Sequential Schmidt truncation

Input an integer $N\ge 1$, nonzero finite-dimensional complex inner-product spaces $H_1,\dots ,H_N$ with fixed ordered orthonormal bases, a normalized vector $\psi \in H_1\otimes \cdots \otimes H_N$ with exact coefficients in those bases, and integer rank caps $D_k\ge 1$ for $1\le k<N$. For $N=1$ the cap list is empty. The algorithm performs the following exact finite procedure.

1. Set the current remainder to ${\psi }^{(0)}:=\psi$, the retained left space before the first cut to $S_0:=\mathbb{C}$, and an empty list of site tensors and discarded squared norms.

2. For each $k=1,\dots ,N-1$, reshape the current remainder as a matrix whose row index is the retained left index in $S_{k-1}$ together with the physical basis index in $H_k$, and whose column index is the ordered product basis of $H_{k+1}\otimes \cdots \otimes H_N$. Compute an exact singular value decomposition. Order the positive singular values nonincreasingly. Within each repeated positive singular value, form a deterministic orthonormal basis of its left singular subspace by projecting the row-coordinate standard basis vectors in lexicographic order and applying Gram–Schmidt, discarding zero residuals; obtain the corresponding right singular vector as $M^{\ast }u\mathrm{/}s$ for singular value $s>0$. Retain the first $m_k:=\min (D_k,r_k)$ positive singular directions, where $r_k$ is the number of positive singular values, and record ${\delta }_k:={\sum }_{j=m_k+1}^{r_k}{s}_{k,j}^2$ from this current unnormalized remainder. Reshape the retained left singular vectors into a site tensor $A_k^{i_k}$ with left index in $S_{k-1}$ and right index in $S_k:={\mathbb{C}}^{m_k}$; pass the retained singular values times the adjoints of the retained right singular vectors as the next current remainder ${\psi }^{(k)}$. Since the input has norm one and $D_k\ge 1$, every current remainder is nonzero and each $m_k\ge 1$.

3. After the last cut, reshape the remaining array as the final site tensor $A_N^{i_N}$, use outer bond dimensions $D_0=D_N=1$ with scalar boundary columns $\mathrm{\ell }=r=(1)$, and return the contracted unnormalized open MPS $\varphi :={\sum }_{i_1,\dots ,i_N}{\mathrm{\ell }}^{\ast }{A}_1^{i_1}\cdots A_N^{i_N}r\mathrm{\mid }{i}_1⟩\otimes \cdots \otimes \mathrm{\mid }{i}_N⟩\in H_1\otimes \cdots \otimes H_N$, the list $({\delta }_1,\dots ,{\delta }_{N-1})$, and the bond dimensions $(m_1,\dots ,m_{N-1})$. Each $\mathrm{\mid }{i}_j⟩$ runs through the fixed ordered orthonormal basis of $H_j$. For $N>1$, the compressed remainder ${\psi }^{(N-1)}\in S_{N-1}\otimes H_N$ supplies the entries of $A_N^{i_N}$; its retained left index is contracted with the preceding site tensors to form $\varphi$. For $N=1$, return $\varphi =\psi$ and empty lists with the same scalar boundaries. If $\mathrm{\parallel }\varphi {\mathrm{\parallel }}^2=0$, return a norm-zero flag and do not normalize. Otherwise an optional normalized copy is $\varphi \mathrm{/}\mathrm{\parallel }\varphi \mathrm{\parallel }$; the returned $\varphi$ and discarded norms always refer to the unnormalized sweep output.

All coefficients and SVDs are exact mathematical objects; no floating-point threshold is used. The tie rule fixes a choice even when a rank cap cuts through a degenerate singular-value subspace.

Finite open MPS and injective translation invariant tensor

Schmidt decomposition