Entropy of entanglement


The entropy of entanglement is a measure of the degree of quantum entanglement for a many-body quantum state. Given a density matrix of a composite quantum system, thought of as describing more than a subsystem, it is possible to perform a "partial trace" operation, obtaining a so-called "reduced" density matrix describing knowledge of the state of a subsystem. A density matrix describes a mixed state if and only if its Von Neumann entropy is non-zero. When a density matrix is obtained via partial trace over one subsystem of a composite state, this is equivalent to the object described by the reduced density matrix being entangled to a part of the broader system that was traced over. The entanglement entropy of a subsystem is thus defined as the Von Neumann entropy of the reduced density matrix of the subsystem, traced over other subsystems.
More mathematically; if a state describing two subsystems A and B is a separable state, then the reduced density matrix is 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 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 ; 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.
The entanglement entropy can be expressed using the singular values of the Schmidt decomposition of the state. Any pure state can be written as where and are orthonormal states in subsystem and subsystem respectively. The entropy of entanglement is simply
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 or 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:
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 in 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 , 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.