Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state
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| Format: | Preprint |
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2024
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| _version_ | 1866910743866114048 |
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| author | Girotti, Federico Poletti, Damiano |
| author_facet | Girotti, Federico Poletti, Damiano |
| contents | Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems. Moreover, they represent the noncommutative generalization of classical Orsntein-Uhlenbeck semigroups; analogously to the classical case, GQMSs are uniquely determined by a "drift" matrix $\mathbf{Z}$ and a "diffusion" matrix $\mathbf{C}$, together with a displacement vector $\mathbfζ$. In this work, we completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states; as a side result, we are able to characterize quadratic Hamiltonians admitting a ground state. Moreover, we study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of $\mathbf{Z}$. Then, we prove that environment-induced decoherence takes place and that the dynamics approaches an Hamiltonian closed evolution for long times; we are also able to determine the speed at which this happens. Finally, we study convergence of ergodic means and recurrence and transience of the semigroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state Girotti, Federico Poletti, Damiano Functional Analysis Quantum Physics 46L55, 37L40, 81V73 Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems. Moreover, they represent the noncommutative generalization of classical Orsntein-Uhlenbeck semigroups; analogously to the classical case, GQMSs are uniquely determined by a "drift" matrix $\mathbf{Z}$ and a "diffusion" matrix $\mathbf{C}$, together with a displacement vector $\mathbfζ$. In this work, we completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states; as a side result, we are able to characterize quadratic Hamiltonians admitting a ground state. Moreover, we study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of $\mathbf{Z}$. Then, we prove that environment-induced decoherence takes place and that the dynamics approaches an Hamiltonian closed evolution for long times; we are also able to determine the speed at which this happens. Finally, we study convergence of ergodic means and recurrence and transience of the semigroup. |
| title | Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state |
| topic | Functional Analysis Quantum Physics 46L55, 37L40, 81V73 |
| url | https://arxiv.org/abs/2412.10020 |