{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T14:10:14Z","timestamp":1756303814294,"version":"3.44.0"},"reference-count":141,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T00:00:00Z","timestamp":1756252800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Engineering and Physical Sciences Research Council","award":["EP\/T022140\/1"],"award-info":[{"award-number":["EP\/T022140\/1"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>We propose a two step strategy for estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a \"pattern counting&amp;apos;&amp;apos; estimator computed as a simple additive functional of the outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cramer-Rao bound in the limit of large output size. Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs&amp;apos; quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of \\cite{GiGoGu} this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of \\cite{DayouCounting}.<\/jats:p>","DOI":"10.22331\/q-2025-08-27-1835","type":"journal-article","created":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T13:42:00Z","timestamp":1756302120000},"page":"1835","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator"],"prefix":"10.22331","volume":"9","author":[{"given":"Federico","family":"Girotti","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Nottingham, United Kingdom"},{"name":"Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK"},{"name":"Department of Mathematics, Polytechnic University of Milan, Milan, Piazza L. da Vinci 32, 20133, Italy"}]},{"given":"Alfred","family":"Godley","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Nottingham, United Kingdom"},{"name":"Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK"}]},{"given":"Madalin","family":"Guta","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Nottingham, United Kingdom"},{"name":"Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK"}]}],"member":"9598","published-online":{"date-parts":[[2025,8,27]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"F. Girotti, A. Godley, and M. Gu\u0163\u0103, J. Phys. A: Mathematical and Theoretical 57, 245304 (2024).","DOI":"10.1088\/1751-8121\/ad4c2b"},{"key":"1","doi-asserted-by":"publisher","unstructured":"D. Yang, S. F. Huelga, and M. B. Plenio, Phys. Rev. X 13, 031012 (2023).","DOI":"10.1103\/PhysRevX.13.031012"},{"key":"2","unstructured":"C. W. Helstrom, Quantum detection and estimation theory (Academic Press, 1976)."},{"key":"3","doi-asserted-by":"publisher","unstructured":"A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, 2011).","DOI":"10.1007\/978-88-7642-378-9"},{"key":"4","doi-asserted-by":"publisher","unstructured":"M. Hayashi, Asymptotic theory of quantum statistical inference: Selected papers (World Scientific Publishing Co., 2005).","DOI":"10.1142\/5630"},{"key":"5","doi-asserted-by":"publisher","unstructured":"M. G. Paris, International Journal of Quantum Information 7, 125 (2008).","DOI":"10.1142\/S0219749909004839"},{"key":"6","doi-asserted-by":"publisher","unstructured":"G. T\u00f3th and I. Apellaniz, Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).","DOI":"10.1088\/1751-8113\/47\/42\/424006"},{"key":"7","doi-asserted-by":"publisher","unstructured":"R. Demkowicz-Dobrza\u0144ski, W. Gorecki, and M. Gu\u0163\u0103, Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020).","DOI":"10.1088\/1751-8121\/ab8ef3"},{"key":"8","doi-asserted-by":"publisher","unstructured":"K. Banaszek, M. Cramer, and D. Gross, New Journal of Physics 15, 125020 (2013).","DOI":"10.1088\/1367-2630\/15\/12\/125020"},{"key":"9","doi-asserted-by":"publisher","unstructured":"F. Albarelli, M. Barbieri, M. Genoni, and I. Gianani, Physics Letters A 384, 126311 (2020).","DOI":"10.1016\/j.physleta.2020.126311"},{"key":"10","doi-asserted-by":"publisher","unstructured":"J. S. Sidhu and P. Kok, AVS Quantum Science 2, 014701 (2020).","DOI":"10.1116\/1.5119961"},{"key":"11","doi-asserted-by":"publisher","unstructured":"V. Giovannetti, S. Lloyd, and L. Maccone, Physical Review Letters 96, 010401 (2005).","DOI":"10.1103\/PhysRevLett.96.010401"},{"key":"12","doi-asserted-by":"publisher","unstructured":"A. Fujiwara and H. Imai, Journal of Physics A: Mathematical and Theoretical 41, 255304 (2008).","DOI":"10.1088\/1751-8113\/41\/25\/255304"},{"key":"13","doi-asserted-by":"publisher","unstructured":"V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222 (2011).","DOI":"10.1038\/nphoton.2011.35"},{"key":"14","doi-asserted-by":"publisher","unstructured":"B. Escher, R. de Matos Filho, and L. Davidovich, Nature Phys. 7, 406\u2013411 (2011).","DOI":"10.1038\/nphys1958"},{"key":"15","doi-asserted-by":"publisher","unstructured":"R. Demkowicz-Dobrza\u0144ski, J. Ko\u0142ody\u0144ski, and M. Gu\u0163\u0103, Nature Communications 3, 1063 (2012).","DOI":"10.1038\/ncomms2067"},{"key":"16","doi-asserted-by":"publisher","unstructured":"D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D. O. Soares-Pinto, I. S. Oliveira, and G. Adesso, Phys. Rev. Lett. 112, 210401 (2014).","DOI":"10.1103\/PhysRevLett.112.210401"},{"key":"17","doi-asserted-by":"publisher","unstructured":"A. Smirne, J. Ko\u0142ody\u0144ski, S. F. Huelga, and R. Demkowicz-Dobrza\u0144ski, Phys. Rev. Lett. 116, 120801 (2016).","DOI":"10.1103\/PhysRevLett.116.120801"},{"key":"18","doi-asserted-by":"publisher","unstructured":"L. Seveso, M. A. C. Rossi, and M. G. A. Paris, Phys. Rev. A 95, 012111 (2017).","DOI":"10.1103\/PhysRevA.95.012111"},{"key":"19","doi-asserted-by":"publisher","unstructured":"J. F. Haase, A. Smirne, J. Kolodynski, R. Demkowicz-Dobrzanski, and S. F. Huelga, New Journal of Physics 20, 053009 (2018).","DOI":"10.1088\/1367-2630\/aab67f"},{"key":"20","doi-asserted-by":"publisher","unstructured":"S. Pang and T. A. Brun, Physical Review A - Atomic, Molecular, and Optical Physics 90, 022117 (2014).","DOI":"10.1103\/PHYSREVA.90.022117"},{"key":"21","doi-asserted-by":"publisher","unstructured":"M. A. C. Rossi, F. Albarelli, D. Tamascelli, and M. G. Genoni, Phys. Rev. Lett. 125, 200505 (2020).","DOI":"10.1103\/PhysRevLett.125.200505"},{"key":"22","doi-asserted-by":"publisher","unstructured":"S. Zhou and L. Jiang, PRX Quantum 2, 010343 (2021).","DOI":"10.1103\/PRXQuantum.2.010343"},{"key":"23","doi-asserted-by":"publisher","unstructured":"M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).","DOI":"10.1103\/PhysRevX.6.031033"},{"key":"24","doi-asserted-by":"publisher","unstructured":"M. Tsang, Phys. Rev. A 104, 052411 (2021).","DOI":"10.1103\/PhysRevA.104.052411"},{"key":"25","doi-asserted-by":"publisher","unstructured":"C. Lupo, Z. Huang, and P. Kok, Phys. Rev. Lett. 124, 080503 (2020).","DOI":"10.1103\/PhysRevLett.124.080503"},{"key":"26","doi-asserted-by":"publisher","unstructured":"L. J. Fiderer, T. Tufarelli, S. Piano, and G. Adesso, PRX Quantum 2, 020308 (2021).","DOI":"10.1103\/PRXQuantum.2.020308"},{"key":"27","doi-asserted-by":"publisher","unstructured":"C. Oh, S. Zhou, Y. Wong, and L. Jiang, Phys. Rev. Lett. 126, 120502 (2021).","DOI":"10.1103\/PhysRevLett.126.120502"},{"key":"28","doi-asserted-by":"publisher","unstructured":"M. Tsang, H. M. Wiseman, and C. Caves, Phys. Rev. Lett. 106, 090401 (2011).","DOI":"10.1103\/PhysRevLett.106.090401"},{"key":"29","doi-asserted-by":"publisher","unstructured":"D. W. Berry, M. Tsang, M. J. Hall, and H. M. Wiseman, Physical Review X 5 (2015), 10.1103\/PhysRevX.5.031018.","DOI":"10.1103\/PhysRevX.5.031018"},{"key":"30","doi-asserted-by":"publisher","unstructured":"S. Ng, S. Z. Ang, T. A. Wheatley, H. Yonezawa, A. Furusawa, E. H. Huntington, and M. Tsang, Phys. Rev. A 93, 042121 (2016).","DOI":"10.1103\/PhysRevA.93.042121"},{"key":"31","doi-asserted-by":"publisher","unstructured":"L. M. Norris, G. A. Paz-Silva, and L. Viola, Phys. Rev. Lett. 116, 150503 (2016).","DOI":"10.1103\/PhysRevLett.116.150503"},{"key":"32","doi-asserted-by":"publisher","unstructured":"H. Shi and Q. Zhuang, npj Quantum Inf 9, 27 (2023).","DOI":"10.1038\/s41534-023-00693-w"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Y. Sung, F. Beaudoin, L. M. Norris, F. Yan, D. K. Kim, J. Y. Qiu, U. von L\u00fcpke, J. L. Yoder, T. P. Orlando, S. Gustavsson, et al., Nature communications 10, 3715 (2019).","DOI":"10.1038\/s41467-019-11699-4"},{"key":"34","doi-asserted-by":"publisher","unstructured":"M. Tsang, Phys. Rev. A 107, 012611 (2023).","DOI":"10.1103\/PhysRevA.107.012611"},{"key":"35","doi-asserted-by":"publisher","unstructured":"J. P. Dowling and K. P. Seshadreesan, Journal of Lightwave Technology 33, 2359 (2015).","DOI":"10.1109\/JLT.2014.2386795"},{"key":"36","doi-asserted-by":"publisher","unstructured":"C. L. Degen, F. Reinhard, and P. Cappellaro, Reviews of Modern Physics 89, 035002 (2017).","DOI":"10.1103\/RevModPhys.89.035002"},{"key":"37","doi-asserted-by":"publisher","unstructured":"L. Pezz\u00e8, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. 90, 035005 (2018).","DOI":"10.1103\/RevModPhys.90.035005"},{"key":"38","doi-asserted-by":"publisher","unstructured":"C. D. Marciniak, T. Feldker, I. Pogorelov, R. Kaubruegger, D. V. Vasilyev, R. van Bijnen, P. Schindler, P. Zoller, R. Blatt, and T. Monz, Nature 603, 604 (2022).","DOI":"10.1038\/s41586-022-04435-4"},{"key":"39","doi-asserted-by":"publisher","unstructured":"A. Zwick and G. A. \u00c1lvarez, Journal of Magnetic Resonance Open 16-17, 100113 (2023).","DOI":"10.1016\/j.jmro.2023.100113"},{"key":"40","doi-asserted-by":"publisher","unstructured":"J. Robinson, M. Miklos, and Y. e. a. Tso, Nat. Phys. 20, 208\u2013213 (2024).","DOI":"10.1038\/s41567-023-02310-1"},{"key":"41","doi-asserted-by":"publisher","unstructured":"J. A. Jones, S. D. Karlen, J. Fitzsimons, A. Ardavan, S. C. Benjamin, G. A. D. Briggs, and J. J. Morton, Science 324, 1166 (2009).","DOI":"10.1126\/SCIENCE.1170730"},{"key":"42","doi-asserted-by":"publisher","unstructured":"J. Amor\u00f3s-Binefa and J. Ko\u0142ody\u0144ski, New Journal of Physics 23, 123030 (2021).","DOI":"10.1088\/1367-2630\/ac3b71"},{"key":"43","doi-asserted-by":"publisher","unstructured":"J. B. Brask, R. Chaves, and J. Kolodynski, Physical Review X 5, 031010 (2015).","DOI":"10.1103\/PHYSREVX.5.031010"},{"key":"44","doi-asserted-by":"publisher","unstructured":"F. Albarelli, M. A. C. Rossi, M. G. A. Paris, and M. G. Genoni, New J. Phys. 19, 123011 (2017).","DOI":"10.1088\/1367-2630\/aa9840"},{"key":"45","doi-asserted-by":"publisher","unstructured":"N. Aslam, H. Zhou, and E. e. a. Urbach, Nat Rev Phys 5, 157\u2013169 (2023).","DOI":"10.1038\/s42254-023-00558-3"},{"key":"46","doi-asserted-by":"publisher","unstructured":"L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Physical Review Letters 114, 220405 (2015).","DOI":"10.1103\/PHYSREVLETT.114.220405"},{"key":"47","doi-asserted-by":"publisher","unstructured":"M. Mehboudi, A. Sanpera, and L. A. Correa, Journal of Physics A: Mathematical and Theoretical 52, 303001 (2019).","DOI":"10.1088\/1751-8121\/AB2828"},{"key":"48","doi-asserted-by":"publisher","unstructured":"B. P. Abbott et al., Physical Review Letters 116, 061102 (2016).","DOI":"10.1103\/PhysRevLett.116.061102"},{"key":"49","doi-asserted-by":"publisher","unstructured":"M. Tse et al., Physical Review Letters 123, 231107 (2019).","DOI":"10.1103\/PhysRevLett.123.231107"},{"key":"50","doi-asserted-by":"publisher","unstructured":"M. Bailes et al., Nature Reviews Physics 2021 3:5 3, 344 (2021).","DOI":"10.1038\/s42254-021-00303-8"},{"key":"51","doi-asserted-by":"publisher","unstructured":"N. Christensen and R. Meyer, Reviews of Modern Physics 94 (2022), 10.1103\/REVMODPHYS.94.025001.","DOI":"10.1103\/REVMODPHYS.94.025001"},{"key":"52","doi-asserted-by":"publisher","unstructured":"V. Belavkin, Theoretical and Mathematical Physics 26, 213 (1976).","DOI":"10.1007\/BF01032091"},{"key":"53","doi-asserted-by":"publisher","unstructured":"S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).","DOI":"10.1103\/PhysRevLett.72.3439"},{"key":"54","doi-asserted-by":"publisher","unstructured":"S. Haroche and J.-M. Raimond, Exploring the Quantum: atoms, cavities and photons (Oxford University Press, 2006).","DOI":"10.1093\/acprof:oso\/9780198509141.001.0001"},{"key":"55","doi-asserted-by":"publisher","unstructured":"F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, Physics Reports 954, 1 (2022).","DOI":"10.1016\/j.physrep.2022.01.001."},{"key":"56","doi-asserted-by":"publisher","unstructured":"D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Info. Comput. 7, 401\u2013430 (2007).","DOI":"10.26421\/QIC7.5-6-1"},{"key":"57","doi-asserted-by":"publisher","unstructured":"C. Sch\u00f6n, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf, Phys. Rev. Lett. 95, 110503 (2005).","DOI":"10.1103\/PhysRevLett.95.110503"},{"key":"58","doi-asserted-by":"publisher","unstructured":"M. Fannes, B. Nachtergaele, and R. F. Werner, Communications in Mathematical Physics 144, 443 (1992).","DOI":"10.1007\/BF02099178"},{"key":"59","doi-asserted-by":"publisher","unstructured":"M. Fannes, B. Nachtergaele, and R. F. Werner, Journal of Functional Analysis 120, 511 (1994).","DOI":"10.1006\/jfan.1994.1041"},{"key":"60","doi-asserted-by":"publisher","unstructured":"S. Attal and Y. Pautrat, Ann. Henri Poincar\u00e9 7, 59 (2006a).","DOI":"10.1007\/s00023-005-0242-8"},{"key":"61","doi-asserted-by":"publisher","unstructured":"F. Verstraete and J. I. Cirac, Phys. Rev. Lett. 104, 190405 (2010).","DOI":"10.1103\/PhysRevLett.104.190405"},{"key":"62","doi-asserted-by":"publisher","unstructured":"J. Gough, Lett. Math. Phys. 67, 207 (2004).","DOI":"10.1023\/B:MATH.0000035039.56638.e1"},{"key":"63","doi-asserted-by":"publisher","unstructured":"S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865 (1997).","DOI":"10.1103\/PhysRevLett.79.3865"},{"key":"64","doi-asserted-by":"publisher","unstructured":"F. Benatti, S. Alipour, and A. T. Rezakhani, New J. Phys. 16, 015023 (2014).","DOI":"10.1088\/1367-2630\/16\/1\/015023"},{"key":"65","doi-asserted-by":"publisher","unstructured":"P. Sekatski, M. Skotiniotis, J. Ko\u0142ody\u0144ski, and W. D\u00fcr, Quantum 1, 27 (2017).","DOI":"10.22331\/q-2017-09-06-27"},{"key":"66","doi-asserted-by":"publisher","unstructured":"R. Demkowicz-Dobrza\u0144ski, J. Czajkowski, and P. Sekatski, Phys. Rev. X 7, 041009 (2017).","DOI":"10.1103\/PhysRevX.7.041009"},{"key":"67","doi-asserted-by":"publisher","unstructured":"S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Nat Commun 9, 78 (2018).","DOI":"10.1038\/s41467-017-02510-3"},{"key":"68","doi-asserted-by":"publisher","unstructured":"W. G\u00f3recki, S. Zhou, L. Jiang, and R. Demkowicz-Dobrza\u0144ski, Quantum 4, 288 (2020).","DOI":"10.22331\/q-2020-07-02-288"},{"key":"69","unstructured":"C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed. (Springer-Verlag, Berlin and New York, 2004)."},{"key":"70","doi-asserted-by":"publisher","unstructured":"H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).","DOI":"10.1017\/CBO9780511813948"},{"key":"71","doi-asserted-by":"publisher","unstructured":"J. Gough and M. R. James, IEEE Trans. Automat. Control 54, 2530 (2009).","DOI":"10.1109\/TAC.2009.2031205"},{"key":"72","doi-asserted-by":"publisher","unstructured":"J. Combes, J. Kerckhoff, and M. Sarovar, Adv. Phys. X 2, 784 (2017).","DOI":"10.1080\/23746149.2017.1343097"},{"key":"73","doi-asserted-by":"publisher","unstructured":"V. P. Belavkin, Found. Phys. 24, 685 (1994).","DOI":"10.1007\/BF02054669"},{"key":"74","doi-asserted-by":"publisher","unstructured":"J. Dalibard, Y. Castin, and K. M\u00f8lmer, Phys. Rev. Lett. 68, 580 (1992).","DOI":"10.1103\/PhysRevLett.68.580"},{"key":"75","doi-asserted-by":"publisher","unstructured":"H. M. Wiseman and G. J. Milburn, Phys. Rev. A 47, 642 (1993).","DOI":"10.1103\/PhysRevA.47.642"},{"key":"76","doi-asserted-by":"publisher","unstructured":"L. Bouten, R. van Handel, and M. R. James, SIAM J. Control Optim. 46, 2199 (2007).","DOI":"10.1137\/060651239"},{"key":"77","doi-asserted-by":"publisher","unstructured":"H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).","DOI":"10.1007\/978-3-540-47620-7"},{"key":"78","doi-asserted-by":"publisher","unstructured":"H. Mabuchi, Quantum Semiclass. Opt. 8, 1103 (1996).","DOI":"10.1088\/1355-5111\/8\/6\/002"},{"key":"79","doi-asserted-by":"publisher","unstructured":"J. Gambetta and H. M. Wiseman, Phys. Rev. A 64, 042105 (2001).","DOI":"10.1103\/PhysRevA.64.042105"},{"key":"80","doi-asserted-by":"publisher","unstructured":"D. W. Berry and H. M. Wiseman, Phys. Rev. A 65, 043803 (2002).","DOI":"10.1103\/PhysRevA.65.043803"},{"key":"81","doi-asserted-by":"publisher","unstructured":"D. T. Pope, H. M. Wiseman, and N. K. Langford, Phys. Rev. A 70, 043812 (2004).","DOI":"10.1103\/PhysRevA.70.043812"},{"key":"82","doi-asserted-by":"publisher","unstructured":"J. F. Ralph, K. Jacobs, and C. D. Hill, Phys. Rev. A 84, 052119 (2011).","DOI":"10.1103\/PhysRevA.84.052119"},{"key":"83","doi-asserted-by":"publisher","unstructured":"B. Chase and J. M. Geremia, Phys. Rev. A 79, 022314 (2009).","DOI":"10.1103\/PhysRevA.79.022314"},{"key":"84","doi-asserted-by":"publisher","unstructured":"P. Six, P. Campagne-Ibarcq, L. Bretheau, B. Huard, and P. Rouchon, in 2015 54th IEEE Conference on Decision and Control (CDC) (2015) pp. 7742\u20137748.","DOI":"10.1109\/CDC.2015.7403443"},{"key":"85","doi-asserted-by":"publisher","unstructured":"K. Macieszczak, M. Guta, I. Lesanovsky, and J. P. Garrahan, Physical Review A 93, 022103 (2016).","DOI":"10.1103\/PhysRevA.93.022103"},{"key":"86","doi-asserted-by":"publisher","unstructured":"F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, Quantum 2, 110 (2018).","DOI":"10.22331\/q-2018-12-03-110"},{"key":"87","doi-asserted-by":"publisher","unstructured":"T. Ilias, D. Yang, S. F. Huelga, and M. B. Plenio, PRX Quantum 3, 010354 (2022).","DOI":"10.1103\/PRXQuantum.3.010354"},{"key":"88","doi-asserted-by":"publisher","unstructured":"M. B. Plenio and S. F. Huelga, Phys. Rev. A 93, 032123 (2016).","DOI":"10.1103\/PhysRevA.93.032123"},{"key":"89","doi-asserted-by":"publisher","unstructured":"S. Gammelmark and K. M\u00f8lmer, Phys. Rev. A 87, 032115 (2013).","DOI":"10.1103\/PhysRevA.87.032115"},{"key":"90","doi-asserted-by":"publisher","unstructured":"A. Negretti and K. M\u00f8lmer, New J. Phys. 15, 125002 (2013).","DOI":"10.1088\/1367-2630\/15\/12\/125002"},{"key":"91","doi-asserted-by":"publisher","unstructured":"A. H. Kiilerich and K. M\u00f8lmer, Phys. Rev. A 94, 032103 (2016).","DOI":"10.1103\/PhysRevA.94.032103"},{"key":"92","doi-asserted-by":"publisher","unstructured":"J. F. Ralph, S. Maskell, and K. Jacobs, Phys. Rev. A 96, 052306 (2017).","DOI":"10.1103\/PhysRevA.96.052306"},{"key":"93","doi-asserted-by":"publisher","unstructured":"C. Zhang, K. Zhou, W. Feng, and X.-Q. Li, Phys. Rev. A 99, 022114 (2019).","DOI":"10.1103\/PhysRevA.99.022114"},{"key":"94","doi-asserted-by":"publisher","unstructured":"M. Tsang, Phys. Rev. Lett. 102, 250403 (2009a).","DOI":"10.1103\/PhysRevLett.102.250403"},{"key":"95","doi-asserted-by":"publisher","unstructured":"M. Tsang, Phys. Rev. A 80, 033840 (2009b).","DOI":"10.1103\/PhysRevA.80.033840"},{"key":"96","doi-asserted-by":"publisher","unstructured":"M. Tsang, Phys. Rev. A 81, 013824 (2010).","DOI":"10.1103\/PhysRevA.81.013824"},{"key":"97","doi-asserted-by":"publisher","unstructured":"I. Guevara and H. Wiseman, Phys. Rev. Lett. 115, 180407 (2015).","DOI":"10.1103\/PhysRevLett.115.180407"},{"key":"98","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103 and N. Yamamoto, IEEE Trans. Automat. Contr. 61, 921 (2016).","DOI":"10.1109\/TAC.2015.2448491"},{"key":"99","doi-asserted-by":"publisher","unstructured":"M. G. Genoni, Phys. Rev. A 95, 012116 (2017).","DOI":"10.1103\/PhysRevA.95.012116"},{"key":"100","doi-asserted-by":"publisher","unstructured":"M. Levitt and M. Gu\u0163\u0103, Phys. Rev. A 95, 033825 (2017).","DOI":"10.1103\/PhysRevA.95.033825"},{"key":"101","doi-asserted-by":"publisher","unstructured":"M. Levitt, M. Gu\u0163\u0103, and H. I. Nurdin, Automatica 90, 255 (2018).","DOI":"10.1016\/j.automatica.2017.12.037"},{"key":"102","doi-asserted-by":"publisher","unstructured":"J. P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 104, 160601 (2010).","DOI":"10.1103\/PhysRevLett.104.160601"},{"key":"103","doi-asserted-by":"publisher","unstructured":"M. van Horssen and M. Gu\u0163\u0103, Journal of Mathematical Physics 56, 022109 (2015).","DOI":"10.1063\/1.4907995"},{"key":"104","doi-asserted-by":"publisher","unstructured":"D. Burgarth, V. Giovannetti, A. N. Kato, and K. Yuasa, New Journal of Physics 17, 113055 (2015).","DOI":"10.1088\/1367-2630\/17\/11\/113055"},{"key":"105","doi-asserted-by":"publisher","unstructured":"F. Carollo, R. L. Jack, and J. P. Garrahan, Phys. Rev. Lett. 122, 130605 (2019).","DOI":"10.1103\/PhysRevLett.122.130605"},{"key":"106","doi-asserted-by":"publisher","unstructured":"T. Benoist, L. H\u00e4nggli, and C. Rouz\u00e9, Quantum 6, 772 (2022).","DOI":"10.22331\/q-2022-08-04-772"},{"key":"107","doi-asserted-by":"publisher","unstructured":"F. Girotti, J. P. Garrahan, and M. Gu\u0163\u0103, Annales Henri Poincar\u00e9 (2023), 10.1007\/s00023-023-01286-1.","DOI":"10.1007\/s00023-023-01286-1"},{"key":"108","doi-asserted-by":"publisher","unstructured":"A. Fallani, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, PRX Quantum 3, 020310 (2022).","DOI":"10.1103\/PRXQuantum.3.020310"},{"key":"109","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103, Physical Review A 83, 062324 (2011).","DOI":"10.1103\/PhysRevA.83.062324"},{"key":"110","doi-asserted-by":"publisher","unstructured":"S. Gammelmark and K. M\u00f8lmer, Phys. Rev. Lett. 112, 170401 (2014).","DOI":"10.1103\/PhysRevLett.112.170401"},{"key":"111","doi-asserted-by":"publisher","unstructured":"C. Catana, L. Bouten, and M. Gu\u0163\u0103, Journal of Physics A: Mathematical and Theoretical 48, 365301 (2015).","DOI":"10.1088\/1751-8113\/48\/36\/365301"},{"key":"112","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103 and J. Kiukas, Communications in Mathematical Physics 335, 1397 (2015).","DOI":"10.1007\/S00220-014-2253-0"},{"key":"113","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103 and J. Kiukas, Journal of Mathematical Physics 58, 052201 (2017).","DOI":"10.1063\/1.4982958"},{"key":"114","doi-asserted-by":"publisher","unstructured":"A. Godley and M. Gu\u0163\u0103, Quantum 7, 973 (2023).","DOI":"10.22331\/q-2023-04-06-973"},{"key":"115","doi-asserted-by":"publisher","unstructured":"K. Stannigel, P. Rabl, and P. Zoller, New J. Phys. 12, 063014 (2012).","DOI":"10.1088\/1367-2630\/14\/6\/063014"},{"key":"116","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103 and J. Kahn, Physical Review A 73, 052108 (2006).","DOI":"10.1103\/PhysRevA.73.052108"},{"key":"117","doi-asserted-by":"publisher","unstructured":"J. Kahn and M. Gu\u0163\u0103, Communications in Mathematical Physics 289, 597 (2009).","DOI":"10.1007\/s00220-009-0787-3"},{"key":"118","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103, B. Janssens, and J. Kahn, Commun. Math. Phys. 277, 127 (2008).","DOI":"10.1007\/s00220-007-0357-5"},{"key":"119","doi-asserted-by":"publisher","unstructured":"C. Butucea, M. Gu\u0163\u0103, and M. Nussbaum, Annals Statist. 46, 3676 (2016).","DOI":"10.1214\/17-AOS1672"},{"key":"120","doi-asserted-by":"publisher","unstructured":"M. Gu\u0163\u0103 and A. Jencova, Communications in Mathematical Physics 276, 341 (2007).","DOI":"10.1007\/s00220-007-0340-1"},{"key":"121","doi-asserted-by":"publisher","unstructured":"R. D. Gill and M. Gu\u0163\u0103, IMS Collections 9, 105 (2013).","DOI":"10.1214\/12-IMSCOLL909"},{"key":"122","doi-asserted-by":"publisher","unstructured":"K. Yamagata, A. Fujiwara, and R. D. Gill, The Annals of Statistics 41, 2197 (2013).","DOI":"10.1214\/13-AOS1147"},{"key":"123","doi-asserted-by":"publisher","unstructured":"A. Fujiwara and K. Yamagata, Bernoulli 26, 2105 (2020).","DOI":"10.3150\/19-BEJ1185"},{"key":"124","doi-asserted-by":"publisher","unstructured":"A. Fujiwara and K. Yamagata, Ann. Statist. 51, 1159 (2023).","DOI":"10.1214\/23-aos2285"},{"key":"125","doi-asserted-by":"publisher","unstructured":"H. Yuen and M. Lax, IEEE Trans. Inf. Theory 19, 740 (1973).","DOI":"10.1109\/TIT.1973.1055103"},{"key":"126","doi-asserted-by":"publisher","unstructured":"S. L. Braunstein, C. M. Caves, and G. J. Milburn, Annals of Physics 247, 135 (1996).","DOI":"10.1006\/aphy.1996.0040"},{"key":"127","doi-asserted-by":"publisher","unstructured":"R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000).","DOI":"10.1103\/PhysRevA.61.042312"},{"key":"128","doi-asserted-by":"publisher","unstructured":"L. Pezze and A. Smerzi, Atom Interferometry 188, 691 (2014).","DOI":"10.3254\/978-1-61499-448-0-691"},{"key":"129","doi-asserted-by":"publisher","unstructured":"L. Pezz\u00e8, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Phys. Rev. Lett. 119, 130504 (2017).","DOI":"10.1103\/PhysRevLett.119.130504"},{"key":"130","doi-asserted-by":"publisher","unstructured":"J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019).","DOI":"10.1088\/1751-8121\/ab5d4d"},{"key":"131","doi-asserted-by":"publisher","unstructured":"S. Attal and Y. Pautrat, Ann. Henri Poincar\u00e9 7, 59 ,(2006b).","DOI":"10.1007\/s00023-005-0242-8"},{"key":"132","doi-asserted-by":"publisher","unstructured":"D. Petz, Communications in Mathematical Physics 105, 123\u2013131 (1986).","DOI":"10.1007\/BF01212345"},{"key":"133","doi-asserted-by":"publisher","unstructured":"D. Petz, Quarterly Journal of Mathematics 39, 97\u2013108 (1988).","DOI":"10.1093\/qmath\/39.1.97"},{"key":"134","doi-asserted-by":"publisher","unstructured":"H. Barnum and E. Knill, Journal of Mathematical Physics 43, 2097\u20132106 (2002).","DOI":"10.1063\/1.1459754"},{"key":"135","doi-asserted-by":"publisher","unstructured":"M. Tsang, ``Quantum reversal: a general theory of coherent quantum absorbers,&apos;&apos; Quantum 9, 1650 (2025).","DOI":"10.22331\/q-2025-02-26-1650"},{"key":"136","doi-asserted-by":"publisher","unstructured":"J. R. Johansson, P. D. Nation, and F. Nori, Comp. Phys. Comm. 184, 1234 (2013).","DOI":"10.1016\/j.cpc.2012.11.019"},{"key":"137","doi-asserted-by":"publisher","unstructured":"J. M. Radcliffe, J. Phys. A: Gen. Phys. 4, 313 (1971).","DOI":"10.1088\/0305-4470\/4\/3\/009"},{"key":"138","doi-asserted-by":"publisher","unstructured":"A. W. van der Vaart, Asymptotic Statistics Cambridge University Press, (1998).","DOI":"10.1017\/CBO9780511802256"},{"key":"139","doi-asserted-by":"publisher","unstructured":"G. D. Lin, Journal of Statistical Distributions and Applications 4, 5 (2017).","DOI":"10.1186\/s40488-017-0059-2"},{"key":"140","doi-asserted-by":"publisher","unstructured":"K. Schm\u00fcdgen, ``Ten lectures on the moment problem,&apos;&apos; arXiv:2008.12698 [math.FA] (2020).","DOI":"10.48550\/arXiv.2008.12698"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-08-27-1835\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T13:42:04Z","timestamp":1756302124000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-08-27-1835\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,27]]},"references-count":141,"URL":"https:\/\/doi.org\/10.22331\/q-2025-08-27-1835","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,27]]},"article-number":"1835"}}