Irreversibility, Loschmidt Echo, and Thermodynamic Uncertainty Relation

Entropy production characterizes irreversibility. This viewpoint allows us to consider the thermodynamic uncertainty relation, which states that a higher precision can be achieved at the cost of higher entropy production, as a relation between precision and irreversibility. Considering the original and perturbed dynamics, we show that the precision of an arbitrary counting observable in continuous measurement of quantum Markov processes is bounded from below by the Loschmidt echo between the two dynamics, representing the irreversibility of quantum dynamics. When considering particular perturbed dynamics, our relation leads to several thermodynamic uncertainty relations, indicating that our relation provides a unified perspective on classical and quantum thermodynamic uncertainty relations.

Figure 1

Quantification of irreversibility. (a) Entropy production $\sigma$ in classical Markov processes, defined by a log ratio between ${\mathcal{P}}_F(\mathrm{\Gamma })$, the probability for observing a trajectory $\mathrm{\Gamma }$ in the forward process, and ${\mathcal{P}}_R(\overline{\mathrm{\Gamma }})$, the probability for observing a time-reversed trajectory $\overline{\mathrm{\Gamma }}$ in the reversed process. (b) Loschmidt echo $\eta$ in quantum dynamics, which is the fidelity between two states, $e^{-iH\tau }|\mathrm{\Psi }(0)⟩$ and $e^{-i{H}_{\star }\tau }|\mathrm{\Psi }(0)⟩$ at $t=\tau$. These states are obtained through forward-time evolution induced by $H$ and $H_{\star }$, respectively. (c) An interpretation of the Loschmidt echo $\eta$ as the fidelity between two states, $|\mathrm{\Psi }(0)⟩$ and $e^{i{H}_{\star }\tau }{e}^{-iH\tau }|\mathrm{\Psi }(0)⟩$, at $t=0$. The latter state is obtained through forward-time evolution by $H$ and the subsequent reversed-time evolution by $H_{\star }$.

Figure 2

Illustration of continuous measurement model for $N=4$. The initial states of $S$ and $E$ are $|{\psi }_S⟩$ and $|0_3,0_2,0_1,0_0⟩$, respectively. The environment subspace $|0_k⟩$ interacts with $S$ during an interval $[t_k,t_{k+1}]$. Finally, at $t=\tau$, $E$ is measured with a set of projectors $\{|\mathit{m}⟩⟨\mathit{m}|{\}}_{\mathit{m}}$. Suppose that the measurement record is $\mathit{m}=[1_3,0_2,0_1,1_0]$. Then, the state of the principal system undergoes two jump events in the intervals $[t_0,t_1]$ and $[t_3,t_4]$.