Geometrical Bounds on Irreversibility in Open Quantum Systems

The Clausius inequality has deep implications for reversibility and the arrow of time. Quantum theory is able to extend this result for closed systems by inspecting the trajectory of the density matrix on its manifold. Here we show that this approach can provide an upper and lower bound to the irreversible entropy production for open quantum systems as well. These provide insights on how the information on the initial state is forgotten through a thermalization process. Limits of the applicability of our bounds are discussed and demonstrated in a quantum photonic simulator.

Figure 1

Principal features of our scheme: (a) the quantum system is initialized in a given state ${\rho }_0$, then interacts with a reservoir at temperature $T$. The evolution induced by the interactions brings the system in a state $\rho (t)$ and produces irreversible entropy. (b) Studying the trajectory on the state space allows for information-geometric considerations delivering a lower bound on the produced entropy in terms of the geodesic length $\mathcal{L}$.

Figure 2

Quantum simulation via quantum logic gates. (a) Experimental apparatus. The system and the ancilla photons are generated via a spontaneous parametric down-conversion (SPDC) process through a $\beta$ barium-borate (BBO) crystal pumped with a cw laser (80 mW at 405 nm). The SPDC is type I, frequency degenerate (bandwidth of the parametric fluorescence greater than 30 nm). To ensure indistinguishability of the photons, these are spectrally filtered to a narrow bandwidth (7.5 nm) and selected by single-mode fibers. The qubits are injected in the photonic setup which performs the logical operations depicted in (b) and (c) representing, respectively, an amplitude damping (AD) and an inverse amplitude damping (IAD) channel. Their combination allows for reconstructing a generalized amplitude damping (GAD) channel, simulating the thermalization dynamics of the qubit illustrated in the main text [29]. The measured nonclassical interference visibility is 0.78, to be compared with the expected value 0.80. List of abbreviations: polarizing beam splitter (PBS), partially polarizing beam splitter (PPBS), half wave plate (HWP), quarter wave plate (QWP), quantum state tomography (QST).

Figure 3

Lower (green line) and upper (red line) bounds to the irreversible entropy production for a GAD channel induced by a bosonic reservoir, as a function of time $\tau$. The initial state is $|D⟩=1/\sqrt{2}(|H⟩+|V⟩)$ in (a) and $|V⟩$ in (b). The solid lines are theoretical predictions; experimental points have been obtained from the density matrices retrieved through quantum state tomography. Vertical errors have been obtained through a Monte Carlo routine that takes into account the Poissonian statistics of the measured counts, while horizontal errors depend on the uncertainty on the rotation $R(\theta )$ [29].

Figure 4

Asymptotic irreversible entropy production as a function of the temperature $T$ for the thermalization of the system $\mathcal{A}$. The dotted curves show the predictions associated to different pure states $|H⟩$ (red), $|D⟩$ (green) and $|V⟩$ (blue). The solid curves show the geometric lower bound (5) in the thermalization limit for the same three cases, with the same color code, which provides a quite a good prediction for $\mathrm{\Delta }{S}_{\mathcal{A}}^{\mathrm{irr}}(t\to \infty )$ for almost all values of $T$.