# What states are satisfying an entropic area law and why do they satisfy it? More specificly why do matrix product states satisfy it?

I am currently reading some papers concerning the question why the density matrix renormalization group (DMRG) method is working well for simulating one dimensional systems and bad for higher dimensional systems. Usually the literature claims that certain states created by DMRG, the matrix product states, satisfying an "entropic area law" are simulatable in one dimension and not in higher dimensions.

As far as I understand obeying an "entropic area law" means that the entanglement entropy of a reduced subsystem is proportional to the border of the subsystem and not the volume of the subsystem. It is unclear to me, why this is a reasonable assumption.

I know of one big review article concerning the subject (http://arxiv.org/abs/0808.3773). Unfortunately especially for the matrix product states it is said that the entropic area law, follows trivially from the definition, but i can not follow that. Regarding the other examples given in the article, i am somewhat overwhelmed and find it hard to see what they have in common and where the differences are.

What states feature an entropic area law and more importantly why do they exhibit it?

More specifically why does the ground state of a matrix product state exhibit the property, while other states don't (I read that matrix product states do that, but I don't see why)?

P.s.:

I am also thankful for counter examples, that could help illustrate the qualitative difference between states satisfying the are law and states not satisfying it.

Finally, why do Matrix Product States (MPS) satisfy the area law? First, consider the bipartite state $$\vert\psi\rangle = \sum_{ij} (\sum_{\alpha=1}^{D} a^i_\alpha b^j_\alpha) \vert i\rangle\vert j\rangle\ .$$ This state has Schmidt rank $D$, i.e., its entanglement is bounded by $\log\,D$: It satisfies an area law. Now, an MPS with a cut at position $s$ is exactly of this form, with $i=(i_1,\dots,i_s)$, $j=(i_{s+1},\dots,i_N)$, and $$a^{(i_1\dots i_s)} = A^{[1],i_1}\cdots A^{[s],i_s}$$ and $$b^{(i_{s+1}\dots i_N)} = A^{[s+1],i_{s+1}}\cdots A^{[N],i_N}$$ (here, $A^{[1],i_1}$ is a $1\times D$ matrix, $A^{[N],i_N}$ is a $D\times 1$ matrix, and the others are $D\times D$ matrices).