A direct algebraic proof for the non-positivity of Liouvillian eigenvalues in Markovian quantum dynamics

• URL: http://arxiv.org/abs/2504.02256v1
• Date: Thu, 03 Apr 2025 03:54:25 GMT
• Title: A direct algebraic proof for the non-positivity of Liouvillian eigenvalues in Markovian quantum dynamics
• Authors: Yikang Zhang, Thomas Barthel,
• Abstract summary: Markovian open quantum systems are described by the Lindblad master equation $partial_trho =mathcalL(rho)$.<n>For systems with a finite-dimensional Hilbert space, it is a fundamental property of the Liouvillian, that the real-parts of all its eigenvalues are non-positive.
• Score: 2.973419031093673
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Markovian open quantum systems are described by the Lindblad master equation $\partial_t\rho =\mathcal{L}(\rho)$, where $\rho$ denotes the system's density operator and $\mathcal{L}$ the Liouville super-operator, which is also known as the Liouvillian. For systems with a finite-dimensional Hilbert space, it is a fundamental property of the Liouvillian, that the real-parts of all its eigenvalues are non-positive which, in physical terms, corresponds to the stability of the system. The usual argument for this property is indirect, using that $\mathcal{L}$ generates a quantum channel and that quantum channels are contractive. We provide a direct algebraic proof based on the Lindblad form of Liouvillians.

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