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2022

Universal relation for operator complexity

Fan, Zhong-Ying

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We study Krylov complexity $C_K$ and operator entropy $S_K$ in operator growth.

We find that for a variety of systems, including chaotic ones and integrable theories, the two quantities always enjoy a logarithmic relation $S_K\sim \log{C_K}$ at long times, where dissipative behavior emerges in unitary evolution.
Otherwise, the relation does not hold any longer.
Universality of the relation is deeply connected to irreversibility of operator growth.

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Built on

J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling,
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M. Howard Lee, Ergodic Theory, Infinite Products, and Long Time Behavior in Hermitian Models
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S. K. Jian, B. Swingle and Z. Y. Xian, Complexity growth of operators in the SYK model and in JT gravity,
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2008
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VS Viswanath and Gerhard Müller, The Recursion Method: Applications to Many-body Dynamics
2008
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E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Operator complexity: a journey to the edge of Krylov space,
Original
2009
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