Second law of thermodynamics in closed quantum many-body systems

Abstract

The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.

Figures (2)

• Figure 1: Illustration of how each primitive macroscopic subsystem grows as $L$ is increased ($L=8$, $16$, $32$ from left to right) in the case of $d=2$ and $K=4$. We can define a macroscopic subsystem $\mathcal{S}_L$ as a union of certain primitive macroscopic subsystems. For example, we can consider a proper macroscopic subsystem corresponding to the shaded region as a union of $\mathcal{S}_L^{(6)},\mathcal{S}_L^{(7)},$ and $\mathcal{S}_L^{(10)}$.
• Figure 2: Locality $(\ell)$ dependence of $S_{\mathrm{vN}}[\rho_{L|\ell}^{\mathrm{tot}}]/\ell^d$ (green filled circles) and the entanglement entropy density $S_{\mathrm{vN}}[\rho_{L|\ell}^{\bm{r}=\bm{0}}]/\ell^d$ (purple square) in the one-dimensional transverse-field Ising model at $J=g=1$ and $\beta=1$ with $L=512$ (see text for details). Both quantities are obtained from $10^5$ METTS samples, where the error bars represent the standard deviation. The light-blue line indicates the exact thermodynamic entropy density in the thermodynamic limit. The inset shows the $L$ dependence of the standard deviations at fixed $\ell=8$ for both entropy densities. Imaginary-time evolution is carried out using a second-order Trotter decomposition with step size $\delta\beta=0.01$ and truncation threshold $10^{-14}$.

Theorems & Definitions (90)

• Definition 1: Proper sequences of macroscopic subsystems
• Definition 2: Local observables
• Definition 3: Additive observables
• Definition 4: Proper sequences of additive observables
• Definition 5: Macroscopic state
• Definition 6: Macroscopic equivalence
• Definition 7: Normal macroscopic state
• Definition 8: iMATE
• Definition 9: Normal iMATE
• Example 1: Typical sequence of METTS represents iMATE, informal version of Proposition \ref{['proposition:METTS_Justification_Decay_pL']}