Concept in quantum physics
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
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 consists of
particles. A bipartition of the system is a partition which divides 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:
![{\displaystyle {\mathcal {S}}(\rho _{A})=-\operatorname {Tr} [\rho _{A}\operatorname {log} \rho _{A}]=-\operatorname {Tr} [\rho _{B}\operatorname {log} \rho _{B}]={\mathcal {S}}(\rho _{B})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0bd22e963c256be0cac8920c00999aee344442)
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:
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:
![{\displaystyle {\mathcal {S}}_{\alpha }(\rho _{A})={\frac {1}{1-\alpha }}\operatorname {log} \operatorname {tr} (\rho _{A}^{\alpha })={\mathcal {S}}_{\alpha }(\rho _{B})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cef0e9faed49a5db4024ed053e3e968efe936479)
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
![{\displaystyle H=(p_{A}^{2}+p_{B}^{2})/2+\omega _{1}^{2}(q_{A}^{2}+q_{B}^{2})/{2}+{\omega _{2}^{2}(q_{A}-q_{B})^{2}}/{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63d85581576de35f2312aee98ff1b96492ab40a6)
With
, the system's pure ground state density matrix is
, which in position basis is
. Then
[2]
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.
[3]
References/sources