On the evolution of operator complexity beyond scrambling
- 26 Downloads
We study operator complexity on various time scales with emphasis on those much larger than the scrambling period. We use, for systems with a large but finite number of degrees of freedom, the notion of K-complexity employed in  for infinite systems. We present evidence that K-complexity of ETH operators has indeed the character associated with the bulk time evolution of extremal volumes and actions. Namely, after a period of exponential growth during the scrambling period the K-complexity increases only linearly with time for exponentially long times in terms of the entropy, and it eventually saturates at a constant value also exponential in terms of the entropy. This constant value depends on the Hamiltonian and the operator but not on any extrinsic tolerance parameter. Thus K-complexity deserves to be an entry in the AdS/CFT dictionary. Invoking a concept of K-entropy and some numerical examples we also discuss the extent to which the long period of linear complexity growth entails an efficient randomization of operators.
KeywordsAdS-CFT Correspondence Random Systems
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
- M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000).Google Scholar
- M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
- V.S. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Physics, Springer Verlag (1994).Google Scholar
- A. Peres, Ergodicity and mixing in quantum theory. I, Phys. Rev. A 30 (1984) 504 [INSPIRE].
- M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051].
- M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A 32 (1999) 1163 [cond-mat/9809360].
- I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, (2000).Google Scholar