Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation

• URL: http://arxiv.org/abs/2605.01494v1
• Date: Sat, 02 May 2026 15:31:10 GMT
• Title: Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation
• Abstract summary: We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation.<n>We show how these operations can be performed efficiently in the TT/MPS format.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.

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