward to compute; it requires only performing a partial trace, then computing eigenvalues of the result. The drawback of the entropy is that it only qualifies as an entanglement monotone for initially pure states.

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