A direct algebraic proof for the non-positivity of Liouvillian spectral values in Markovian quantum dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917428475199488 |
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| author | Zhang, Yikang Barthel, Thomas |
| author_facet | Zhang, Yikang Barthel, Thomas |
| contents | Markovian open quantum systems are described by the Lindblad master equation $\partial_tρ=\mathcal{L}(ρ)$, where $ρ$ 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. Analogously, for infinite-dimensional Hilbert spaces, the Liouvillian as a map on trace-class operators only has spectral values with non-positive real parts. The usual arguments for these properties are 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A direct algebraic proof for the non-positivity of Liouvillian spectral values in Markovian quantum dynamics Zhang, Yikang Barthel, Thomas Quantum Physics Statistical Mechanics Mathematical Physics Markovian open quantum systems are described by the Lindblad master equation $\partial_tρ=\mathcal{L}(ρ)$, where $ρ$ 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. Analogously, for infinite-dimensional Hilbert spaces, the Liouvillian as a map on trace-class operators only has spectral values with non-positive real parts. The usual arguments for these properties are 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. |
| title | A direct algebraic proof for the non-positivity of Liouvillian spectral values in Markovian quantum dynamics |
| topic | Quantum Physics Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2504.02256 |