# Truncating the bond dimension of an MPS -- how good is the approximation?

$$\newcommand{\complex}{\mathbb{C}}\newcommand{\ket}[1]{|#1\rangle}$$ Let $$\ket{\psi}\in(\complex^d)^{\otimes n}$$ be a pure quantum state. It is well-known that $$\ket{\psi}$$ is a matrix product state with bond dimension $$r$$ if and only if $$\ket{\psi}$$ has Schmidt rank $$r$$ with respect to the bipartite cuts $$(1 | 2,\dots, n), (1,2|3,\dots,n),\dots,(1,\dots,n-1|n)$$. It therefore seems natural to expect that if the sum of all but the first $$r$$ singular values of $$\ket{\psi}$$ with respect to each of these bipartite cuts is at most $$\epsilon$$, then $$\ket{\psi}$$ is $$f(\epsilon)$$-close to a matrix product state with bond dimension $$r$$, where $$f$$ is some (non-trivial) function that might also depend on $$n$$ and $$r$$. I am looking for references that compute such a function $$f$$.

It states that the total error in approximating a state $$\lvert\psi\rangle$$ by one with Schmidt rank $$r$$ in every cut, $$\lvert\psi_D\rangle$$, is $$\| \lvert\psi\rangle - \lvert \psi_D \rangle \|^2 \le (N-1) \delta\ ,$$ where $$\delta$$ is the sum of the square of the singular values.
(It is worth noting that this bound scales better in $$N$$ than what one would get from the triangle inequality of the norm; the reason is that the errors from different truncations are orthogonal, see also the proof discussed below.)