The Canonical Forms of Matrix Product States in Infinite-Dimensional Hilbert Spaces

TL;DR

This paper proves that any element in the tensor product of separable infinite-dimensional Hilbert spaces can be represented as an infinite-dimensional matrix product state, extending finite-dimensional MPS theory to infinite dimensions.

Contribution

It establishes the canonical form of infinite-dimensional MPS using singular value decomposition and Schmidt decomposition, providing a foundational framework for infinite-dimensional quantum states.

Findings

Any element in the tensor product space can be expressed as an infinite-dimensional MPS.

Constructs explicit MPS representations for eigenstates of coupled harmonic oscillators.

Connects tensor product spaces with Hilbert-Schmidt operators in infinite dimensions.

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.