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

• URL: http://arxiv.org/abs/2502.12934v1
• Date: Tue, 18 Feb 2025 15:16:04 GMT
• Title: The Canonical Forms of Matrix Product States in Infinite-Dimensional Hilbert Spaces
• Authors: Niilo Heikkinen,
• Abstract summary: 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 construction of infinite-dimensional MPS (idMPS) is analogous to the well-known finite-dimensional construction in terms of singular value decompositions of matrices.
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• 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.

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