Relationship between Hartley entropy and local dimension

Question

I am recently reading a paper about entanglement entropy. It mentions that if we consider a 1D spin chain and write a pure state in the matrix product state:

\begin{align} |\psi\rangle = A^{\sigma_1}A^{\sigma_2}\cdots A^{\sigma_i}A^{\sigma_{i+1}}\cdots A^{\sigma_N}\left|\sigma_1\cdots\sigma_N\right\rangle \end{align}

And we treat spins $1,\dots,i$ to be subsystem A and spins $i+1,\dots,N$ to be subsystem B, then we can define the $n$-th Renyi entropy between A and B:

\begin{align} \rho &= \left|\psi\right\rangle\left\langle \psi\right|\\ \rho_B &= \text{Tr}_A\rho\\ S_n(B) &= \frac{1}{1-n}\log\text{Tr}\rho_B^n \end{align}

For the zeroth Renyi entropy, it equals

\begin{align} S_0(B) =\log r \end{align}

Here $r$ is the number of nonzero eigenvalues of $\rho_B$ and it is also the Schmidt rank. But the paper also says that $r$ equals $d_{i,i+1}$, the local dimension between $A^{\sigma_i}$ and $A^{\sigma_{i+1}}$ in MPS. Why is it true? How to prove it?

References:

1. https://doi.org/10.1103/PhysRevX.7.031016 Appendix A Eqn.A11.

Answer 1

It is only true that $r\le d_{i,i+1}$.

This is easy to see: First, write the matrix products using explicit sums over the matrix indices. Then, the only summation index which couples the subsystems A and B is the index from the matrix product $A^{\sigma_i}A^{\sigma_{i+1}}$, which can take $d_{i,i+1}$ values.

Thus, the state can be written as $$ \sum_{k=1}^{d_{i,i+1}} |a_k\rangle|b_k\rangle\ . $$ The number of terms in the sum upper bounds the number of terms in the Schmidt decomposition. Thus, $r\le d_{i,i+1}$.

Of course, if you pose additional conditions on the MPS (like a "nice" canonical form), it might also hold that $r=d_{i,i+1}$.