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Notes on complexity growth rate, grand potential and partition function

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Abstract

We examine the complexity/volume conjecture and further investigate the possible connections between complexity and partition function. The complexity/partition function relation is then utilized to study the complexity of the thermofield double state of extended SYK models for various conditions.The difference between the complexity/partition function relation with the complexity/action duality comes from the fact that partition function can be evaluated from more than one saddle point, so they differ at most at the non-perturbative level. We further analyze free energy and the growth rate of complexity in the neutral AdS-Vaidya black hole formed by collapsing an uncharged spherically symmetric thin shell of null fluid. The relation between the late-time complexity growth rate and black hole/SYK wormhole phase transition is also discussed. Finally, we check the Lloyd bound with our proposal.

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Notes

  1. This in turn implies \(d {\dot{C}}\ge 0\). The variation of the complex growth rate then has a property similar to entropy. Actually, the second law of complexity states [44]: If the computational complexity is less than maximum, then with overwhelming likelihood it will increase, both into the future and into the past.

  2. Note that complexity is related to the action via the CA duality. One can prove the equivalence of the CA duality and our proposal by noting that \(dC/dt \sim T \ln {\mathcal {Z}}\) and in the Euclidean signature, one has \(t \sim 1/T\), so that \(C \sim A\). The action A in the CA duality can be regarded as the quantum effective action in the saddle. However, \(\ln {\mathcal {Z}}\) can receive contribution from more one saddles. The difference between our proposal and the CA duality only at the non-perturbative level.

  3. In Refs. [71, 72, 74], the authors proved that in the case of light shockwave, the switchback effect will be reflected in the full-time evolution of complexity.

  4. Since time is not an operator in quantum mechanics, the energy-time uncertainty relationship does not mean that time must be uncertain, but it must take time to evolve to an orthogonal state.

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Acknowledgements

We would like to thank Hong Lü, Song He, Keun-Young Kim, Runqiu Yang and Qingbing Wang for helpful discussions. This work is partly supported by NSFC (No. 11875184 & No. 11805117).

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  1. Department of Physics, College of Sciences, Shanghai University, Shanghai, 200444, People’s Republic of China

    Wei Sun

  2. Shanghai Key Laboratory of High Temperature Superconductors, Shanghai University, Shanghai, 200444, People’s Republic of China

    Xian-Hui Ge

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Sun, W., Ge, XH. Notes on complexity growth rate, grand potential and partition function. Gen Relativ Gravit 54, 46 (2022). https://doi.org/10.1007/s10714-022-02933-4

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