Final answer:
The canonical form of an MPS encodes all Schmidt decompositions across any bipartition and applies to both open and periodic systems, with appropriate adjustments for the latter.
Step-by-step explanation:
When expressing a Matrix Product State (MPS) in its canonical form, the representation can indeed encode all Schmidt decompositions between a subsystem and its complement.
This property is not limited to the decompositions between left and right parts of a system but includes any bipartition, such as one site in the bulk of a chain versus the rest of the system. The canonical form provides a systematic way to represent the state such that the information about entanglement (captured by the Schmidt decomposition) is preserved across different bipartitions.
For periodic boundary conditions, an MPS can still be written in a canonical form, though the structure of the tensors may differ due to the cyclical nature of the system. The canonical form is designed to work for open systems, but modifications and careful considerations are necessary to apply it to periodic systems.
The site tensors in both open and periodic systems need to satisfy certain orthogonality conditions to ensure that the Schmidt decomposition is properly encoded.
The requirement for site tensors in an MPS to represent all possible Schmidt decompositions is that they satisfy the orthogonality conditions corresponding to each bipartition. In practical terms, this often implies iteratively applying singular value decomposition during the construction of the MPS to ensure that the state is in its canonical form.