Emergence of a thermal equilibrium in a subsystem of a pure ground state by quantum entanglement

By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters up to 24 sites, 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. Here, the whole system is bipartitoned with the entanglement cut that covers the entire volume of subsystem A. An effective temperature T-A of subsystem A induced by quantum entanglement is not a parameter but can be determined from the entanglement von Neumann entropy S-A and the total energy E-A of subsystem A calculated for the ground state of the whole system. We show that temperature T-A can be derived by minimizing the relative entropy for the reduced density matrix operator of subsystem A and the Gibbs state (i.e., thermodynamic density matrix operator) of subsystem A with respect to the coupling strength between subsystems A and B. Temperature 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 agree numerically with the entanglement entropy S-A and the total energy 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 matches, within a maximally 1.5% error in the finite size clusters studied, the thermodynamic density matrix operator of subsystem A at temperature T-A, despite that these density-matrix operators are different in general. We also find that temperature T-A evaluated from the ground state of the whole system depends insignificantly on the system sizes, which is consistent with the fact that the thermodynamic temperature is an intensive quantity. We argue that quantum fluctuation in an entangled pure state can mimic thermal fluctuation in a subsystem. We also provide two simple but nontrivial analytical examples of free bosons and free fermions for which the two density-matrix operators are exactly the same if the effective temperature T-A is adopted. We furthermore discuss implications and possible applications of our finding.