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In one dimension, the area law and its implications for the approximabilityby Matrix Product States are the key to efficient numerical simulationsinvolving quantum states. Similarly, in simulations involving quantumoperators, the approximability by Matrix Product Operators (in Hilbert-Schmidtnorm) is tied to an operator area law, namely the fact that the Operator SpaceEntanglement Entropy (OSEE)---the natural analog of entanglement entropy foroperators, investigated by Zanardi [Phys. Rev. A 63, 040304(R) (2001)] and byProsen and Pizorn [Phys. Rev. A 76, 032316 (2007)]---, is bounded. In thepresent paper, it is shown that the OSEE can be calculated in two-dimensionalconformal field theory, in a number of situations that are relevant toquestions of simulability of long-time dynamics in one spatial dimension.  It is argued that: (i) thermal density matrices $\rho \propto e^{-\beta H}$and Generalized Gibbs Ensemble density matrices $\rho \propto e^{- H_{\rmGGE}}$ with local $H_{\rm GGE}$ generically obey the operator area law; (ii)after a global quench, the OSEE first grows linearly with time, then decreasesback to its thermal or GGE saturation value, implying that, while the operatorarea law is satisfied both in the initial state and in the asymptoticstationary state at large time, it is strongly violated in the transientregime; (iii) the OSEE of the evolution operator $U(t) = e^{-i H t}$ increaseslinearly with $t$, unless the Hamiltonian is in a localized phase; (iv) localoperators in Heisenberg picture, $\phi(t) = e^{i H t} \phi e^{-i H t}$, have anOSEE that grows sublinearly in time (perhaps logarithmically), however it isunclear whether this effect can be captured in a traditional CFT framework, asthe free fermion case hints at an unexpected breakdown of conformal invariance.

Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1+1d