Fluctuation Theorem for Many-Body Pure Quantum States

We prove the second law of thermodynamics and the nonequilibrium fluctuation theorem for pure quantum states. The entire system obeys reversible unitary dynamics, where the initial state of the heat bath is not the canonical distribution but is a single energy eigenstate that satisfies the eigenstate-thermalization hypothesis. Our result is mathematically rigorous and based on the Lieb-Robinson bound, which gives the upper bound of the velocity of information propagation in many-body quantum systems. The entanglement entropy of a subsystem is shown connected to thermodynamic heat, highlighting the foundation of the information-thermodynamics link. We confirmed our theory by numerical simulation of hard-core bosons, and observed dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our result reveals a universal scenario that the second law emerges from quantum mechanics, and can be experimentally tested by artificial isolated quantum systems such as ultracold atoms.

Figure 1

Schematics of our setup. (a) The total system, where the initial state of B is an energy eigenstate $|E_i⟩$. (b) Subregions used in our argument.

Figure 2

Time dependence of average entropy production $⟨\sigma ⟩$ with parameters ${\gamma }^{\prime }/\omega =0.1$, 1, 4, $\gamma /\omega =1$, and $g/\omega =0.1$. Inset: The lattice structure for the numerical calculation; system $S$ (green) attached to bath $B$ (cyan).

Figure 3

Time dependence of $⟨e^{-\sigma }⟩$ that numerically confirms the IFT. The structure of $S$ and $B$ is the same as that in the inset of Fig. 2. The parameters are given by ${\gamma }^{\prime }/\omega =0.05$, 0.2, 1, $\gamma /\omega =1$, and $g/\omega =0.1$. Inset: Time dependence of the deviation of $⟨e^{-\sigma }⟩$ from unity.