In one dimension, the area law and its implications for the approximability by Matrix Product States are the key to efficient numerical simulations involving quantum states. Similarly, in simulations involving quantum operators, the approximability by Matrix Product Operators (in Hilbert-Schmidt norm) is tied to an operator area law, namely the fact that the Operator Space Entanglement Entropy (OSEE)---the natural analog of entanglement entropy for operators, investigated by Zanardi [Phys. Rev. A 63, 040304(R) (2001)] and by Prosen and Pizorn [Phys. Rev. A 76, 032316 (2007)]---, is bounded. In the present paper, it is shown that the OSEE can be calculated in two-dimensional conformal field theory, in a number of situations that are relevant to questions 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 $rho propto e^{- H_{rm GGE}}$ with local $H_{rm GGE}$ generically obey the operator area law; (ii) after a global quench, the OSEE first grows linearly with time, then decreases back to its thermal or GGE saturation value, implying that, while the operator area law is satisfied both in the initial state and in the asymptotic stationary state at large time, it is strongly violated in the transient regime; (iii) the OSEE of the evolution operator $U(t) = e^{-i H t}$ increases linearly with $t$, unless the Hamiltonian is in a localized phase; (iv) local operators in Heisenberg picture, $phi(t) = e^{i H t} phi e^{-i H t}$, have an OSEE that grows sublinearly in time (perhaps logarithmically), however it is unclear whether this effect can be captured in a traditional CFT framework, as the free fermion case hints at an unexpected breakdown of conformal invariance.
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