Abstract In this work, we prove that any element in the tensor product of separable infinite-dimensional Hilbert spaces can be expressed as a matrix product state (MPS) of possibly infinite bond dimension. The proof is based on the singular value decomposition of compact operators and the connection between tensor products and Hilbert–Schmidt operators via the Schmidt decomposition in infinite-dimensional separable Hilbert spaces. The construction of infinite-dimensional MPS (idMPS) is analogous to the well-known finite-dimensional construction in terms of singular value decompositions of matrices. The infinite matrices in idMPS give rise to operators acting on (possibly infinite-dimensional) auxiliary Hilbert spaces. As an example we explicitly construct an MPS representation for certain eigenstates of a chain of three coupled harmonic oscillators.
In this report, we discuss a necessary and sufficient condition for the existence
of Schmidt decomposition of a pure state in a multipartite Hilbert space. Schmidt
decomposition exists for any pure state in a bipartite Hilbert space. It is extremely
useful since it reduces the number of terms needed to represent a pure state using
orthonormal bases of the respective parts. However, for 3 or more partite Hilbert
spaces, Schmidt decomposition does not exist in general for an arbitrary pure state.
In this thesis, we derive a condition that states whether a state in an n-partite
Hilbert space is Schmidt-decomposable or not, i.e., whether we can represent it
through the orthonormal bases of the respective parts using less number of terms
that is required in general.