Matrix product state

Remarks

Every vector of $\mathcal{H}$ is a matrix product state with $D_k = \min(d^k, d^{N-k})$, obtained by successive singular value decompositions. The least possible $D_k$ for a given $\psi$ is the rank of $\psi$ seen as a $d^k \times d^{N-k}$ matrix. This is the Schmidt rank of $\psi$ across the cut between sites $k$ and $k+1$.

Example: the vector $\lvert 0 \cdots 0 \rangle + \lvert 1 \cdots 1 \rangle$ in $(\C^2)^{\otimes N}$, with basis labels $0, 1$, is a periodic matrix product state with $D = 2$. Take $A_k^{[0]} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $A_k^{[1]} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ for every $k$. The product $A^{[i_1]} \cdots A^{[i_N]}$ is zero unless all $i_k$ are equal, and then its trace is $1$. The ground state of the AKLT model is another standard example, with $d = 3$ and $D = 2$.

Non-example: the same vector, for $N \ge 2$, is not a matrix product state with $D = 1$. With $D = 1$ the coefficient factorises as $A_1^{[i_1]} \cdots A_N^{[i_N]}$, so $\psi$ is a product vector, and the coefficient matrix of $\psi$ across the first cut has rank $2$, not $1$.