Thermal equilibrium emerging in a subsystem of a pure ground state by quantum entanglement

Abstract

By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters, we demonstrate that quantum entanglement between subsystems $A$ and $B$ in a pure ground state of a whole system $A+B$ can induce thermal equilibrium in subsystem $A$. Temperature ${\cal T}_{A}$ of subsystem $A$ is $not$ a parameter but can be determined from the entanglement von Neumann entropy $\mathcal{S}_{A}$ and the total energy $\mathcal{E}_{A}$ of subsystem $A$ calculated for the ground state of the whole system. Temperature ${\cal T}_{A}$ is essentially identical to the thermodynamic temperature, for which the entropy and the internal energy evaluated using the canonical ensemble in statistical mechanics for the isolated subsystem $A$ are almost indistinguishable numerically from the entanglement entropy $\mathcal{S}_{A}$ and the total energy $\mathcal{E}_{A}$ of subsystem $A$. Fidelity calculations ascertain that the reduced density matrix operator of subsystem $A$ for the pure but entangled ground state of the whole system $A+B$ is almost identical to the Gibbs state (i.e., thermodynamic density matrix operator) of subsystem $A$ at temperature ${\cal T}_{A}$. We argue that quantum fluctuation in an entangled pure state can mimic thermal fluctuation in a subsystem. We also provide two simple but nontrivial examples of free bosons and free fermions for which these statements can be exactly proved analytically. We furthermore discuss implications and possible applications of our finding.