Universal Spreading of Conditional Mutual Information in Noisy Random Circuits
Abstract: We study the evolution of conditional mutual information in generic open quantum systems, focusing on one-dimensional random circuits with interspersed local noise. Unlike in noiseless circuits, where conditional mutual information spreads linearly while being bounded by the lightcone, we find that noisy random circuits with an error rate $p$ exhibit superlinear propagation of conditional mutual information, which diverges far beyond the lightcone at a critical circuit depth $t_c \propto p{-1}$. We demonstrate that the underlying mechanism for such rapid spreading is the combined effect of local noise and a scrambling unitary, which selectively removes short-range correlations while preserving long-range correlations. To analytically capture the dynamics of conditional mutual information in noisy random circuits, we introduce a coarse-graining method, and we validate our theoretical results through numerical simulations. Furthermore, we identify a universal scaling law governing the spreading of conditional mutual information.
- J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
- M. Srednicki, Phys. Rev. E 50, 888 (1994).
- P. Hayden and J. Preskill, J. High Energy Phys. 2007, 120 (2007).
- Y. Sekino and L. Susskind, J. High Energy Phys. 2008, 065 (2008).
- W. Brown and O. Fawzi, in 2013 IEEE International Symposium on Information Theory (2013) pp. 346–350.
- W. Brown and O. Fawzi, Commun. Math. Phys. 340, 867 (2015).
- T. Schuster and N. Y. Yao, Phys. Rev. Lett. 131, 160402 (2023).
- M. B. Hastings, Phys. Rev. B 76, 201102 (2007).
- M. S. Leifer and D. Poulin, Annals of Physics 323, 1899 (2008).
- D. Poulin and M. B. Hastings, Phys. Rev. Lett. 106, 080403 (2011).
- F. G. S. L. Brandão and M. J. Kastoryano, Commun. Math. Phys. 365, 1 (2019).
- K. Kato and F. G. S. L. Brandão, Commun. Math. Phys. 370, 117 (2019).
- A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
- M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
- I. H. Kim, Conditional independence in quantum many-body systems, Ph.D. thesis, California Institute of Technology (2013).
- M. Christandl and A. Winter, Journal of Mathematical Physics 45, 829 (2004).
- K. P. Seshadreesan and M. M. Wilde, Phys. Rev. A 92, 042321 (2015).
- O. Fawzi and R. Renner, Commun. Math. Phys. 340, 575 (2015).
- D. Petz, Rev. Math. Phys. 15, 79 (2003).
- J. Preskill, “Lecture notes for physics 219: Quantum Computation,” (2018).
- B. Yoshida and A. Kitaev, “Efficient decoding for the Hayden-Preskill protocol,” (2017), arXiv:1710.03363 .
- B. Yoshida, “Recovery algorithms for Clifford Hayden-Preskill problem,” (2022), arXiv:2106.15628 .
- Note that when we depolarize one of the qubits of a Bell pair 12(|00⟩+|11⟩)12ket00ket11\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | start_ARG 00 end_ARG ⟩ + | start_ARG 11 end_ARG ⟩ ), both qubits are depolarized.
- D. Gottesman, “The Heisenberg Representation of Quantum Computers,” (1998), arXiv:quant-ph/9807006 .
- S. Aaronson and D. Gottesman, Phys. Rev. A 70, 052328 (2004).
- Z. Webb, Quantum Info. Comput. 16, 1379 (2016).
- H. Zhu, Phys. Rev. A 96, 062336 (2017).
- For the numerical results with the finite coarse-graining factor m𝑚mitalic_m, Inorm(A:C|B){I^{\mathrm{norm}}(A:C|B)}italic_I start_POSTSUPERSCRIPT roman_norm end_POSTSUPERSCRIPT ( italic_A : italic_C | italic_B ) has an exponential tail after the initial linear decay. We choose xdecsubscript𝑥decx_{\mathrm{dec}}italic_x start_POSTSUBSCRIPT roman_dec end_POSTSUBSCRIPT based on the linear decay. We also truncate the distorted portions by the boundary. See [38] for the details.
- E. H. Lieb and D. W. Robinson, Commun.Math. Phys. 28, 251 (1972).
- L. Zou and J. Haah, Phys. Rev. B 94, 075151 (2016).
- K. Kato and F. G. S. L. Brandão, Phys. Rev. Research 2, 032005 (2020).
- Y. Zhang and S. Gopalakrishnan, “Nonlocal growth of quantum conditional mutual information under decoherence,” (2024), arXiv:2402.03439 .
- K. M. R. Audenaert and M. B. Plenio, New J. Phys. 7, 170 (2005).
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).
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