  Efficient methods to access the entanglement of a quantum many-body state,
where the complexity generally scales exponentially with the system size $N$,
have long a concern. Here we propose the Schmidt tensor network state (Schmidt
TNS) that efficiently represents the Schmidt decomposition of finite- and even
infinite-size quantum states with nontrivial bipartition boundary. The key idea
is to represent the Schmidt coefficients (i.e., entanglement spectrum) and
transformations in the decomposition to tensor networks (TNs) with
linearly-scaled complexity versus $N$. Specifically, the transformations are
written as the TNs formed by local unitary tensors, and the Schmidt
coefficients are encoded in a positive-definite matrix product state (MPS).
Translational invariance can be imposed on the TNs and MPS for the
infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating
the ground state of the quasi-one-dimensional spin model with geometrical
frustration. Our results show that the MPS encoding the Schmidt coefficients is
weakly entangled even when the entanglement entropy of the decomposed state is
strong. This justifies the efficiency of using MPS to encode the Schmidt
coefficients, and promises an exponential speedup on the full-state sampling
tasks.
