Quantum coarse-grained entropy and thermalization in closed systems

We investigate the detailed properties of observational entropy, introduced by Šafránek et al. [Phys. Rev. A 99, 010101 (2019)] as a generalization of Boltzmann entropy to quantum mechanics. This quantity can involve multiple coarse grainings, even those that do not commute with each other, without losing any of its properties. It is well-defined out of equilibrium, and for some coarse grainings it generically rises to the correct thermodynamic value even in a genuinely isolated quantum system. The quantity contains several other entropy definitions as special cases, it has interesting information-theoretic interpretations, and mathematical properties—such as extensivity and upper and lower bounds—suitable for an entropy. Here we describe and provide proofs for many of its properties, discuss its interpretation and connection to other quantities, and provide numerous simulations and analytic arguments supporting the claims of its relationship to thermodynamic entropy. This quantity may thus provide a clear and well-defined foundation on which to build a satisfactory understanding of the second thermodynamical law in quantum mechanics.

Figure 1

(a) Sketch of Hilbert space, (b) evolution of probabilities, and (c) graph of observational entropy, in 12-dimensional (12D) Hilbert space with subspaces of dimensions 1, 3, and 8, respectively. Blobs in (a) represent the amount of probability projected into each Hilbert subspace, but it should be kept in mind that the right picture is projecting a density matrix that lives in 12D space into these lower-dimensional subspaces; this cannot be depicted here. The blue curve in (b) represents a possible evolution of probabilities $p_i\left(t\right)=\mathrm{tr}\left[{\widehat{P}}_i{\widehat{\rho }}_t\right]$, with density matrix starting in the 1D subspace Hilbert subspace ${\mathcal{H}}_1$. (c) Observational entropy as function of probabilities $p_2$ and $p_3$ (where $p_1=1-p_2-p_3$), and the blue curve is the corresponding entropy $S_{O\left(C\right)}\left({\widehat{\rho }}_t\right)$ from evolution (b). Observational entropy is a strictly concave function; since each corner of its graph represents one of the subspaces, the entropy must increase at least for a short time when starting in one of them.

Figure 2

Evolution of the observational entropy with positional coarse graining of the nonintegrable system of size $L=16$, coarse grained into $p=4$ parts of size $\delta =4$, starting at $t=0$ in a state of $N=4$ particles contained in the left side of the box (sites $\left\{1\text{–}8\right\}$). As time passes, particles expand through the entire box and the observational entropy quickly increases, reaching a value not far from the maximal value $S_{max}=lndim\mathcal{H}$, where $dim\mathcal{H}=\left(\begin{array}{c}L\\ N\end{array}\right)=1820$, depicted by the straight green line.

Figure 3

Schematic depiction of two relevant observational entropies for thermodynamics, illustrated on a model of fermionic chain used in our simulations. We consider situation of $p=m=2$ regions (subsystems or partitions), evolving through Hamiltonian $\widehat{H}={\widehat{H}}^{\left(1\right)}\otimes \widehat{I}+\widehat{I}\otimes {\widehat{H}}^{\left(2\right)}+\epsilon {\widehat{H}}^{\left(\mathrm{int}\right)}$. $S_{xE}$ uses positional coarse graining, which for indistinguishable particles correspond to measuring local particle densities, and then coarse graining in total energy. Factorized observational entropy $S_F$ uses local energy coarse graining, corresponding to measuring energy of the first and the second region, respectively. Dashed curves represent interparticle forces, solid curves represent particle hopping.

Figure 4

(a) The factorized observational entropy $S_F$ (dark blue lines) and observational entropy $S_{xE}$ (light red lines) with nonintegrable dynamics ($t^{\prime }=V^{\prime }=0.96$; full lines) and integrable dynamics ($t^{\prime }=V^{\prime }=0.0$; dashed lines). The system of length $L=8$ with hardwall boundary conditions starts in the 11th energy eigenstate of the reduced Hamiltonian ${\widehat{H}}^{\left(1\text{–}8\right)}$. At $t=0$ the right wall is expanded so that $L=16$ and the system evolve. Coarse graining is given by local Hamiltonians for $S_F$, and local position operators for $S_{xE}$, respectively. We coarse grain into $p=m=2$ partitions, corresponding in FOE to $\mathcal{C}={\mathcal{C}}_{{\widehat{H}}^{\left(1\text{–}8\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(8\text{–}16\right)}}$. The thermodynamic entropies are 7.389708 in the nonintegrable case (straight green line), and 7.301491 in the integrable case (straight green dashed line). Because the system is initially in an energy eigenstate of the reduced Hamiltonian, which corresponds to the coarse graining in FOE, the FOE only has one probability value that is nonzero. Hence this entropy is initially zero. (b) The same quantities as in (a) but with the coarse graining to four partitions, which for FOE corresponds to $\mathcal{C}={\mathcal{C}}_{{\widehat{H}}^{\left(1\text{–}4\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(4\text{–}8\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(8\text{–}12\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(12\text{–}16\right)}}$.

Figure 5

The entanglement entropy between the first and last eight sites is measured as a function of time for the same systems as in Fig. 4. In the nonintegrable case ($t^{\prime }=V^{\prime }=0.96$), the entanglement entropy is expected to asymptote to $1/2$ of the thermodynamic entropy of the complete system (shown as green straight lines), in the limit of large system sizes. The curve for nonintegrable system (full line) is above the integrable one (dashed line).

Figure 6

(a) The factorized observational entropy $S_F$ (dark blue) and observational entropy $S_{xE}$ (light red) that start in a “pure thermal state” with inverse temperature $\beta =1$, in the system of size $L=8$. At $t=30$, the right wall is expanded to double the system size so that $L=16$ and the system continues to evolve. The straight green lines represent the thermodynamic (canonical) entropy $S\left({\widehat{\rho }}_{\text{th}}\right)$ before and after the expansion. We coarse grain into $p=m=4$ partitions that for FOE corresponds to $\mathcal{C}={\mathcal{C}}_{{\widehat{H}}^{\left(1\text{–}4\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(4\text{–}8\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(8\text{–}12\right)}}\otimes {\mathcal{C}}_{{\widehat{H}}^{\left(12\text{–}16\right)}}$. This graph shows nonintegrable dynamics ($t^{\prime }=V^{\prime }=0.96$). (b) The same as (a) but for integrable dynamics ($t^{\prime }=V^{\prime }=0$).

Figure 7

(a) The red and blue curves in the middle show observational entropies $S_{xE}$ (light red) and $S_F$ (dark blue) for microcanonical states (line), random superpositions of neighboring energy eigenstates (crosses), and energy eigenstates (dots), from top to bottom. The lowest (full light green) curve is the microcanonical entropy $S_{\mathrm{micro}}\left(E\right)$ given by logarithm of the density of states. The top (full dark green) curve that has a slightly different shape than the other curves is the thermodynamic (canonical) entropy $S\left({\widehat{\rho }}_{\text{th}}\right)$, with the inverse temperature calculated such that the mean value of energy of the thermal state ${\widehat{\rho }}_{\text{th}}$ corresponds to the energy $E$ depicted the horizontal axis. This graph shows nonintegrable system ($t^{\prime }=V^{\prime }=0.96$). (b) The same but for integrable system ($t^{\prime }=V^{\prime }=0$). (For increased visibility, we decided not to plot observational entropies for microcanonical states.)

Figure 8

FOE of a microcanonical state $S_F\left({\widehat{\rho }}_E^{\mathrm{micro}}\right)$ (line), adjusted FOE of a random PS state $S_F\left({\widehat{\rho }}_E\right)+\overline{{\eta }^{2}ln{\eta }^2}-\overline{{\left|\zeta \right|}^{2}ln{\left|\zeta \right|}^2}$ (crosses), and adjusted FOE of energy eigenstates $S_F\left(\right|E\rangle )+\overline{{\eta }^{2}ln{\eta }^2}$ (dots). Order 1 corrections $\overline{{\eta }^{2}ln{\eta }^2}$ and $\overline{{\left|\zeta \right|}^{2}ln{\left|\zeta \right|}^2}$ have been calculated from the theory, and are given by Eqs. (D29) and (D30). According to the theory, Eqs. (D33) and (D34), the above functions should be approximately equal. The plot shows that the curves nicely overlap, demonstrating that the observed differences between FOEs of different states match precisely our predictions of order 1 corrections. The parameters of the model are the same as in Fig. 7.

Figure 9

Observational entropy of measuring energy with resolution $\mathrm{\Delta }$ and position with resolution $\delta$ for a system of length $L=16$. The system starts contained within the first eight sites with hard wall boundary conditions, in energy eigenstate number 11 of Hamiltonian ${\widehat{H}}^{\left(1\text{–}8\right)}$. At $t=0$ the right wall is expanded so that $L=16$ and the system evolves. We study the integrable system, defined by parameters $t=V=1$ and $t^{\prime }=V^{\prime }=0.0$. The resolution in position is four sites corresponding to $\delta =4$. The resolution in energy is $\mathrm{\Delta }=\frac{E_{max}-E_{min}}{M}$, where $E_{max}$ and $E_{min}$ are maximum and minimum eigenvalues of the Hamiltonian, and $M$ is the number of energy bins. The three lines correspond to different resolutions in measuring energy: $M=1$ (red dashed), $M=8$ (green half-dashed), and $M=64$ (blue dotted). $M=1$ represents an inability to measure energy, and the resulting entropy is then observational entropy coarse grained only in position.