The forward-backward correlation conjecture for matrix product states

Let $\rho^0_{1,j}$ be the reduced density matrix on sites $1$ through $j$ of a matrix product state $\Psi_0$ as in Eq. (mps), and let $D$ and $\xi'$ be the bond-dimension parameter and correlation-length parameter appearing in the matrix product state and Eq. (fwdback). Assume that the forward-backward condition holds for all $j,l,A,B_L$ for the given $\xi'$.

Forward-backward correlation conjecture. There exists a function $f(D_{eff})$ such that

$$S(\rho^0_{1,j})\leq f(D^{\xi'}).$$

Here $S(\rho^0_{1,j})$ denotes the entropy of the reduced state on sites $1$ through $j$. The conjecture asserts that satisfying the forward-backward condition forces a bound on the entanglement entropy depending only on the effective bond-dimension parameter and the correlation-length parameter, supporting an area law for one-dimensional quantum systems.