Abstract
Using spread complexity and spread entropy, we study non-unitary quantum dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos construction for the Krylov basis to the Schrödinger picture. Moreover, we implement an algorithm adapted to complex symmetric Hamiltonians. This reduces the computational memory requirements by half compared to the bi-Lanczos construction. We apply this construction to the one-dimensional tight-binding Hamiltonian subject to repeated measurements at fixed small time intervals, resulting in effective non-unitary dynamics. We find that the spread complexity initially grows with time, followed by an extended decay period and saturation. The choice of initial state determines the saturation value of complexity and entropy. In analogy to measurement-induced phase transitions, we consider a quench between hermitian and non-hermitian Hamiltonian evolution induced by turning on regular measurements at different frequencies. We find that as a function of the measurement frequency, the time at which the spread complexity starts growing increases. This time asymptotes to infinity when the time gap between measurements is taken to zero, indicating the onset of the quantum Zeno effect, according to which measurements impede time evolution.
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Acknowledgments
The authors would like to thank Vijay Balasubramanian, Souvik Banerjee, Pablo Basteiro, Pawel Caputa, Mario Flory, Giuseppe Di Giulio, Taishi Kawamoto, René Meyer, Sara Murciano, Pratik Nandy, Pingal Pratyush Nath, Subir Sachdev, Himanshu Sahu, Aninda Sinha, Zhuo-Yu Xian, Kunal Pal, Kuntal Pal, and Diptarka Das for useful discussions and comments on related topics at various stages. The work of A.B. is supported by the Polish National Science Centre (NCN) grant 2021/42/E/ST2/00234. R.N.D. and J.E. are supported by Germany’s Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 390858490), and by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research centre “ToCoTronics”, Project-ID 258499086—SFB 1170. R.N.D. further acknowledges the support by the Deutscher Akademischer Austauschdienst (DAAD, German Academic Exchange Service) through the funding programme, “Research Grants - Doctoral Programmes in Germany, 2021/22 (57552340)”. This research was also supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science. B.D. acknowledges MHRD, India for Research Fellowship. B.D. would also like to acknowledge the support provided by Max Planck Partner Group grant MAXPLA/PHY/2018577. B.D. would further like to acknowledge the support provided by the MATRICS grant SERB/PHY/2020334.
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Bhattacharya, A., Das, R.N., Dey, B. et al. Spread complexity for measurement-induced non-unitary dynamics and Zeno effect. J. High Energ. Phys. 2024, 179 (2024). https://doi.org/10.1007/JHEP03(2024)179
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DOI: https://doi.org/10.1007/JHEP03(2024)179