Thermalization and irreversibility of an isolated quantum system

Abstract

The irreversibility and thermalization of many-body systems can be attributed to the erasure of spread non-equilibrium state information by local operations. This thermalization mechanism can be demonstrated by the sequence of $\hat{O}_i(t_i)$, where $\hat{O_i}$ is a local operator, $\hat{O_i}(t_i) = e^{i\hat{H}t_i} \hat{O_i} e^{-i\hat{H}t_i}$, $\hat{H}$ is the system Hamiltonian, $t_i$ can take positive, negative, or zero values, and the sequence is arranged according to the subscript $i$. We numerically demonstrate the information erasure of initial non-equilibrium quantum states through such sequence in a one-dimensional Hubbard model. During this process, the system's entanglement entropy increases monotonically toward a stable value. By incorporating this information erasure mechanism into an isolated system, our numerical simulations reveal that in this completely isolated system, a thermalization process emerges.

Figures (4)

• Figure 1: Schematic representation of the simplified gas-like model. The circles denote lattice sites, while the connecting lines indicate possible hopping between neighboring sites. The small spheres represent particles, with different colors distinguishing different particle types. (a) A two-dimensional lattice system containing three types of fermions: red, blue, and green. (b) A one-dimensional ring lattice with a length of six sites. The figure shows an initial non-equilibrium Fock state, where red ($\tau$) and blue ($\upsilon$) particles are placed at sites 1–3.
• Figure 2: The dynamical evolution of entanglement entropy and mutual information. (a) The evolution of entanglement entropy $S_{\rho}$ is presented for different subsystem sizes $A$. The blue, orange, and green lines correspond to $A = \{1,2,3\}$, $\{1,2\}$, and $\{1\}$, respectively. The black dashed line denotes the subsystem entropy predicted by the microcanonical ensemble. (b) The orange line represents the time evolution of the average mutual information between different lattice sites, defined as $\langle I(i:j) \rangle = \frac{1}{N} \sum_{i,j} I(i:j)$ , where $N = C_6^2$ is the number of site pairs. The orange shaded region indicates the standard deviation of mutual information, $\sigma_{I} = \sqrt{\frac{1}{N} \sum_{i,j} (I_{ij} - \langle I_{ij} \rangle)^2}$ . The black dashed line denotes the mutual information predicted by the microcanonical ensemble. The inset shows mutual information fluctuations between different lattice sites $i$ and $j$ within the time range marked by the dashed box.
• Figure 3: Simulation of Information Erasure with $\hat{O}(t)$. In panels (a) and (b), the orange and blue dashed lines represent the entanglement entropy and mutual information evolution with and without local perturbations, respectively. The solid orange and blue lines are their envelopes, while the shaded areas indicate discrepancies between the two cases. The entanglement entropy $S(\rho_A)$ is computed for subsystem $A = \{1, 2, 3\}$. The system evolves forward from $t = 0$ to $250$ and backward from $t = 270$ to $520$, with local operations applied only during $t = 250$ to $270$. Panel (c) depicts the evolution of entanglement entropy and mutual information under the $[\hat{O}^\dagger \hat{O}(t)]^N$ sequence, where $t = 50$, and $\hat{O}$ matches that in panels (a) and (b). Panel (d) shows the standard deviation of $\langle \hat{n}_{i \neq 2} \rangle$ and $\langle \hat{J}_{i} \rangle$ over all possible Fock states as a function of $N$. Panel (e) displays the average value of $\langle \hat{n}_{2,\tau} \rangle$ for all possible Fock states at lattice site 2, with different initial states, as a function of $N$. The blue (orange) curve corresponds to the initial state $|0\rangle_{2,\tau}$ ($|1\rangle_{2,\tau}$), and the shaded region represents the corresponding statistical standard deviation.
• Figure 4: Isolated system model and entanglement entropy dynamics. (a) A perturbative particle $\phi$ is introduced into the model from Fig. \ref{['fig1']}(b). This particle moves only between two lattice sites within the dashed box and starts at lattice site 2. (b) Dynamical evolution of entanglement entropy and mutual information after introducing $\phi$. The entanglement entropy $S(\rho_A)$ is computed for subsystem $A = \{1,2,3\}$, considering only the $\tau$ and $\upsilon$ particles. In (b), the orange line represents the average mutual information between lattice sites 1–6, with the shaded area indicating its standard deviation. (c) The model structure in (a) mapped onto a superconducting qubit chip. Polygons represent superconducting qubits, solid lines denote $XY$ coupling, and dashed lines indicate $ZZ$ coupling. Qubits marked with small balls are initialized in state $|1\rangle$.