correspondingly, SD) in the chain: Step 1: express the 's in a local basis for qubit 2: The vectors are not necessarily normalized. Step 2: write each vector in terms of the at most (Vidal's emphasis) Schmidt vectors and, correspondingly, coefficients : Step 3: make the substitutions and obtain: Repeating the steps 1 to 3, one can construct the whole decomposition of state D. The last 's are a special case, like the first ones, expressing the right-hand Schmidt vectors at the bond in terms of the local basis at the lattice place. As shown in, it is straightforward to obtain the Schmidt decomposition at bond, i.e. , from D. The Schmidt eigenvalues, are given explicitly in D: The Schmidt eigenvectors are simply: and Rationale Now, looking at D, instead of initial terms, there are . Apparently this is just a fancy way of rewriting the coefficients , but in fact there is more to it than that. Assuming that N is even, the Schmidt rank for a bipartite cut in the middle of the chain can have a maximal value of ; in this case we end up with at least coefficients, considering only the ones, slightly more than the initial ! The truth is that the decomposition D is useful when dealing with systems that exhibit a low degree of entanglement, which fortunately is the case with many 1D systems, where the Schmidt coefficients of the ground state decay in an exponential manner with : Therefore, it is possible to take into account only some of the Schmidt coefficients (namely the largest ones), dropping the others and consequently normalizing again the state: where is the number of kept Schmidt coefficients. Let's get away from this abstract picture and refresh ourselves with a concrete example, to emphasize the advantage of making this decomposition. Consider for instance the case of 50 fermions in a ferromagnetic chain, for the sake of simplicity. A dimension of 12, let's say, for the would be a reasonable choice, keeping the discarded eigenvalues at % of the total, as shown by numerical studies, meaning roughly coefficients, as compared to the originally ones. Even if the Schmidt eigenvalues don't have this exponential decay, but they show an algebraic decrease, we can still use D to describe our state . The number of coefficients to account for a faithful description of may be sensibly larger, but still