Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912287645761536 |
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| author | Benoist, Tristan Hautecoeur, Arnaud Pellegrini, Clément |
| author_facet | Benoist, Tristan Hautecoeur, Arnaud Pellegrini, Clément |
| contents | Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap and the peripheral spectrum is described in a precise manner. Next two perturbations of this operator are studied. This allows to derive limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation Principle) for the empirical mean of functions of the Markov chain as well as the Lyapounov exponent of the underlying random dynamical system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03879 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems Benoist, Tristan Hautecoeur, Arnaud Pellegrini, Clément Probability Mathematical Physics Functional Analysis Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap and the peripheral spectrum is described in a precise manner. Next two perturbations of this operator are studied. This allows to derive limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation Principle) for the empirical mean of functions of the Markov chain as well as the Lyapounov exponent of the underlying random dynamical system. |
| title | Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems |
| topic | Probability Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2402.03879 |