Entropy of entanglement

The entropy of entanglement is an entanglement measure for many-body quantum state.

Bipartite entanglement entropy

Bipartite entanglement entropy is defined with respect to a bipartition of a state into two partitions $A$ and $B$.

Von Neumann entanglement entropy

The bipartite Von Neumann entanglement entropy $S$ is defined as the Von Neumann entropy of either of its reduced states; the result is independent of which one we pick. That is, for a pure state $\rho_{AB}= |\Psi\rangle\langle\Psi|_{AB}$, it is given by:

$\mathcal{S}(\rho_A)= -\operatorname{Tr}[\rho_A\operatorname{log}\rho_A] = -\operatorname{Tr}[\rho_B\operatorname{log}\rho_B] = \mathcal{S}(\rho_B)$

where $\rho_{A}=\operatorname{Tr}_B(\rho_{AB})$ and $\rho_{B}=\operatorname{Tr}_A(\rho_{AB})$ 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:

• 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:

• Negativity
• Logarithmic negativity
• Robustness of entanglement^[1]

Renyi entanglement entropies

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

$\mathcal{S}_\alpha (\rho_A) = \frac{1}{1-\alpha} \text{log} \text{Tr} (\rho^\alpha) = \mathcal{S}_\alpha(\rho_B)$

Note that the limit $\alpha\rightarrow 1$, The Renyi entanglement entropy approaches the Von Neumann entanglement 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 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. ^[2]

References/sources

1. http://www.quantiki.org/wiki/Entropy_of_entanglement
2. 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. Compendium of Quantum Physics. Springer. pp. 205–209. doi:10.1007/978-3-540-70626-7_66. ISBN 978-3-540-70626-7.