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arXiv:2012.05449

Time-inhomogeneous Quantum Markov Chains with Decoherence on Finite State Spaces

Chia-Han Chou, Wei-Shih Yang

Published Dec 10, 2020• 0 citations
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Abstract

We introduce and study time-inhomogeneous quantum Markov chains with parameter $ζ\ge 0$ and decoherence parameter $0 \leq p \leq 1$ on finite spaces and their large scale equilibrium properties. Here $ζ$ resembles the inverse temperature in the annealing random process and $p$ is the decoherence strength of the quantum system. Numerical evaluations show that if $ ζ$ is small, then quantum Markov chain is ergodic for all $0 < p \le 1$ and if $ ζ$ is large, then it has multiple limiting distributions for all $0 < p \le 1$. In this paper, we prove the ergodic property in the high temperature region $0 \le ζ\le 1$. We expect that the phase transition occurs at the critical point $ζ_c=1$. For coherence case $p=0$, a critical behavior of periodicity also appears at critical point $ζ_o=2$.

DOI
10.48550/arxiv.2012.05449

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We introduce and study time-inhomogeneous quantum Markov chains with parameter ζ≥0ζ\ge 0ζ≥0 and decoherence parameter 0≤p≤10 \leq p \leq 10≤p≤1 on finite spaces and their large scale equilibrium properties. Here ζζζ resembles the inverse temperature in the annealing random process and ppp is the decoherence strength of the quantum system. Numerical evaluations show that if ζ ζζ is small, then quantum Markov chain is ergodic for all 0<p≤10 < p \le 10<p≤1 and if ζ ζζ is large, then it has multiple limiting distributions for all 0<p≤10 < p \le 10<p≤1. In this paper, we prove the ergodic property in the high temperature region 0≤ζ≤10 \le ζ\le 10≤ζ≤1. We expect that the phase transition occurs at the critical point ζc=1ζ_c=1ζc​=1. For coherence case p=0p=0p=0, a critical behavior of periodicity also appears at critical point ζo=2ζ_o=2ζo​=2.
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A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
Quantum Markov processes are widely used models of the dynamics open quantum systems, a fundamental topic in theoretical and mathematical physics with important applications in experimental realisations of quantum systems such as ultracold atomic gases and new quantum information technologies such as quantum metrology and quantum control. In this thesis we present a mathematical framework which effectively characterises dynamical phase transitions in quantum Markov processes, using the theory of large deviations, by combining insights developed in non-equilibrium dynamics with techniques from quantum information and probability. We provide a natural decomposition for quantum Markov chains into phases, paving the way for the rigorous treatment of critical features of such systems such as phase transitions and phase purification. A full characterisation of dynamical phase transitions beyond properties of the steady state is described in terms of a dynamical perspective through critical behaviour of the quantum jump trajectories. We extend a fundamental result from large deviations for classical Markov chains, the Sanov theorem, to a quantum setting; we prove this Sanov theorem for the output of quantum Markov chains, a result which could be extended to a quantum Donsker-Varadhan theory. We perform an in-depth analysis of the atom maser, an infinite-dimensional quantum Markov process exhibiting various types of critical behaviour: for certain parameters it exhibits strong intermittency in the atom detection counts, and has a bistable stationary state. We show that the atom detection counts satisfy a large deviations principle, and therefore we deal with a phase cross-over rather than a genuine phase transition, although the latter occurs in the limit of infinite pumping rate. As a corollary, we obtain the Central Limit Theorem for the counting process.
Abstract We formalize and prove the extension to finite temperature of a class of quantum phase transitions, acting as condensations in the space of states, recently introduced and discussed at zero temperature (Ostilli and Presilla 2021 J. Phys. A: Math. Theor. 54 055005). In details, we find that if, for a quantum system at canonical thermal equilibrium, one can find a partition of its Hilbert space into two subspaces, and , such that, in the thermodynamic limit, dim ℋ cond / dim ℋ → 0 and the free energies of the system restricted to these subspaces cross each other for some value of the Hamiltonian parameters, then, the system undergoes a first-order quantum phase transition driven by those parameters. The proof is based on an exact probabilistic representation of quantum dynamics at an imaginary time identified with the inverse temperature of the system. We also show that the critical surface has universal features at high and low temperatures.
We study a class of quantum Markov processes that, on the one hand, is inspired by the micromaser experiment in quantum optics and, on the other hand, by classical birth and death processes. We prove some general geometric properties and irreducibility for non-degenerated parameters. Furthermore, we analyze ergodic properties of the corresponding transition operators. For homogeneous birth and death rates we show how these can be fully determined by explicit calculation. As for classical birth and death chains we obtain a rich yet simple class of quantum Markov chains on an infinite space, which allow only local transitions while having divers ergodic properties.