Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems $—$ i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears $—$ purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.
 Y. Li, X. Chen, and M. P. A. Fisher, ``Quantum zeno effect and the many-body entanglement transition,'' Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134.
 B. Skinner, J. Ruhman, and A. Nahum, ``Measurement-induced phase transitions in the dynamics of entanglement,'' Phys. Rev. X 9, 031009 (2019), arXiv:1808.05953.
 M. J. Gullans and D. A. Huse, ``Dynamical purification phase transitions induced by quantum measurements,'' Phys. Rev. X 10, 041020 (2020a), arXiv:1905.05195.
 S. Choi, Y. Bao, X.-L. Qi, and E. Altman, ``Quantum error correction in scrambling dynamics and measurement-induced phase transition,'' Phys. Rev. Lett. 125, 030505 (2019), arXiv:1903.05124.
 F. G. Brandao, A. W. Harrow, and M. Horodecki, ``Local random quantum circuits are approximate polynomial-designs,'' Commun. Math. Phys. 346, 397–434 (2016), arXiv:1208.0692.
 J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, ``Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-clifford gates,'' (2020), arXiv:2002.09524.
 M. J. Gullans and D. A. Huse, ``Scalable probes of measurement-induced criticality,'' Phys. Rev. Lett. 125, 070606 (2020) 125, 070606 (2020b), arXiv:1910.00020.
 X. Cao, A. Tilloy, and A. D. Luca, ``Entanglement in a fermion chain under continuous monitoring,'' SciPost Phys. 7, 24 (2019), arXiv:1804.04638.
 X. Chen, Y. Li, M. P. A. Fisher, and A. Lucas, ``Emergent conformal symmetry in nonunitary random dynamics of free fermions,'' Phys. Rev. Research 2, 033017 (2020), arXiv:2004.09577.
 A. Nahum and B. Skinner, ``Entanglement and dynamics of diffusion-annihilation processes with majorana defects,'' Phys. Rev. Research 2, 023288 (2020), arXiv:1911.11169.
 E. S. Meckes, The random matrix theory of the classical compact groups, Vol. 218 (Cambridge University Press, 2019).
 K. M. R. Audenaert, ``A sharp fannes-type inequality for the von neumann entropy,'' J. Phys. A 40, 8127–8136 (2007), quant-ph/0610146.
 B. Collins and P. Sniady, ``Integration with respect to the Haar measure on unitary, orthogonal and symplectic group,'' Commun. Math. Phys. 264, 773–795 (2006), arXiv:math-ph/0402073.
 Matteo Ippoliti and Vedika Khemani, "Postselection-Free Entanglement Dynamics via Spacetime Duality", Physical Review Letters 126 6, 060501 (2021).
 Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani, "Entanglement Phase Transitions in Measurement-Only Dynamics", Physical Review X 11 1, 011030 (2021).
 Adam Nahum, Sthitadhi Roy, Brian Skinner, and Jonathan Ruhman, "Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory", arXiv:2009.11311.
 Michael J. Gullans, Stefan Krastanov, David A. Huse, Liang Jiang, and Steven T. Flammia, "Quantum coding with low-depth random circuits", arXiv:2010.09775.
 Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli, "Topological order and criticality in (2+1)D monitored random quantum circuits", arXiv:2011.06595.
 Sarang Gopalakrishnan and Michael J. Gullans, "Entanglement and purification transitions in non-Hermitian quantum mechanics", arXiv:2012.01435.
 Chao-Ming Jian, Bela Bauer, Anna Keselman, and Andreas W. W. Ludwig, "Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions", arXiv:2012.04666.
 Oliver Lunt, Marcin Szyniszewski, and Arijeet Pal, "Dimensional hybridity in measurement-induced criticality", arXiv:2012.03857.
 Shengqi Sang, Yaodong Li, Tianci Zhou, Xiao Chen, Timothy H. Hsieh, and Matthew P. A. Fisher, "Entanglement Negativity at Measurement-Induced Criticality", arXiv:2012.00031.
 Yimu Bao, Soonwon Choi, and Ehud Altman, "Symmetry enriched phases of quantum circuits", arXiv:2102.09164.
The above citations are from Crossref's cited-by service (last updated successfully 2021-03-04 00:57:06) and SAO/NASA ADS (last updated successfully 2021-03-04 00:57:08). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.