The bond dimension is the dimension of the truncated matrix product state (MPS). Let us assume that I am simulating some many-body system with high entanglement via the density matrix renormalization group (DMRG). At a conference I was attending a few days ago, someone told me that the bond-dimension increases with the amount of entanglement in the system. Therefore, simulating a highly entangled system with DMRG requires a huge amount of computational time. However, how, exactly is the bond-dimension connected to a system's entanglement?
The entanglement of any region in a matrix product state of bond dimension $D$ is bounded by $S\le 2\log D$. Thus, in order to simulate a system with a lot of entanglement, the bond dimension (and thus the memory and time of the computation) will grow exponentially with the entropy.
Conversely, we know that if for a state $\vert\psi\rangle$ the $\alpha$-Renyi entropy (for $\alpha<1$) of any block is bounded by a constant, then an efficient approximation of $\vert\psi\rangle$ by an MPS exists (i.e., where $D$ scales polynomially in the system size and the inverse precision). This is proven in https://arxiv.org/abs/cond-mat/0505140. The general connection between entropy scaling and the approximability by MPS is discussed in https://arxiv.org/abs/0705.0292, where in particular examples of states with an area law for $\alpha\ge1$ are given which cannot be approximated efficiently by MPS. (Note that for $\alpha=1$, there is (afaik) no translation invariant example.)