Entropy of entanglement

The entropy of entanglement (or entanglement entropy) is a measure of the degree of quantum entanglement between two subsystems constituting a two-part composite quantum system. Given a pure bipartite quantum state of the composite system, it is possible to obtain a reduced density matrix describing knowledge of the state of a subsystem. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for any of the subsystems. If it is non-zero, it indicates the two subsystems are entangled.

Mathematically, if a state describing two subsystems A and B ${\displaystyle |\Psi _{AB}\rangle =|\phi _{A}\rangle \otimes |\phi _{B}\rangle }$ is a product state, then the reduced density matrix ${\displaystyle \rho _{A}=\operatorname {Tr} _{B}|\Psi _{AB}\rangle \langle \Psi _{AB}|=|\phi _{A}\rangle \langle \phi _{A}|}$ is a pure state. Thus, the entropy of the state is zero; similarly, the density matrix of B would also have zero entropy. If the entropy of the reduced density matrix is nonzero, the reduced density matrix is a mixed state, which indicates that the subsystems A and B are entangled.

Entanglement entropy was first proposed by Sorkin as a source for black hole entropy, and remains a candidate. It is thought to have connections to gravity, and the possibility of induced gravity, following the work of Jacobson, and ideas of Sakharov.