Testing quantum master equations for complete positivity: A direct approach

• URL: http://arxiv.org/abs/2410.00353v1
• Date: Tue, 1 Oct 2024 02:51:13 GMT
• Title: Testing quantum master equations for complete positivity: A direct approach
• Authors: Timur V. Tscherbul,
• Abstract summary: We establish a direct mapping between the Liouvillian and Kossakowski matrices of an arbitrary Markovian QME. As an application, we establish complete positivity of the quantum optical Bloch-Redfield QME for a three-level V-system driven by incoherent light. Our approach makes it possible to test QMEs for complete positivity without solving them, and to restore complete positivity by keeping only non-negative eigenvalues of the Kossakowski matrix.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: While quantum master equations (QMEs) are the primary workhorse in quantum information science, quantum optics, spectroscopy, and quantum thermodynamics, verifying complete positivity of the associated $N$-level quantum dynamical maps remains an outstanding challenge for $N\ge 3$. We address this challenge by establishing a direct mapping between the Liouvillian and Kossakowski matrices of an arbitrary Markovian QME. The mapping relies on the Moore-Penrose pseudo-inverse of a rectangular matrix composed of the structure constants of SU$(N)$. As an application, we establish complete positivity of the quantum optical Bloch-Redfield QME for a three-level V-system driven by incoherent light. Our approach makes it possible to test QMEs for complete positivity without solving them, and to restore complete positivity by keeping only non-negative eigenvalues of the Kossakowski matrix.

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