Thermodynamic Uncertainty Relation for General Open Quantum Systems

We derive a thermodynamic uncertainty relation for general open quantum dynamics, described by a joint unitary evolution on a composite system comprising a system and an environment. By measuring the environmental state after the system-environment interaction, we bound the counting observables in the environment by the survival activity, which reduces to the dynamical activity in classical Markov processes. Remarkably, the relation derived herein holds for general open quantum systems with any counting observable and any initial state. Therefore, our relation is satisfied for classical Markov processes with arbitrary time-dependent transition rates and initial states. We apply our relation to continuous measurement and the quantum walk to find that the quantum nature of the system can enhance the precision. Moreover, we can make the lower bound arbitrarily small by employing appropriate continuous measurement.

Figure 1

Principal system $S$ and environment $E$. (a) Basic model. The initial states of $S$ and $E$ are $|\psi ⟩$ and $|0⟩$, respectively. The composite system $S+E$ undergoes a unitary transformation $U$, and $E$ is measured by the observable $\mathcal{G}$. (b) Continuous measurement case ($N=3$). The initial states of the principal system and environment are $|\psi ⟩$ and $|0_2,0_1,0_0⟩$, respectively. The initial substate $|0_k⟩$ interacts with $S$ within the time interval $[t_k,t_{k+1}]$ via a unitary operator $U_{t_k}$. The measurement record is obtained by measuring $E$ at $t=T$. (c) Quantum walk case. The initial states of chirality (principal system) and position (environmental system) are $|\mathsf{R}⟩$ and $|0⟩$, respectively. The principal and environmental systems interact at each step via a unitary operator $\mathcal{U}$. The position is obtained by measuring $E$ at step $t=T$.