Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Quantum Measurements and Quantum Metrology

Ed. by Paternostro, Mauro

Open Access
See all formats and pricing
More options …

Precision Limits in Quantum Metrology with Open Quantum Systems

J. F. Haase
  • Corresponding author
  • Institut für Theoretische Physik, Albert-Einstein-Allee, Universität Ulm, Ulm, Germany
  • Center for Integrated Quantum Science and Technology (IQST), Albert-Einstein-Allee, Universität Ulm, Ulm, Germany
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ A. Smirne
  • Institut für Theoretische Physik, Albert- Einstein-Allee, Universität Ulm, Ulm, Germany
  • Center for Integrated Quantum Science and Technology (IQST), Albert- Einstein-Allee, Universität Ulm, Ulm, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ S. F. Huelga
  • Institut für Theoretische Physik, Albert- Einstein-Allee, Universität Ulm, Ulm, Germany
  • Center for Integrated Quantum Science and Technology (IQST), Albert- Einstein-Allee, Universität Ulm, Ulm, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ J. Kołodynski
  • ICFO-Institut de Ciènces Fotòniques, The Barcelona Institute of Science and Technology, Castelldefels ( Barcelona), Spain
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ R. Demkowicz-Dobrzanski
Published Online: 2018-10-05 | DOI: https://doi.org/10.1515/qmetro-2018-0002


The laws of quantum mechanics allow to perform measurements whose precision supersedes results predicted by classical parameter estimation theory. That is, the precision bound imposed by the central limit theorem in the estimation of a broad class of parameters, like atomic frequencies in spectroscopy or external magnetic field in magnetometry, can be overcomewhen using quantum probes. Environmental noise, however, generally alters the ultimate precision that can be achieved in the estimation of an unknown parameter. This tutorial reviews recent theoretical work aimed at obtaining general precision bounds in the presence of an environment.We adopt a complementary approach,wherewe first analyze the problem within the general framework of describing the quantum systems in terms of quantum dynamical maps and then relate this abstract formalism to a microscopic description of the system’s dissipative time evolution.We will show that although some forms of noise do render quantum systems standard quantum limited, precision beyond classical bounds is still possible in the presence of different forms of local environmental fluctuations.

Keywords: frequency estimation; open quantum systems; precision limits


  • [1] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).Google Scholar

  • [2] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993).Google Scholar

  • [3] G. Tóth and I. Apellaniz, J. Phys. AMath. and Theo. 47, 424006 (2014).Google Scholar

  • [4] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kołodynski, Progress in Optics 60, 345 (2015).Google Scholar

  • [5] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, ArXiv e-prints (2016), arXiv:1609.01609 [quant-ph] .Google Scholar

  • [6] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).Google Scholar

  • [7] D. Budker and M. Romalis, Nat. Phys. 3, 227 (2007).CrossrefGoogle Scholar

  • [8] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, Nat. Phys. 4, 810 (2008).Google Scholar

  • [9] D. J.Wineland, J. J. Bollinger,W. M. Itano, and F. L. Moore, Phys. Rev. A 46, R6797 (1992).CrossrefGoogle Scholar

  • [10] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994).Google Scholar

  • [11] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Rev. Mod. Phys. 87, 637 (2015).Google Scholar

  • [12] B. P. Abbott et. al. (LIGO Scienti_c Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).Google Scholar

  • [13] C. M. Caves, Phys. Rev. D 23, 1693 (1981).Google Scholar

  • [14] M. G. Genoni, S. Olivares, and M. G. A. Paris, Phys. Rev. Lett. 106, 153603 (2011).Google Scholar

  • [15] R. Schnabel, Phys. Rep. 684, 1 (2017).Google Scholar

  • [16] The LIGO Scienti_c Collaboration, Nat. Phys. 7, 962 (2011).Google Scholar

  • [17] J. Aasi, J. Abadie, B. P. Abbott, and et. al., Nat. Photonics 7, 613 (2013).CrossrefGoogle Scholar

  • [18] R. Demkowicz-Dobrzanski, K. Banaszek, and R. Schnabel, Phys. Rev. A 88, 041802 (2013).Google Scholar

  • [19] J. P. Dowling and K. P. Seshadreesan, J. Light. Technol. 33, 2359 (2015).Google Scholar

  • [20] W. P. Schleich, K. S. Ranade, C. Anton, M. Arndt, M. Aspelmeyer, M. Bayer, G. Berg, T. Calarco, H. Fuchs, E. Giacobino, M. Grassl, P. Hänggi, W. M. Heckl, I.-V. Hertel, S. Huelga, F. Jelezko, B. Keimer, J. P. Kotthaus, G. Leuchs, N. Lütkenhaus, U. Maurer, T. Pfau, M. B. Plenio, E. M. Rasel, O. Renn, C. Silberhorn, J. Schiedmayer, D. Schmitt-Landsiedel, K. Schönhammer, A. Ustinov, P. Walther, H. Weinfurter, E. Welzl, R. Wiesendanger, S. Wolf, A. Zeilinger, and P. Zoller, Appl. Phys. B 122, 130 (2016).Google Scholar

  • [21] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865 (1997).CrossrefGoogle Scholar

  • [22] B. M. Escher, R. L. deMatos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011).Google Scholar

  • [23] R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, Nat. Commun. 3, 1063 (2012).Google Scholar

  • [24] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, Nature 455, 648 (2008).Google Scholar

  • [25] N. Zhao, J. Honert, B. Schmid, M. Klas, J. Isoya, M. Markham, D. Twitchen, F. Jelezko, R.-B. Liu, H. Fedder, and J. Wrachtrup, Nat Nano 7, 657 (2012).CrossrefGoogle Scholar

  • [26] J. Cai, F. Jelezko, M. B. Plenio, and A. Retzker, New J. Phys 15, 013020 (2013).Google Scholar

  • [27] Y. Romach, C.Müller, T. Unden, L. J. Rogers, T. Isoda, K. M. Itoh, M. Markham, A. Stacey, J. Meijer, S. Pezzagna, B. Naydenov, L. P. McGuinness, N. Bar-Gill, and F. Jelezko, Phys. Rev. Lett. 114, 017601 (2015).Google Scholar

  • [28] A. Smirne, J. Kołodynski, S. F. Huelga, and R. Demkowicz- Dobrzanski, Phys. Rev. Lett. 116, 120801 (2016).Google Scholar

  • [29] J. F. Haase, A. Smirne, J. Kołodynski, R. Demkowicz-Dobrzanski, and S. F. Huelga, New J. Phys 20, 053009 (2018).Google Scholar

  • [30] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, ArXiv e-prints (2017), arXiv:1701.05152 [quant-ph] .Google Scholar

  • [31] A. Górecka, F. A. Pollock, P. Liuzzo-Scorpo, R. Nichols, G. Adesso, and K. Modi, ArXiv e-prints (2017), arXiv:1712.08142 [quant-ph] .Google Scholar

  • [32] S. Boixo, S. T. Flammia, C. M. Caves, and J. Geremia, Phys. Rev. Lett. 98, 090401 (2007).Google Scholar

  • [33] Á. Rivas and A. Luis, New J. Phys 14, 093052 (2012).Google Scholar

  • [34] R. Demkowicz-Dobrzanski, J. Czajkowski, and P. Sekatski, Phys. Rev. X 7, 041009 (2017).Google Scholar

  • [35] U. Dorner, New J. Phys 14, 043011 (2012).Google Scholar

  • [36] J. Jeske, J. H. Cole, and S. F. Huelga, New J. Phys 16, 073039 (2014).Google Scholar

  • [37] L.-S. Guo, B.-M. Xu, J. Zou, and B. Shao, Sci. Rep. 6, 33254 (2016).Google Scholar

  • [38] R. Yousefjani, S. Salimi, and A. Khorashad, Ann. Phys. 381, 80 (2017).Google Scholar

  • [39] R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodynski, and A. Acín, Phys. Rev. Lett. 111, 120401 (2013).Google Scholar

  • [40] M. A. Taylor and W. P. Bowen, Phys. Rep. 615, 1 (2016).Google Scholar

  • [41] R. Schnabel, N.Mavalvala, D. E. McClelland, and P. K. Lam, Nat. Commun. 1, 121 (2010).Google Scholar

  • [42] M. Szczykulska, T. Baumgratz, and A. Datta, Adv. Phys. X 1, 621 (2016).Google Scholar

  • [43] J. Ma, X. Wang, C. Sun, and F. Nori, Phys. Rep. 509, 89 (2011).Google Scholar

  • [44] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Rev. Mod. Phys. 81, 1051 (2009).Google Scholar

  • [45] J. P. Dowling, Contemp. Phys. 49, 125 (2008).Google Scholar

  • [46] The existence of the inverse function is not guaranteed, however it exists locally apart from pathological cases, e.g., r(ph)=const.Google Scholar

  • [47] N. F. Ramsey, Phys. Rev. 78, 695 (1950).CrossrefGoogle Scholar

  • [48] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A 47, 3554 (1993).Google Scholar

  • [49] J. S. Bell, Rev. Mod. Phys. 38, 447 (1966).Google Scholar

  • [50] S. Kochen and E. Specker, Indiana Univ. Math. J. 17, 59 (1968).Google Scholar

  • [51] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).CrossrefGoogle Scholar

  • [52] V. Giovannetti, S. Lloyd, and L.Maccone, Nat. Photonics 5, 222 (2011).CrossrefGoogle Scholar

  • [53] S. Dooley,W. J.Munro, and K. Nemoto, Phys. Rev. A 94, 052320 (2016).Google Scholar

  • [54] A. van den Bos, Parameter estimation for scientists and engineers (Wiley-Interscience, 2007).Google Scholar

  • [55] D. M. Greenberger, M. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Fundamental Theories of Physics, Vol. 37, edited by M. Kafatos (Springer Netherlands, 1989) pp. 69-72.Google Scholar

  • [56] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996).CrossrefGoogle Scholar

  • [57] C.W. Helstrom, Quantumdetection and estimation theory (Academic press, 1976).Google Scholar

  • [58] S. L. Braunstein, C. M. Caves, and G. Milburn, Ann. Phys. 247, 135 (1996).Google Scholar

  • [59] L. Pezzé and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009).Google Scholar

  • [60] R. Demkowicz-Dobrzanski and L.Maccone, Phys. Rev. Lett. 113, 250801 (2014).Google Scholar

  • [61] R. Augusiak, J. Kołodynski, A. Streltsov, M. N. Bera, A. Acín, and M. Lewenstein, Phys. Rev. A 94, 012339 (2016).Google Scholar

  • [62] H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, 2002).Google Scholar

  • [63] Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction (SpringerBriefs in Physics, 2012).Google Scholar

  • [64] Á. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014).Google Scholar

  • [65] J. Kołodynski and R. Demkowicz-Dobrzanski, New J. Phys 15, 073043 (2013).Google Scholar

  • [66] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).Google Scholar

  • [67] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016).Google Scholar

  • [68] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).Google Scholar

  • [69] M.-D. Choi, Linear Algebra Its Appl. 10, 285 (1975).Google Scholar

  • [70] A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403 (2010).Google Scholar

  • [71] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009).Google Scholar

  • [72] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).CrossrefGoogle Scholar

  • [73] L. Seveso, M. A. C. Rossi, and M. G. A. Paris, Phys. Rev. A 95, 012111 (2017).Google Scholar

  • [74] G. Kitagawa, Int. J. Control 25, 745 (1977).CrossrefGoogle Scholar

  • [75] Note that here it is straightforward to derive the connection to the Heisenberg inequality: Instead of estimating !0 one directly measures the interrogation time, i.e. all derivatives in Eq. (36) are taken with respect to t instead of !0. The result FQ = 4∆2H in combination with the QCRB in Eq. (40) yields the desired energy-time Heisenberg inequality.Google Scholar

  • [76] M. G. A. Paris, Int. J. Quantum Inf. 07, 125 (2009).CrossrefGoogle Scholar

  • [77] A. Fujiwara, Phys. Rev. A 63, 042304 (2001).Google Scholar

  • [78] S. Alipour and A. T. Rezakhani, Phys. Rev. A 91, 042104 (2015).Google Scholar

  • [79] A. Fujiwara and H. Imai, J. Phys. A: Math. Theor. 41, 255304 (2008).Google Scholar

  • [80] Note that a norm is naturally induced by the Hilbert-Schmidt scalar product, A · B = tr[A†B].Google Scholar

  • [81] R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar

  • [82] S. Alipour, M. Mehboudi, and A. T. Rezakhani, Phys. Rev. Lett. 112, 120405 (2014).Google Scholar

  • [83] M. Guµa and A. Jencová, Commun.Math. Phys. 276, 341 (2007).Google Scholar

  • [84] A. Fujiwara, J. Phys. A: Math. Gen. 39, 12489 (2006).Google Scholar

  • [85] K.Macieszczak, M. Fraas, and R. Demkowicz-Dobrzanski, New J. Phys. 16, 113002 (2014).Google Scholar

  • [86] M. Jarzyna and R. Demkowicz-Dobrzanski, New J. Phys. 17, 013010 (2015).Google Scholar

  • [87] D. W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman, Phys. Rev. A 80, 052114 (2009).Google Scholar

  • [88] D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, Phys. Rev. A 81, 012305 (2010).Google Scholar

  • [89] H.-P. Breuer, B. Kappler, and F. Petruccione, Ann. Phys. 291, 36 (2001).Google Scholar

  • [90] S. Maniscalco, F. Intravaia, J. Piilo, and A. Messina, J Opt B Quantum and Semiclassical Opt 6, S98 (2004).Google Scholar

  • [91] C. Fleming, N. I. Cummings, C. Anastopoulos, and B. L. Hu, J. Phys. A Math. and Theo. 43, 405304 (2010).Google Scholar

  • [92] We may stress here, that these terms are neglected after the partial trace over the environmental degrees have been performed. Neglecting fast oscillating terms on the level of the Hamiltonian corresponds to the rotating-wave-approximation. These approximations are not equivalent in general, see for example [90].Google Scholar

  • [93] J. B. Brask, R. Chaves, and J. Kołodynski, Phys. Rev. X 5, 031010 (2015).Google Scholar

  • [94] P. Szankowski, M. Trippenbach, and J. Chwedenczuk, Phys. Rev. A 90, 063619 (2014).Google Scholar

  • [95] A. S. Holevo, Rep. Math. Phys. 32, 211 (1993).CrossrefGoogle Scholar

  • [96] A. S. Holevo, J. Math. Phys. 37, 1812 (1996).Google Scholar

  • [97] B. Vacchini, Lect. Notes Phys. 787, 39 (2010).Google Scholar

  • [98] M. Lostaglio, K. Korzekwa, and A. Milne, Phys. Rev. A 96, 032109 (2017).CrossrefGoogle Scholar

  • [99] Y. Matsuzaki, S. C. Benjamin, and J. Fitzsimons, Phys. Rev. A 84, 012103 (2011).Google Scholar

  • [100] A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 109, 233601 (2012).Google Scholar

  • [101] K. Macieszczak, Phys. Rev. A 92, 010102 (2015).Google Scholar

  • [102] J. F. Haase, P. J. Vetter, T. Unden, A. Smirne, J. Rosskopf, B. Naydenov, A. Stacey, F. Jelezko, M. B. Plenio, and S. F. Huelga, Phys. Rev. Lett. 121, 060401 (2018).Google Scholar

  • [103] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, Phys. Rev. Lett. 104, 133601 (2010).Google Scholar

  • [104] X.-M. Lu, X. Wang, and C. P. Sun, Phys. Rev. A 82, 042103 (2010).Google Scholar

  • [105] P. L. Richards, J. Appl. Phys 76, 1 (1994).Google Scholar

  • [106] G. D. Boreman, Basic electro-optics for electrical engineers, Vol. 31 (SPIE Press, 1998).Google Scholar

  • [107] A. D’Amico and C. D. Natale, IEEE Sensors Journal 1, 183 (2001).Google Scholar

  • [108] J. Fraden, Handbook of modern sensors: physics, designs, and applications (Springer Science & Business Media, 2004).Google Scholar

  • [109] A. Luis, Phys. Rev. A 65, 025802 (2002).Google Scholar

  • [110] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Nature 450, 393 (2007).Google Scholar

  • [111] F. Benatti and D. Braun, Phys. Rev. A 87, 012340 (2013).Google Scholar

  • [112] N. Killoran, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 112, 150501 (2014).Google Scholar

  • [113] Thereby one has to keep in mind that this replacement concerns the derivative in the FI, while the functional dependence of the applied channel may not be a one-to-one replacement. For example, in the scenario of phase estimation one is interested to find the product λ= !0t.Google Scholar

  • [114] W. K. Wootters, Phys. Rev. D 23, 357 (1981).Google Scholar

  • [115] G. Gibbons, J. Geom. Phys. 8, 147 (1992).Google Scholar

  • [116] I. Bengtsson and K. Zyczkowski, Geometry of quantum states: An introduction to quantum entanglement (Cambridge University Press, 2006).Google Scholar

  • [117] T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, Phys. Rev. Lett. 106, 130506 (2011).Google Scholar

  • [118] M. Skotiniotis, P. Sekatski, andW. Dür, New J. Phys 17, 073032 (2015).Google Scholar

  • [119] M. M. Rams, P. Sierant, O. Dutta, P. Horodecki, and J. Zakrzewski, Phys. Rev. X 8, 021022 (2018).Google Scholar

  • [120] M. Beau and A. del Campo, Phys. Rev. Lett. 119, 010403 (2017).Google Scholar

  • [121] S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan, and C. M. Caves, Phys. Rev. A 77, 012317 (2008).Google Scholar

  • [122] S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, Phys. Rev. Lett. 101, 040403 (2008).Google Scholar

  • [123] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, Nature 471, 486 (2011).Google Scholar

  • [124] C. Macchiavello, S. F. Huelga, J. I. Cirac, A. K. Ekert, and M. B. Plenio, “Computing and measurement,” (Kluwer Academic, Plenum Publishers, New York, 2000) Chap. Decoherence and Quantum Error Correction in Frequency Standards, Quantum Communications.Google Scholar

  • [125] J. Preskill, e-print arXiv:quant-ph/0010098 (2000), quantph/ 0010098 .Google Scholar

  • [126] P. Sekatski, M. Skotiniotis, J. Kołodynski, andW. Dür, Quantum 1, 27 (2017).Google Scholar

  • [127] S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Nat. Commun. 9, 78 (2018).Google Scholar

  • [128] P. Sekatski, M. Skotiniotis, andW. Dür, New J. Phys 18, 073034 (2016).Google Scholar

  • [129] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).Google Scholar

  • [130] P. W. Shor, Phys. Rev. A 52, R2493 (1995).Google Scholar

  • [131] D. A. Herrera-Martí, T. Gefen, D. Aharonov, N. Katz, and A. Retzker, Phys. Rev. Lett. 115, 200501 (2015).Google Scholar

  • [132] G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Phys. Rev. Lett. 112, 150801 (2014).Google Scholar

  • [133] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Phys. Rev. Lett. 112, 150802 (2014).Google Scholar

  • [134] T. Unden, P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Plenio, M. Markham, D. Twitchen, A. Stacey, I. Lovchinsky, A. O. Sushkov, M. D. Lukin, A. Retzker, B. Naydenov, L. P. McGuinness, and F. Jelezko, Phys. Rev. Lett. 116, 230502 (2016).Google Scholar

  • [135] M. B. Plenio and S. F. Huelga, Phys. Rev. A 93, 032123 (2016).Google Scholar

  • [136] T. Gefen, D. A. Herrera-Martí, and A. Retzker, Phys. Rev. A 93, 032133 (2016).Google Scholar

  • [137] O. Oreshkov and T. A. Brun, Phys. Rev. A 76, 022318 (2007).Google Scholar

  • [138] F. Reiter, A. S. Sørensen, P. Zoller, and C. A.Muschik, Nat. Commun. 8, 1822 (2017).Google Scholar

  • [139] X.-M. Lu, S. Yu, and C. H. Oh, Nat. Commun. 6, 7282 (2015).Google Scholar

  • [140] S. Gammelmark and K. Mølmer, Phys. Rev. Lett. 112, 170401 (2014).Google Scholar

  • [141] M. G. Genoni, Phys. Rev. A 95, 012116 (2017).Google Scholar

  • [142] F. Albarelli, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, ArXiv e-prints (2018), arXiv:1803.05891 [quant-ph].Google Scholar

  • [143] T. Kapourniotis and A. Datta, ArXiv e-prints (2018), arXiv:1807.04267 [quant-ph] .Google Scholar

  • [144] T. Gefen, F. Jelezko, and A. Retzker, Phys. Rev. A 96, 032310 (2017).Google Scholar

  • [145] S. Pang and A. N. Jordan, Nat. Commun. 8, 14695 (2017).Google Scholar

  • [146] J. Yang, S. Pang, and A. N. Jordan, Phys. Rev. A 96, 020301 (2017).Google Scholar

  • [147] S. Schmitt, T. Gefen, F. M. Stürner, T. Unden, G. Wol_, C. Müller, J. Scheuer, B. Naydenov, M. Markham, S. Pezzagna, J. Meijer, I. Schwarz, M. Plenio, A. Retzker, L. P. McGuinness, and F. Jelezko, Science 356, 832 (2017).Google Scholar

  • [148] J. M. Boss, K. S. Cujia, J. Zopes, and C. L. Degen, Science 356, 837 (2017).Google Scholar

  • [149] M. Naghiloo, A. N. Jordan, and K. W. Murch, Phys. Rev. Lett. 119, 180801 (2017).Google Scholar

About the article

Received: 2018-08-15

Accepted: 2018-09-06

Published Online: 2018-10-05

Published in Print: 2016-08-01

Citation Information: Quantum Measurements and Quantum Metrology, Volume 5, Issue 1, Pages 13–39, ISSN (Online) 2299-114X, DOI: https://doi.org/10.1515/qmetro-2018-0002.

Export Citation

© by J. F. Haase et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Francesco Albarelli, Matteo A. C. Rossi, Dario Tamascelli, and Marco G. Genoni
Quantum, 2018, Volume 2, Page 110

Comments (0)

Please log in or register to comment.
Log in