Statistical entropy of quantum systems

Abstract: Let $D_1$ and $D_2$ be the Hilbert space dimensions of two subsystems of a quantum system of total Hilbert space dimension $D=D_1D_2$. In the thermodynamic limit (with $1\ll D_1 \ll D_2$), we know from the works of Page and Sen that the average von Neumann (VN) entropy of the first subsystem is $\mathbb{E}({S}^{sb}_{VN})=\ln(D_1)+O(D_1/D_2)$ if the full system is in a random pure state. Here, it is argued that this result can be strengthened for a thermalized quantum system. Consider the subspace $\mathcal{H}E$ of the total Hilbert space corresponding to a narrow shell around the energy $E$. We find that the result of Page and Sen holds for each of these subspaces, that is, the VN entropy, when averaged over the states in $\mathcal{H}_E$, is given by $\overline{S}^{sb}{VN} \approx \ln \widetilde{d}_1$, where $\widetilde{d}_1$ represents the dimension of the effective Hilbert space of the first subsystem relevant to $\mathcal{H}_E$. If $d_E = \dim{(\mathcal{H}_E)}$, we estimate that $\widetilde{d}_1 = D_1^\gamma$, where $\gamma = \ln (d_E) / \ln (D)$. This finding has significant implications, as it suggests an equivalence between the VN entropy and the thermodynamic (TH) entropy of a subsystem within a much larger thermalized quantum system. For completeness, we also discuss in this work the issue of equivalence between the VN entropy and TH entropy for isolated system (as a whole) and open system. For numerical demonstration of our important results, we here consider a one-dimensional spin-1/2 chain with next-nearest neighbor interactions.