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Entropy of entanglement
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<p> The entropy of entanglement (or entanglement entropy) is a <a href="/facts/Entanglement_monotone/JBhby2Cm">measure</a> of the degree of <a href="/facts/Quantum_entanglement/afzslwN0">quantum entanglement</a> between two subsystems constituting a two-part composite <a href="/facts/Quantum_system/BRc3Mzgr">quantum system</a>. Given a <a href="/facts/Pure_state/7jbxd7Dx">pure</a> bipartite <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> of the composite system, it is possible to obtain a <a href="/facts/Reduced_density_matrix/afzslwN0">reduced density matrix</a> describing knowledge of the state of a subsystem. The entropy of entanglement is the <a href="/facts/Von_Neumann_entropy/nzeJKowg">Von Neumann entropy</a> of the reduced density matrix for any of the subsystems. If it is non-zero, it indicates the two subsystems are entangled. </p><p>More mathematically; if a state describing two subsystems <i>A</i> and <i>B</i> | Ψ A B ⟩ = | ϕ A ⟩ | ϕ B ⟩ {\displaystyle |\Psi _{AB}\rangle =|\phi _{A}\rangle |\phi _{B}\rangle } is a separable state, then the reduced density matrix ρ A = Tr B | Ψ A B ⟩ ⟨ Ψ A B | = | ϕ A ⟩ ⟨ ϕ A | {\displaystyle \rho _{A}=\operatorname {Tr} _{B}|\Psi _{AB}\rangle \langle \Psi _{AB}|=|\phi _{A}\rangle \langle \phi _{A}|} is a <a href="/facts/Pure_state/7jbxd7Dx">pure state</a>. Thus, the entropy of the state is zero. Similarly, the density matrix of <i>B</i> would also have 0 entropy. A reduced density matrix having a non-zero entropy is therefore a signal of the existence of entanglement in the system. </p><p>Entanglement entropy was first proposed as a source for <a href="/facts/Black_hole_thermodynamics/LghBnFWD">black hole entropy</a>, and remains a candidate. It is now thought to have connections to gravity, and the possibility it is <a href="/facts/Induced_gravity/r1yCsfEl">induced</a>, following the work of <a href="/facts/Ted_Jacobson/HCmcrMjm">Jacobson</a>, and the original ideas of <a href="/facts/Andrei_Sakharov/fryrhYEK">Andrei Sakharov</a>. </p>