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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)?


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.

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up vote 4 down vote accepted

You are asking quite a few questions, so let me try to go step by step.

First, an area law is a very special property among quantum states: If you pick a random state, it will have almost maximal entropy (i.e. a volume rather than an area scaling). So essentially any state would be a counterexample ;-)

On the other hand, ground states appearing in nature typically have very low entangement, which scales like the area (possibly with some logarithmic correction). Intuitively, one can understand this by the fact that the system tries to minimize the energy of the local Hamiltonian terms by creating local entanglement. (Though this is not rigorous and not entirely correct.)

Most importantly, an area law has been rigorously proven by Hastings for ground states of gapped one-dimensional Hamiltonians (This result has been improved since, and it has been shown that an area law is also implied by exponentially decaying correlations.) While this is also believed to hold for many gapless systems (up to a correction), there exists examples which are gapless and have an entropy which scales algebraically with the block size. Similar things are believed to be true in 2D, but there is no rigorous proof.

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).

Let me finally note that an important point why the area law is interesting is that states which satisfy an area law can be approximated efficiently by MPS (and thus simulated using DMRG).

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thanks for the detailed answer, this was very helpful to me! –  ckrk Apr 10 '13 at 17:36
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