Entropy of entanglement
The entropy of entanglement (or entanglement entropy) is an entanglement measure for a many-body quantum state. If a state is a separable state, then the reduced density matrix is a pure state, thus the entropy of the state is zero. Similar result holds for . The nonzero value of the entropy of the reduced density matrix therefore is a signal of the existence of entanglement.
Bipartite entanglement entropy
Suppose that a quantum system consist of particles. A bipartition of the system is a partition which divide the system into two parts and , containing and particles respectively with . Bipartite entanglement entropy is defined with respect to this bipartition.
Von Neumann entanglement entropy
The bipartite Von Neumann entanglement entropy is defined as the Von Neumann entropy of either of its reduced states, since they are of the same value (can be proved from Schmidt decomposition of the state with respect to the bipartition); the result is independent of which one we pick. That is, for a pure state , it is given by:
where and are the reduced density matrices for each partition.
Many entanglement measures reduce to the entropy of entanglement when evaluated on pure states. Among those are:
- Distillable entanglement
- Entanglement cost
- Entanglement of Formation
- Relative entropy of entanglement
- Squashed entanglement
Some entanglement measures that do not reduce to the entropy of entanglement are:
Renyi entanglement entropies
The Renyi entanglement entropies are also defined in terms of the reduced density matrices, and a Renyi index . It is defined as the Rényi entropy of the reduced density matrices:
Note that the limit , The Renyi entanglement entropy approaches the Von Neumann entanglement entropy.
Example with coupled harmonic oscillators
Consider two coupled quantum harmonic oscillators, with positions and , momenta and , and system Hamiltonian
With , the system's pure ground state density matrix is , which in position basis is . Then 
Since happens to be precisely equal to the density matrix of a single quantum harmonic oscillator of frequency at thermal equilibrium with temperature ( such that where is the Boltzmann constant) , the eigenvalues of are for nonnegative integers . The Von Neumann Entropy is thus
Similarly the Renyi entropy .
Area law of bipartite entanglement entropy
A quantum state satisfies an area law if the leading term of the entanglement entropy grows at most proportionally with the boundary between the two partitions. Area laws are remarkably common for ground states of local gapped quantum many-body systems. This has important applications, one such application being that it greatly reduces the complexity of quantum many-body systems. The density matrix renormalization group and matrix product states, for example, implicitly rely on such area laws. 
- Entropy and area Mark Srednicki Phys. Rev. Lett. 71, 666 – Published 2 August 1993 https://arxiv.org/pdf/hep-th/9303048.pdf
- Eisert, J.; Cramer, M.; Plenio, M. B. (February 2010). "Colloquium: Area laws for the entanglement entropy". Reviews of Modern Physics. 82 (1): 277–306. arXiv:0808.3773. Bibcode:2010RvMP...82..277E. doi:10.1103/RevModPhys.82.277.
- Janzing, Dominik (2009). "Entropy of Entanglement". In Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.). Compendium of Quantum Physics. Springer. pp. 205–209. doi:10.1007/978-3-540-70626-7_66. ISBN 978-3-540-70626-7.
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