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, i.e. the subsystem is in a mixed state, it indicates the two subsystems are entangled.
More mathematically; if a state describing two subsystems "A" and "B" formula_1is a separable state, then the reduced density matrix formula_2is a pure state. Thus, the entropy of the state is zero. Similarly, the density matrix of "B" 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.
Bipartite entanglement entropy.
Suppose that a quantum system consists of formula_3particles. A bipartition of the system is a partition which divides the system into two parts formula_4 and formula_5, containing formula_6 and formula_7 particles respectively with formula_8. Bipartite entanglement entropy is defined with respect to this bipartition.
Von Neumann entanglement entropy.
The bipartite Von Neumann entanglement entropy formula_9 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 formula_10, it is given by:
where formula_12 and formula_13 are the reduced density matrices for each partition.
The entanglement entropy can be expressed using the singular values of the Schmidt decomposition of the state. Any pure state can be written as formula_14 where formula_15 and formula_16 are orthonormal states in subsystem formula_4 and subsystem formula_5 respectively. The entropy of entanglement is simply:
formula_19
This form of writing the entropy makes it explicitly clear that the entanglement entropy is the same regardless of whether one computes partial trace over the formula_4 or formula_5 subsystem.
Many entanglement measures reduce to the entropy of entanglement when evaluated on pure states. Among those are:
Some entanglement measures that do not reduce to the entropy of entanglement are:
Renyi entanglement entropies.
The Renyi entanglement entropies formula_22 are also defined in terms of the reduced density matrices, and a Renyi index formula_23. It is defined as the Rényi entropy of the reduced density matrices:
Note that in the limit formula_25, The Renyi entanglement entropy approaches the Von Neumann entanglement entropy.
Example with coupled harmonic oscillators.
Consider two coupled quantum harmonic oscillators, with positions formula_26 and formula_27, momenta formula_28 and formula_29, and system Hamiltonian
With formula_31, the system's pure ground state density matrix is formula_32, which in position basis is formula_33. Then 
formula_34
Since formula_35 happens to be precisely equal to the density matrix of a single quantum harmonic oscillator of frequency formula_36 at thermal equilibrium with temperature formula_37 ( such that formula_38 where formula_39 is the Boltzmann constant), the eigenvalues of formula_35 are formula_41 for nonnegative integers formula_42. The Von Neumann Entropy is thus
Similarly the Renyi entropy formula_44.
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.