Operator space entanglement entropy in a transverse Ising chain

The efficiency of time-dependent density matrix renormalization group methods is intrinsically connected to the rate of entanglement growth. We introduce a measure of entanglement in the space of operators and show, for a transverse Ising spin-$1∕2$ chain, that the simulation of observables, contrary to the simulation of typical pure quantum states, is efficient for initial local operators. For initial operators with a finite index in Majorana representation, the operator space entanglement entropy saturates with time to a level which is calculated analytically, while for initial operators with infinite index the growth of operator space entanglement entropy is shown to be logarithmic.