# Entropy of entanglement

The entropy of entanglement (or entanglement entropy) is an entanglement measure for a many-body quantum state. If a state ${\displaystyle |\Psi _{AB}\rangle =|\phi _{A}\rangle |\phi _{B}\rangle }$is a separable state, then the reduced density matrix ${\displaystyle \rho _{A}=\operatorname {Tr} _{B}|\Psi _{AB}\rangle \langle \Psi _{AB}|=|\phi _{A}\rangle \langle \phi _{A}|}$is a pure state, thus the entropy of the state is zero. Similar result holds for ${\displaystyle \rho _{B}}$. 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 ${\displaystyle N}$particles. A bipartition of the system is a partition which divide the system into two parts ${\displaystyle A}$ and ${\displaystyle B}$, containing ${\displaystyle k}$ and ${\displaystyle l}$ particles respectively with ${\displaystyle k+l=N}$. Bipartite entanglement entropy is defined with respect to this bipartition.

### Von Neumann entanglement entropy

The bipartite Von Neumann entanglement entropy ${\displaystyle S}$ 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 ${\displaystyle \rho _{AB}=|\Psi \rangle \langle \Psi |_{AB}}$, it is given by:

${\displaystyle {\mathcal {S}}(\rho _{A})=-\operatorname {Tr} [\rho _{A}\operatorname {log} \rho _{A}]=-\operatorname {Tr} [\rho _{B}\operatorname {log} \rho _{B}]={\mathcal {S}}(\rho _{B})}$

where ${\displaystyle \rho _{A}=\operatorname {Tr} _{B}(\rho _{AB})}$ and ${\displaystyle \rho _{B}=\operatorname {Tr} _{A}(\rho _{AB})}$ 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:

Some entanglement measures that do not reduce to the entropy of entanglement are:

### Renyi entanglement entropies

The Renyi entanglement entropies ${\displaystyle {\mathcal {S}}_{\alpha }}$ are also defined in terms of the reduced density matrices, and a Renyi index ${\displaystyle \alpha \geq 0}$. It is defined as the Rényi entropy of the reduced density matrices:

${\displaystyle {\mathcal {S}}_{\alpha }(\rho _{A})={\frac {1}{1-\alpha }}\operatorname {log} \operatorname {tr} (\rho _{A}^{\alpha })={\mathcal {S}}_{\alpha }(\rho _{B})}$

Note that the limit ${\displaystyle \alpha \rightarrow 1}$, The Renyi entanglement entropy approaches the Von Neumann entanglement entropy.

## Example with coupled harmonic oscillators

Consider two coupled quantum harmonic oscillators, with positions ${\displaystyle q_{A}}$ and ${\displaystyle q_{B}}$, momenta ${\displaystyle p_{A}}$ and ${\displaystyle p_{B}}$, and system Hamiltonian

${\displaystyle H=(p_{A}^{2}+p_{B}^{2})/2+\omega _{1}^{2}(q_{A}^{2}+q_{B}^{2})/{2}+{\omega _{2}^{2}(q_{A}-q_{B})^{2}}/{2}}$

With ${\displaystyle \omega _{\pm }^{2}=\omega _{1}^{2}+\omega _{2}^{2}\pm \omega _{2}^{2}}$, the system's pure ground state density matrix is ${\displaystyle \rho _{AB}=|0\rangle \langle 0|}$, which in position basis is ${\displaystyle \langle q_{A},q_{B}|\rho _{AB}|q_{A}',q_{B}'\rangle \propto \exp \left(-{\omega _{+}(q_{A}+q_{B})^{2}}/{2}-{\omega _{-}(q_{A}-q_{B})^{2}}/{2}-{\omega _{+}(q'_{A}+q'_{B})^{2}}/{2}-{\omega _{-}(q'_{A}-q'_{B})^{2}}/{2}\right)}$. Then [2]

${\displaystyle \langle q_{A}|\rho _{A}|q_{A}'\rangle \propto \exp \left({\frac {2(\omega _{+}-\omega _{-})^{2}q_{A}q_{A}'-(8\omega _{+}\omega _{-}+(\omega _{+}-\omega _{-})^{2})(q_{A}^{2}+q_{A}'^{2})}{8(\omega _{+}+\omega _{-})}}\right)}$

Since ${\displaystyle \rho _{A}}$ happens to be precisely equal to the density matrix of a single quantum harmonic oscillator of frequency ${\displaystyle \omega \equiv {\sqrt {\omega _{+}\omega _{-}}}}$ at thermal equilibrium with temperature ${\displaystyle T}$ ( such that ${\displaystyle \omega /k_{B}T=\cosh ^{-1}\left({\frac {8\omega _{+}\omega _{-}+(\omega _{+}-\omega _{-})^{2}}{(\omega _{+}-\omega _{-})^{2}}}\right)}$ where ${\displaystyle k_{B}}$ is the Boltzmann constant) , the eigenvalues of ${\displaystyle \rho _{A}}$ are ${\displaystyle \lambda _{n}=(1-e^{-\omega /k_{B}T})e^{-n\omega /k_{B}T}}$ for nonnegative integers ${\displaystyle n}$. The Von Neumann Entropy is thus

${\displaystyle -\sum _{n}\lambda _{n}\ln(\lambda _{n})={\frac {\omega /k_{B}T}{e^{\omega /k_{B}T}-1}}-\ln(1-e^{-\omega /k_{B}T})}$.

Similarly the Renyi entropy ${\displaystyle S_{\alpha }(\rho _{A})={\frac {(1-e^{-\omega /k_{B}T})^{\alpha }}{1-e^{-\alpha \omega /k_{B}T}}}/(1-\alpha )}$.

## 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

1. ^ http://www.quantiki.org/wiki/Entropy_of_entanglement
2. ^ Entropy and area Mark Srednicki Phys. Rev. Lett. 71, 666 – Published 2 August 1993 https://arxiv.org/pdf/hep-th/9303048.pdf
3. ^ 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.