quantum-computing

Measuring Rényi entropy with an Echo Protocol

Yi-Neng Zhou, Robin Löwenberg, and Julian Sonner

28 min read

0 likes

⚡ Quantum Brief

AI Audio Summary

0:00 / 0:00

Click to play

Quantum News · Media Library

Understand this faster with AI

Ask ChatGPT

Research with Perplexity

Explain with Claude

Discuss on Grok

AbstractWe present efficient and practical protocols to measure the second Rényi entropy, whose exponential is known as the purity. Our approach is based on expressing the purity in terms of transition probabilities generated by an echo-type forward-backward evolution sequence, making it applicable to quantum many-body systems. Notably, our approach does not rely on random-noise averaging, a feature that can be extended to protocols to measure out-of-time-order correlation functions, as we demonstrate. By way of example, we show that our protocols can be practically implemented in superconducting qubit-based platforms, as well as in cavity-QED trapped ultra-cold gases.► BibTeX data@article{Zhou2026measuringrenyi, doi = {10.22331/q-2026-06-30-2146}, url = {https://doi.org/10.22331/q-2026-06-30-2146}, title = {Measuring {R}{\'{e}}nyi entropy with an {E}cho {P}rotocol}, author = {Zhou, Yi-Neng and L{\"{o}}wenberg, Robin and Sonner, Julian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2146}, month = jun, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Reviews of Modern Physics 81, 865–942 (2009). https://doi.org/10.1103/revmodphys.81.865 [2] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. ``Entanglement in many-body systems''. Rev. Mod. Phys. 80, 517–576 (2008). https://doi.org/10.1103/RevModPhys.80.517 [3] J. Eisert, M. Cramer, and M. B. Plenio. ``Colloquium: Area laws for the entanglement entropy''. Reviews of Modern Physics 82, 277–306 (2010). https://doi.org/10.1103/revmodphys.82.277 [4] Nicolas Laflorencie. ``Quantum entanglement in condensed matter systems''. Physics Reports 646, 1–59 (2016). https://doi.org/10.1016/j.physrep.2016.06.008 [5] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium : Many-body localization, thermalization, and entanglement''. Reviews of Modern Physics 91 (2019). https://doi.org/10.1103/revmodphys.91.021001 [6] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''.

Cambridge University Press. Cambridge (2010). https://doi.org/10.1017/CBO9780511976667 [7] Martin B. Plenio and Shashank S. Virmani. ``An introduction to entanglement measures''. Quantum Information & Computation 7, 001–051 (2007). arXiv:quant-ph/0504163. https://doi.org/10.26421/QIC7.1-2-1 arXiv:quant-ph/0504163 [8] Pasquale Calabrese and John Cardy. ``Entanglement entropy and quantum field theory''. Journal of Statistical Mechanics: Theory and Experiment 2004, P06002 (2004). https://doi.org/10.1088/1742-5468/2004/06/p06002 [9] Shinsei Ryu and Tadashi Takayanagi. ``Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence''.

Physical Review Letters 96 (2006). https://doi.org/10.1103/physrevlett.96.181602 [10] Pasquale Calabrese and John Cardy. ``Entanglement entropy and conformal field theory''. Journal of Physics A: Mathematical and Theoretical 42, 504005 (2009). https://doi.org/10.1088/1751-8113/42/50/504005 [11] Tatsuma Nishioka. ``Entanglement entropy: Holography and renormalization group''. Rev. Mod. Phys. 90, 035007 (2018). https://doi.org/10.1103/RevModPhys.90.035007 [12] Tobias J. Osborne and Michael A. Nielsen. ``Entanglement in a simple quantum phase transition''. Physical Review A 66 (2002). https://doi.org/10.1103/physreva.66.032110 [13] A. Osterloh, Luigi Amico, G. Falci, and Rosario Fazio. ``Scaling of entanglement close to a quantum phase transition''. Nature 416, 608–610 (2002). https://doi.org/10.1038/416608a [14] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev. ``Entanglement in quantum critical phenomena''.

Physical Review Letters 90 (2003). https://doi.org/10.1103/physrevlett.90.227902 [15] Jens H. Bardarson, Frank Pollmann, and Joel E. Moore. ``Unbounded growth of entanglement in models of many-body localization''.

Physical Review Letters 109 (2012). https://doi.org/10.1103/physrevlett.109.017202 [16] A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller. ``Measuring entanglement growth in quench dynamics of bosons in an optical lattice''.

Physical Review Letters 109 (2012). https://doi.org/10.1103/physrevlett.109.020505 [17] Norbert Schuch, Michael M. Wolf, Frank Verstraete, and J. Ignacio Cirac. ``Entropy scaling and simulability by matrix product states''.

Physical Review Letters 100 (2008). https://doi.org/10.1103/physrevlett.100.030504 [18] Alexei Kitaev and John Preskill. ``Topological entanglement entropy''.

Physical Review Letters 96 (2006). https://doi.org/10.1103/physrevlett.96.110404 [19] Michael Levin and Xiao-Gang Wen. ``Detecting topological order in a ground state wave function''.

Physical Review Letters 96 (2006). https://doi.org/10.1103/physrevlett.96.110405 [20] Hong-Chen Jiang, Zhenghan Wang, and Leon Balents. ``Identifying topological order by entanglement entropy''. Nature Physics 8, 902–905 (2012). https://doi.org/10.1038/nphys2465 [21] Ruihua Fan, Pengfei Zhang, Huitao Shen, and Hui Zhai. ``Out-of-time-order correlation for many-body localization''. Science Bulletin 62, 707–711 (2017). https://doi.org/10.1016/j.scib.2017.04.011 [22] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford. ``A bound on chaos''. Journal of High Energy Physics 2016 (2016). https://doi.org/10.1007/jhep08(2016)106 [23] Stephen H. Shenker and Douglas Stanford. ``Black holes and the butterfly effect''. Journal of High Energy Physics 2014 (2014). https://doi.org/10.1007/jhep03(2014)067 [24] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. ``Measuring entanglement entropy in a quantum many-body system''. Nature 528, 77–83 (2015). https://doi.org/10.1038/nature15750 [25] Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, and Markus Greiner. ``Quantum thermalization through entanglement in an isolated many-body system''. Science 353, 794–800 (2016). https://doi.org/10.1126/science.aaf6725 [26] Zhi-Kang Lin, Yao Zhou, Bin Jiang, Bing-Quan Wu, Li-Mei Chen, Xiao-Yu Liu, Li-Wei Wang, Peng Ye, and Jian-Hua Jiang. ``Measuring entanglement entropy and its topological signature for phononic systems''. Nature Communications 15 (2024). https://doi.org/10.1038/s41467-024-45887-8 [27] N. M. Linke, S. Johri, C. Figgatt, K. A. Landsman, A. Y. Matsuura, and C. Monroe. ``Measuring the rényi entropy of a two-site fermi-hubbard model on a trapped ion quantum computer''. Physical Review A 98 (2018). https://doi.org/10.1103/physreva.98.052334 [28] Tiff Brydges, Andreas Elben, Petar Jurcevic, Benoı̂t Vermersch, Christine Maier, Ben P. Lanyon, Peter Zoller, Rainer Blatt, and Christian F. Roos. ``Probing rényi entanglement entropy via randomized measurements''. Science 364, 260–263 (2019). https://doi.org/10.1126/science.aau4963 [29] Mohammadamin Tajik, Ivan Kukuljan, Spyros Sotiriadis, Bernhard Rauer, Thomas Schweigler, Federica Cataldini, João Sabino, Frederik Møller, Philipp Schüttelkopf, Si-Cong Ji, Dries Sels, Eugene Demler, and Jörg Schmiedmayer. ``Verification of the area law of mutual information in a quantum field simulator''. Nature Physics 19 (2023). https://doi.org/10.1038/s41567-023-02027-1 [30] Jochen Braumüller, Amir H. Karamlou, Yariv Yanay, Bharath Kannan, David Kim, Morten Kjaergaard, Alexander Melville, Bethany M. Niedzielski, Youngkyu Sung, Antti Vepsäläinen, Roni Winik, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, Charles Tahan, and William D. Oliver. ``Probing quantum information propagation with out-of-time-ordered correlators''. Nature Physics 18, 172–178 (2021). https://doi.org/10.1038/s41567-021-01430-w [31] Jun Li, Ruihua Fan, Hengyan Wang, Bingtian Ye, Bei Zeng, Hui Zhai, Xinhua Peng, and Jiangfeng Du. ``Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator''. Physical Review X 7 (2017). https://doi.org/10.1103/physrevx.7.031011 [32] Geoffrey Penington. ``Entanglement Wedge Reconstruction and the Information Paradox''. JHEP 09, 002 (2020). arXiv:1905.08255. https://doi.org/10.1007/JHEP09(2020)002 arXiv:1905.08255 [33] Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield. ``The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole''. JHEP 12, 063 (2019). arXiv:1905.08762. https://doi.org/10.1007/JHEP12(2019)063 arXiv:1905.08762 [34] Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. ``The entropy of Hawking radiation''. Rev. Mod. Phys. 93, 035002 (2021). arXiv:2006.06872. https://doi.org/10.1103/RevModPhys.93.035002 arXiv:2006.06872 [35] Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. ``Replica Wormholes and the Entropy of Hawking Radiation''. JHEP 05, 013 (2020). arXiv:1911.12333. https://doi.org/10.1007/JHEP05(2020)013 arXiv:1911.12333 [36] Asher Peres. ``Stability of quantum motion in chaotic and regular systems''. Phys. Rev. A 30, 1610–1615 (1984). https://doi.org/10.1103/PhysRevA.30.1610 [37] H.M Pastawski, P.R Levstein, G Usaj, J Raya, and J Hirschinger. ``A nuclear magnetic resonance answer to the boltzmann–loschmidt controversy?''. Physica A: Statistical Mechanics and its Applications 283, 166–170 (2000). https://doi.org/10.1016/S0378-4371(00)00146-1 [38] Rodolfo A. Jalabert and Horacio M. Pastawski. ``Environment-independent decoherence rate in classically chaotic systems''. Phys. Rev. Lett. 86, 2490–2493 (2001). https://doi.org/10.1103/PhysRevLett.86.2490 [39] Thomas Gorin, Tomaž Prosen, Thomas H. Seligman, and Marko Žnidarič. ``Dynamics of loschmidt echoes and fidelity decay''. Physics Reports 435, 33–156 (2006). https://doi.org/10.1016/j.physrep.2006.09.003 [40] Ph. Jacquod and C. Petitjean. ``Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom''. Advances in Physics 58, 67–196 (2009). arXiv:https://doi.org/10.1080/00018730902831009. https://doi.org/10.1080/00018730902831009 arXiv:https://doi.org/10.1080/00018730902831009 [41] A. Goussev, R. A. Jalabert, H. M. Pastawski, and D. Ariel Wisniacki. ``Loschmidt echo''. Scholarpedia 7, 11687 (2012). https://doi.org/10.4249/scholarpedia.11687 [42] Zhiyan Ding, Yongtao Zhan, John Preskill, and Lin Lin. ``End-to-end efficient quantum thermal and ground state preparation made simple'' (2025). arXiv:2508.05703. arXiv:2508.05703 [43] Xinfang Nie, Ze Zhang, Xiuzhu Zhao, Tao Xin, Dawei Lu, and Jun Li. ``Detecting scrambling via statistical correlations between randomized measurements on an nmr quantum simulator'' (2019). arXiv:1903.12237. arXiv:1903.12237 [44] Manoj K. Joshi, Andreas Elben, Benoît Vermersch, Tiff Brydges, Christine Maier, Peter Zoller, Rainer Blatt, and Christian F. Roos. ``Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions''.

Physical Review Letters 124 (2020). https://doi.org/10.1103/physrevlett.124.240505 [45] Bin Yan, Lukasz Cincio, and Wojciech H. Zurek. ``Information scrambling and loschmidt echo''. Phys. Rev. Lett. 124, 160603 (2020). https://doi.org/10.1103/PhysRevLett.124.160603 [46] Ruihua Fan, Pengfei Zhang, Huitao Shen, and Hui Zhai. ``Out-of-Time-Order Correlation for Many-Body Localization''. Science Bulletin 62, 707–711 (2017). https://doi.org/10.1016/j.scib.2017.04.011 [47] Antonio Anna Mele. ``Introduction to haar measure tools in quantum information: A beginner' s tutorial''. Quantum 8, 1340 (2024). https://doi.org/10.22331/q-2024-05-08-1340 [48] Yasuhiro Sekino and Leonard Susskind. ``Fast Scramblers''. Journal of High Energy Physics 2008, 065–065 (2008). https://doi.org/10.1088/1126-6708/2008/10/065 [49] Philipp Uhrich, Soumik Bandyopadhyay, Nick Sauerwein, Julian Sonner, Jean-Philippe Brantut, and Philipp Hauke. ``A cavity quantum electrodynamics implementation of the sachdev–ye–kitaev model'' (2023). arXiv:2303.11343. arXiv:2303.11343 [50] Rahel Baumgartner, Pietro Pelliconi, Soumik Bandyopadhyay, Francesca Orsi, Nick Sauerwein, Philipp Hauke, Jean-Philippe Brantut, and Julian Sonner. ``Quantum simulation of the sachdev-ye-kitaev model using time-dependent disorder in optical cavities'' (2025). arXiv:2411.17802. arXiv:2411.17802 [51] Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, Steven S. Gubser, and Monika Schleier-Smith. ``Treelike interactions and fast scrambling with cold atoms''.

Physical Review Letters 123, 130601 (2019). https://doi.org/10.1103/PhysRevLett.123.130601 [52] Avikar Periwal, Eric S. Cooper, Philipp Kunkel, Julian F. Wienand, Emily J. Davis, and Monika Schleier-Smith. ``Programmable interactions and emergent geometry in an array of atom clouds''. Nature 600, 630–635 (2021). https://doi.org/10.1038/s41586-021-04156-0 [53] Emily J. Davis, Gregory Bentsen, Lukas Homeier, Tracy Li, and Monika H. Schleier-Smith. ``Photon-mediated spin-exchange dynamics of spin-1 atoms''.

Physical Review Letters 122 (2019). https://doi.org/10.1103/physrevlett.122.010405 [54] Michał Karski, Leonid Förster, Jai-Min Choi, Andreas Steffen, Noomen Belmechri, Wolfgang Alt, Dieter Meschede, and Artur Widera. ``Imprinting patterns of neutral atoms in an optical lattice using magnetic resonance techniques''. New Journal of Physics 12, 065027 (2010). https://doi.org/10.1088/1367-2630/12/6/065027 [55] S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, and D. Meschede. ``Coherence properties and quantum state transportation in an optical conveyor belt''. Phys. Rev. Lett. 91, 213002 (2003). https://doi.org/10.1103/PhysRevLett.91.213002 [56] Steven S. Gubser, Johannes Knaute, Sarthak Parikh, Andreas Samberg, and Przemek Witaszczyk. ``p-Adic AdS/CFT''. Communications in Mathematical Physics 352, 1019–1059 (2017). https://doi.org/10.1007/s00220-016-2813-6 [57] Jordan Cotler, Soonwon Choi, Alexander Lukin, Hrant Gharibyan, Tarun Grover, M. Eric Tai, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, Adam M. Kaufman, Markus Greiner, Hannes Pichler, and Patrick Hayden. ``Quantum virtual cooling''. Phys. Rev. X 9, 031013 (2019). https://doi.org/10.1103/PhysRevX.9.031013 [58] Scott Aaronson. ``Shadow tomography of quantum states'' (2018). arXiv:1711.01053. arXiv:1711.01053 [59] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nature Physics 16, 1050–1057 (2020). https://doi.org/10.1038/s41567-020-0932-7 [60] Hsin-Yuan Huang, Michael Broughton, Jordan Cotler, Sitan Chen, Jerry Li, Masoud Mohseni, Hartmut Neven, Ryan Babbush, Richard Kueng, John Preskill, and Jarrod R. McClean. ``Quantum advantage in learning from experiments''. Science 376, 1182–1186 (2022). https://doi.org/10.1126/science.abn7293 [61] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You. ``Classical shadow tomography with locally scrambled quantum dynamics''. Phys. Rev. Res. 5, 023027 (2023). https://doi.org/10.1103/PhysRevResearch.5.023027 [62] Fernando G.S.L. Brandão, Wissam Chemissany, Nicholas Hunter-Jones, Richard Kueng, and John Preskill. ``Models of quantum complexity growth''. PRX Quantum 2, 030316 (2021). https://doi.org/10.1103/PRXQuantum.2.030316 [63] Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. ``Quantum Pseudoentanglement''. Leibniz Int. Proc. Inf. 287, 2:1–2:21 (2024). arXiv:2211.00747. https://doi.org/10.4230/LIPIcs.ITCS.2024.2 arXiv:2211.00747 [64] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, and Beni Yoshida. ``Chaos, complexity, and random matrices''. Journal of High Energy Physics 2017 (2017). https://doi.org/10.1007/jhep11(2017)048 [65] Robert de Mello Koch, Jia-Hui Huang, Chen-Te Ma, and Hendrik J.R. Van Zyl. ``Spectral form factor as an otoc averaged over the heisenberg group''. Physics Letters B 795, 183–187 (2019). https://doi.org/10.1016/j.physletb.2019.06.025 [66] Aurelio Romero-Bermúdez, Koenraad Schalm, and Vincenzo Scopelliti. ``Regularization dependence of the otoc. which lyapunov spectrum is the physical one?''. Journal of High Energy Physics 2019 (2019). https://doi.org/10.1007/jhep07(2019)107 [67] Jonah Kudler-Flam, Laimei Nie, and Shinsei Ryu. ``Conformal field theory and the web of quantum chaos diagnostics''. Journal of High Energy Physics 2020 (2020). https://doi.org/10.1007/jhep01(2020)175 [68] Daniele Toniolo and Sougato Bose. ``Dynamical ${\alpha}$-rényi entropies of local hamiltonians grow at most linearly in time''. Phys. Rev. X 15, 031046 (2025). https://doi.org/10.1103/ycdh-z8zf [69] Yi-Neng Zhou, Liang Mao, and Hui Zhai. ``Rényi entropy dynamics and lindblad spectrum for open quantum systems''.

Physical Review Research 3 (2021). https://doi.org/10.1103/physrevresearch.3.043060 [70] Yi-Neng Zhou and Chang Liu. ``Generalized Loschmidt echo and information scrambling in open systems''. SciPost Phys. 20, 072 (2026). https://doi.org/10.21468/SciPostPhys.20.3.072 [71] Yi-Neng Zhou, Tian-Gang Zhou, and Pengfei Zhang. ``General properties of the spectral form factor in open quantum systems''. Frontiers of Physics 19 (2024). https://doi.org/10.1007/s11467-024-1406-7 [72] A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller. ``Rényi entropies from random quenches in atomic hubbard and spin models''. Phys. Rev. Lett. 120, 050406 (2018). https://doi.org/10.1103/PhysRevLett.120.050406 [73] L. García-Álvarez, I. L. Egusquiza, L. Lamata, A. del Campo, J. Sonner, and E. Solano. ``Digital quantum simulation of minimal $\mathrm{AdS}/\mathrm{CFT}$''. Phys. Rev. Lett. 119, 040501 (2017). https://doi.org/10.1103/PhysRevLett.119.040501 [74] S. J. van Enk and C. W. J. Beenakker. ``Measuring $\mathrm{Tr}{{\rho}}^{n}$ on single copies of ${\rho}$ using random measurements''. Phys. Rev. Lett. 108, 110503 (2012). https://doi.org/10.1103/PhysRevLett.108.110503 [75] B. Vermersch, A. Elben, L.M. Sieberer, N.Y. Yao, and P. Zoller. ``Probing scrambling using statistical correlations between randomized measurements''. Physical Review X 9 (2019). https://doi.org/10.1103/physrevx.9.021061 [76] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller. ``Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states''. Physical Review A 99 (2019). https://doi.org/10.1103/physreva.99.052323 [77] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller. ``The randomized measurement toolbox''.

Nature Reviews Physics 5, 9–24 (2022). https://doi.org/10.1038/s42254-022-00535-2 [78] Christian Beck and Friedrich Schögl. ``Thermodynamics of chaotic systems: An introduction''.

Cambridge University Press. (1993). url: https://doi.org/10.1017/CBO9780511524585. https://doi.org/10.1017/CBO9780511524585 [79] T. Gorin, T. Prosen, T. H. Seligman, and M. Žnidarič. ``Dynamics of Loschmidt echoes and fidelity decay''. Physics Reports 435, 33–156 (2006). arXiv:quant-ph/0607050. https://doi.org/10.1016/j.physrep.2006.09.003 arXiv:quant-ph/0607050 [80] Martin Gärttner, Justin G. Bohnet, Arghavan Safavi-Naini, Michael L. Wall, John J. Bollinger, and Ana Maria Rey. ``Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet''. Nature Physics 13, 781–786 (2017). arXiv:1608.08938. https://doi.org/10.1038/nphys4119 arXiv:1608.08938 [81] Claudia Marina Sánchez, Ken Xuan Wei, Leandro Clemente, Richard Kueng, Horacio M. Pastawski, J. Ignacio Cirac, and Ana Karina Chattah. ``Perturbation independent decay of the Loschmidt echo in a many-body system''.

Physical Review Letters 124, 030601 (2020). https://doi.org/10.1103/PhysRevLett.124.030601 [82] Jochen Braumüller, Amir H. Karamlou, Yariv Yanay, Bharath Kannan, David Kim, Morten Kjaergaard, Alexander Melville, Bethany M. Niedzielski, Youngkyu Sung, Antti Vepsäläinen, Roni Winik, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, Charles Tahan, and William D. Oliver. ``Probing quantum information propagation with out-of-time-ordered correlators''. Nature Physics 18, 172–178 (2022). arXiv:2102.11751. https://doi.org/10.1038/s41567-021-01430-w arXiv:2102.11751 [83] S. K. Zhao, Zi-Yong Ge, Zhongcheng Xiang, G. M. Xue, H. S. Yan, Z. T. Wang, Zhan Wang, H. K. Xu, F. F. Su, Z. H. Yang, He Zhang, Yu-Ran Zhang, Xue-Yi Guo, Kai Xu, Ye Tian, H. F. Yu, D. N. Zheng, Heng Fan, and S. P. Zhao. ``Probing operator spreading via Floquet engineering in a superconducting circuit''.

Physical Review Letters 129, 160602 (2022). arXiv:2108.01276. https://doi.org/10.1103/PhysRevLett.129.160602 arXiv:2108.01276 [84] Brian Swingle, Gregory Bentsen, Monika Schleier-Smith, and Patrick Hayden. ``Measuring the scrambling of quantum information''. Physical Review A 94, 040302 (2016). arXiv:1602.06271. https://doi.org/10.1103/PhysRevA.94.040302 arXiv:1602.06271Cited by[1] Devjyoti Tripathy, Juzar Thingna, and Sebastian Deffner, "Quantum speed limit for the OTOC from an open systems perspective", arXiv:2509.13519, (2025). [2] Amit Vikram, Edwin Chaparro, Muhammad Miskeen Khan, Andrew Lucas, Chris Akers, and Ana Maria Rey, "Bidirectional teleportation using scrambling dynamics: a practical protocol", arXiv:2601.15536, (2026). [3] Michał Piotrak, Thomas Colas, Ana Alonso-Serrano, and Alessio Serafini, "Quantum estimation of cosmological parameters", Journal of High Energy Physics 2026 2, 199 (2026). [4] Yi-Neng Zhou and Chang Liu, "Generalized Loschmidt echo and information scrambling in open systems", SciPost Physics 20 3, 072 (2026). [5] Devjyoti Tripathy, Federico Centrone, and Sebastian Deffner, "Quantum timekeeping and the dynamics of scrambling in critical systems", arXiv:2603.13016, (2026). [6] Rahel Lea Baumgartner, Pietro Pelliconi, Soumik Bandyopadhyay, Francesca Orsi, Philipp Hauke, Jean-Philippe Brantut, and Julian Sonner, "Quantum simulation using Trotterized disorder Hamiltonians in a single-mode optical cavity", arXiv:2512.13774, (2025). [7] Devjyoti Tripathy, Juzar Thingna, and Sebastian Deffner, "Quantum speed limit for the out-of-time-ordered correlator from an open-system perspective", Physical Review A 113 1, L010402 (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-06-30 12:07:40). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-30 12:07:33: Could not fetch cited-by data for 10.22331/q-2026-06-30-2146 from Crossref. This is normal if the DOI was registered recently.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. AbstractWe present efficient and practical protocols to measure the second Rényi entropy, whose exponential is known as the purity. Our approach is based on expressing the purity in terms of transition probabilities generated by an echo-type forward-backward evolution sequence, making it applicable to quantum many-body systems. Notably, our approach does not rely on random-noise averaging, a feature that can be extended to protocols to measure out-of-time-order correlation functions, as we demonstrate. By way of example, we show that our protocols can be practically implemented in superconducting qubit-based platforms, as well as in cavity-QED trapped ultra-cold gases.► BibTeX data@article{Zhou2026measuringrenyi, doi = {10.22331/q-2026-06-30-2146}, url = {https://doi.org/10.22331/q-2026-06-30-2146}, title = {Measuring {R}{\'{e}}nyi entropy with an {E}cho {P}rotocol}, author = {Zhou, Yi-Neng and L{\"{o}}wenberg, Robin and Sonner, Julian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2146}, month = jun, year = {2026} }► References [1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Reviews of Modern Physics 81, 865–942 (2009). https://doi.org/10.1103/revmodphys.81.865 [2] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. ``Entanglement in many-body systems''. Rev. Mod. Phys. 80, 517–576 (2008). https://doi.org/10.1103/RevModPhys.80.517 [3] J. Eisert, M. Cramer, and M. B. Plenio. ``Colloquium: Area laws for the entanglement entropy''. Reviews of Modern Physics 82, 277–306 (2010). https://doi.org/10.1103/revmodphys.82.277 [4] Nicolas Laflorencie. ``Quantum entanglement in condensed matter systems''. Physics Reports 646, 1–59 (2016). https://doi.org/10.1016/j.physrep.2016.06.008 [5] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium : Many-body localization, thermalization, and entanglement''. Reviews of Modern Physics 91 (2019). https://doi.org/10.1103/revmodphys.91.021001 [6] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''.

Physical Review Letters 100 (2008). https://doi.org/10.1103/physrevlett.100.030504 [18] Alexei Kitaev and John Preskill. ``Topological entanglement entropy''.

Physical Review Letters 96 (2006). https://doi.org/10.1103/physrevlett.96.110404 [19] Michael Levin and Xiao-Gang Wen. ``Detecting topological order in a ground state wave function''.

Nature Reviews Physics 5, 9–24 (2022). https://doi.org/10.1038/s42254-022-00535-2 [78] Christian Beck and Friedrich Schögl. ``Thermodynamics of chaotic systems: An introduction''.

Read Original

Source Information

Source: Quantum Journal

Website: https://quantum-journal.org/feed/

Discussion

0 professional contributions

Be the first to add a constructive contribution.

Measuring Rényi entropy with an Echo Protocol

Tags

Source Information