Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems

We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.

Figure 1

(a) Example of a quantum reset process: single particle subject to coherent and dissipative hopping. Here, $\mathrm{\Omega }=1/2$ and $\gamma =1$. (b) Rate function $I_1$ for the flux $Q^L$ (full black curve) and bound from Eq. (12) (dashed red curve). Inset: Behavior close to the mean corresponding to the TUR Eq. (13). (c) An estimate (upper bound) of the entanglement rate function $I_S(\mathcal{S})$ (dashed red curve), from Eq. (7). Inset: The bound is compared with the exact $I_S(\mathcal{S})$ (full black curve) in the quadratic regime analogous to Eq. (13). $I_S(\mathcal{S})$ is not shown for all $\mathcal{S}$, as its computation requires nontrivial minimization of Eq. (7). However, bounds and estimates can be obtained from simple Ansätze. The largest possible value for ${\mathcal{S}}_{\tau }$ is $\approx 0.3863$, achieved by long trajectories without incoherent jumps.