1QuIC, École Polytechnique de Bruxelles, CP 165/59, Université Libre de Bruxelles, 1050 Brussels, Belgium
2RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia
3Department of Nuclear Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan
4Quantum Technology Group, Department of Science and Industry Systems, University of South-Eastern Norway, 3616 Kongsberg, Norway
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Measurement error and disturbance, in the presence of conservation laws, are analysed in general operational terms. We provide novel quantitative bounds demonstrating necessary conditions under which accurate or non-disturbing measurements can be achieved, highlighting an interesting interplay between incompatibility, unsharpness, and coherence. From here we obtain a substantial generalisation of the Wigner-Araki-Yanase (WAY) theorem. Our findings are further refined through the analysis of the fixed-point set of the measurement channel, some extra structure of which is characterised here for the first time.
Popular summary
In the presence of additive conserved quantities such as energy, charge, or angular momentum, there are restrictions on both accurate and non-disturbing measurements of some observables. A classic result on this topic is the Wigner-Araki-Yanase (WAY) theorem which dates back to the $50$s/$60$s, and states that when the measurement interaction is unitary, then the only sharp observables (corresponding to self-adjoint operators) that admit accurate or non-disturbing measurements are those that commute with the conserved quantity.
In this paper, we generalize the WAY theorem by addressing the question of accurate or non-disturbing measurements (in the presence of conservation laws) for observables represented by POVMs (positive operator valued measures) and measurement interactions represented by quantum channels. We find that in order to achieve accurate or non-disturbing measurements for observables that do not commute with the conserved quantity, the observables cannot be sharp, and the measuring apparatus must be prepared in a state with a large coherence in the conserved quantity. In the spirit of the original WAY theorem, we therefore find both a no-go result which prohibits precise measurement and manipulation of individual quantum objects, and a positive counterpart which delineates conditions under which good measurements can be achieved.
► BibTeX data
► References
[1] P. Busch, G. Cassinelli, and P. J. Lahti, Found. Phys. 20, 757 (1990).
https://doi.org/10.1007/BF01889690
[2] M. Ozawa, Phys. Rev. A 67, 042105 (2003).
https://doi.org/10.1103/PhysRevA.67.042105
[3] P. Busch, in Quantum Reality, Relativ. Causality, Closing Epistemic Circ. (Springer, Dordrecht, 2009) pp. 229–256.
https://doi.org/10.1007/978-1-4020-9107-0_13
[4] T. Heinosaari and M. M. Wolf, J. Math. Phys. 51, 092201 (2010).
https://doi.org/10.1063/1.3480658
[5] M. Tsang and C. M. Caves, Phys. Rev. Lett. 105, 123601 (2010).
https://doi.org/10.1103/PhysRevLett.105.123601
[6] M. Tsang and C. M. Caves, Phys. Rev. X 2, 1 (2012).
https://doi.org/10.1103/PhysRevX.2.031016
[7] L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, Phys. Rev. Lett. 109, 100404 (2012).
https://doi.org/10.1103/PhysRevLett.109.100404
[8] J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo, Phys. Rev. Lett. 111, 090506 (2013).
https://doi.org/10.1103/PhysRevLett.111.090506
[9] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Science (80-. ). 339, 178 (2013).
https://doi.org/10.1126/science.1226897
[10] P. Busch, P. Lahti, and R. F. Werner, Phys. Rev. Lett. 111, 160405 (2013).
https://doi.org/10.1103/PhysRevLett.111.160405
[11] P. Busch, P. Lahti, and R. F. Werner, Rev. Mod. Phys. 86, 1261 (2014).
https://doi.org/10.1103/RevModPhys.86.1261
[12] F. Kaneda, S.-Y. Baek, M. Ozawa, and K. Edamatsu, Phys. Rev. Lett. 112, 020402 (2014).
https://doi.org/10.1103/PhysRevLett.112.020402
[13] M. S. Blok, C. Bonato, M. L. Markham, D. J. Twitchen, V. V. Dobrovitski, and R. Hanson, Nat. Phys. 10, 189 (2014).
https://doi.org/10.1038/nphys2881
[14] T. Shitara, Y. Kuramochi, and M. Ueda, Phys. Rev. A 93, 032134 (2016).
https://doi.org/10.1103/PhysRevA.93.032134
[15] C. B. Møller, R. A. Thomas, G. Vasilakis, E. Zeuthen, Y. Tsaturyan, M. Balabas, K. Jensen, A. Schliesser, K. Hammerer, and E. S. Polzik, Nature 547, 191 (2017).
https://doi.org/10.1038/nature22980
[16] I. Hamamura and T. Miyadera, J. Math. Phys. 60, 082103 (2019).
https://doi.org/10.1063/1.5109446
[17] C. Carmeli, T. Heinosaari, T. Miyadera, and A. Toigo, Found. Phys. 49, 492 (2019).
https://doi.org/10.1007/s10701-019-00255-1
[18] K.-D. Wu, E. Bäumer, J.-F. Tang, K. V. Hovhannisyan, M. Perarnau-Llobet, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 125, 210401 (2020).
https://doi.org/10.1103/physrevlett.125.210401
[19] G. M. D’Ariano, P. Perinotti, and A. Tosini, Quantum 4, 363 (2020).
https://doi.org/10.22331/q-2020-11-16-363
[20] A. C. Ipsen, Found. Phys. 52, 20 (2022).
https://doi.org/10.1007/s10701-021-00534-w
[21] T. Heinosaari, T. Miyadera, and M. Ziman, J. Phys. A Math. Theor. 49, 123001 (2016).
https://doi.org/10.1088/1751-8113/49/12/123001
[22] O. Gühne, E. Haapasalo, T. Kraft, J.-P. Pellonpää, and R. Uola, Rev. Mod. Phys. 95, 011003 (2023).
https://doi.org/10.1103/RevModPhys.95.011003
[23] E. P. Wigner, Zeitschrift für Phys. A Hadron. Nucl. 133, 101 (1952).
https://doi.org/10.1007/BF01948686
[24] P. Busch, (2010), arXiv:1012.4372.
arXiv:1012.4372
[25] H. Araki and M. M. Yanase, Phys. Rev. 120, 622 (1960).
https://doi.org/10.1103/PhysRev.120.622
[26] L. Loveridge and P. Busch, Eur. Phys. J. D 62, 297 (2011).
https://doi.org/10.1140/epjd/e2011-10714-3
[27] T. Miyadera and H. Imai, Phys. Rev. A 74, 024101 (2006).
https://doi.org/10.1103/PhysRevA.74.024101
[28] G. Kimura, B. Meister, and M. Ozawa, Phys. Rev. A 78, 032106 (2008).
https://doi.org/10.1103/PhysRevA.78.032106
[29] P. Busch and L. Loveridge, Phys. Rev. Lett. 106, 110406 (2011).
https://doi.org/10.1103/PhysRevLett.106.110406
[30] P. Busch and L. D. Loveridge, in Symmetries Groups Contemp. Phys. (WORLD SCIENTIFIC, 2013) pp. 587–592.
https://doi.org/10.1142/9789814518550_0083
[31] A. Łuczak, Open Syst. Inf. Dyn. 23, 1 (2016).
https://doi.org/10.1142/S123016121650013X
[32] M. Tukiainen, Phys. Rev. A 95, 012127 (2017).
https://doi.org/10.1103/PhysRevA.95.012127
[33] H. Tajima and H. Nagaoka, (2019), arXiv:1909.02904.
arXiv:1909.02904
[34] S. Sołtan, M. Frączak, W. Belzig, and A. Bednorz, Phys. Rev. Res. 3, 013247 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013247
[35] M. Ozawa, Phys. Rev. Lett. 89, 3 (2002a).
https://doi.org/10.1103/PhysRevLett.89.057902
[36] T. Karasawa and M. Ozawa, Phys. Rev. A 75, 032324 (2007).
https://doi.org/10.1103/PhysRevA.75.032324
[37] T. Karasawa, J. Gea-Banacloche, and M. Ozawa, J. Phys. A Math. Theor. 42, 225303 (2009).
https://doi.org/10.1088/1751-8113/42/22/225303
[38] M. Ahmadi, D. Jennings, and T. Rudolph, New J. Phys. 15, 013057 (2013).
https://doi.org/10.1088/1367-2630/15/1/013057
[39] J. Åberg, Phys. Rev. Lett. 113, 150402 (2014).
https://doi.org/10.1103/PhysRevLett.113.150402
[40] H. Tajima, N. Shiraishi, and K. Saito, Phys. Rev. Res. 2, 043374 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043374
[41] L. Loveridge, T. Miyadera, and P. Busch, Found. Phys. 48, 135 (2018).
https://doi.org/10.1007/s10701-018-0138-3
[42] L. Loveridge, J. Phys. Conf. Ser. 1638, 012009 (2020).
https://doi.org/10.1088/1742-6596/1638/1/012009
[43] N. Gisin and E. Zambrini Cruzeiro, Ann. Phys. 530, 1700388 (2018).
https://doi.org/10.1002/andp.201700388
[44] M. Navascués and S. Popescu, Phys. Rev. Lett. 112, 140502 (2014).
https://doi.org/10.1103/PhysRevLett.112.140502
[45] M. H. Mohammady and J. Anders, New J. Phys. 19, 113026 (2017).
https://doi.org/10.1088/1367-2630/aa8ba1
[46] M. H. Mohammady and A. Romito, Quantum 3, 175 (2019).
https://doi.org/10.22331/q-2019-08-19-175
[47] G. Chiribella, Y. Yang, and R. Renner, Phys. Rev. X 11, 021014 (2021).
https://doi.org/10.1103/PhysRevX.11.021014
[48] M. H. Mohammady, Phys. Rev. A 104, 062202 (2021a).
https://doi.org/10.1103/PhysRevA.104.062202
[49] P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Theoretical and Mathematical Physics (Springer International Publishing, Cham, 2016).
https://doi.org/10.1007/978-3-319-43389-9
[50] P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics, Lecture Notes in Physics Monographs, Vol. 31 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995).
https://doi.org/10.1007/978-3-540-49239-9
[51] P. Busch, P. J. Lahti, and Peter Mittelstaedt, The Quantum Theory of Measurement, Lecture Notes in Physics Monographs, Vol. 2 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1996).
https://doi.org/10.1007/978-3-540-37205-9
[52] T. Heinosaari and M. Ziman, The Mathematical language of Quantum Theory (Cambridge University Press, Cambridge, 2011).
https://doi.org/10.1017/CBO9781139031103
[53] B. Janssens, Lett. Math. Phys. 107, 1557 (2017).
https://doi.org/10.1007/s11005-017-0953-z
[54] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1987).
https://doi.org/10.1007/978-3-662-02520-8
[55] O. Bratteli, P. E. T. Jorgensen, A. Kishimoto, and R. F. Werner, J. Oper. Theory 43, 97 (2000).
https://www.jstor.org/stable/24715231
[56] E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
https://doi.org/10.1007/BF01647093
[57] M. Ozawa, Phys. Rev. A 62, 062101 (2000).
https://doi.org/10.1103/PhysRevA.62.062101
[58] M. Ozawa, Phys. Rev. A 63, 032109 (2001).
https://doi.org/10.1103/PhysRevA.63.032109
[59] J.-P. Pellonpää, J. Phys. A Math. Theor. 46, 025302 (2013a).
https://doi.org/10.1088/1751-8113/46/2/025302
[60] J.-P. Pellonpää, J. Phys. A Math. Theor. 46, 025303 (2013b).
https://doi.org/10.1088/1751-8113/46/2/025303
[61] G. Lüders, Ann. Phys. 518, 663 (2006).
https://doi.org/10.1002/andp.20065180904
[62] M. Ozawa, J. Math. Phys. 25, 79 (1984).
https://doi.org/10.1063/1.526000
[63] P. Busch and J. Singh, Phys. Lett. A 249, 10 (1998).
https://doi.org/10.1016/S0375-9601(98)00704-X
[64] P. Busch, M. Grabowski, and P. J. Lahti, Found. Phys. 25, 1239 (1995b).
https://doi.org/10.1007/BF02055331
[65] P. J. Lahti, P. Busch, and P. Mittelstaedt, J. Math. Phys. 32, 2770 (1991).
https://doi.org/10.1063/1.529504
[66] M. M. Yanase, Phys. Rev. 123, 666 (1961).
https://doi.org/10.1103/PhysRev.123.666
[67] M. Ozawa, Phys. Rev. Lett. 88, 050402 (2002b).
https://doi.org/10.1103/PhysRevLett.88.050402
[68] I. Marvian and R. W. Spekkens, Nat. Commun. 5, 3821 (2014).
https://doi.org/10.1038/ncomms4821
[69] C. Cı̂rstoiu, K. Korzekwa, and D. Jennings, Phys. Rev. X 10, 041035 (2020).
https://doi.org/10.1103/PhysRevX.10.041035
[70] D. Petz and C. Ghinea, Quantum Probab. Relat. Top. (World Scientific, Singapore, 2011) pp. 261–281.
https://doi.org/10.1142/9789814338745_0015
[71] A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017).
https://doi.org/10.1103/RevModPhys.89.041003
[72] R. Takagi, Sci. Rep. 9, 14562 (2019).
https://doi.org/10.1038/s41598-019-50279-w
[73] I. Marvian, Phys. Rev. Lett. 129, 190502 (2022).
https://doi.org/10.1103/PhysRevLett.129.190502
[74] G. Tóth and D. Petz, Phys. Rev. A 87, 032324 (2013).
https://doi.org/10.1103/PhysRevA.87.032324
[75] S. Yu, (2013), arXiv:1302.5311.
arXiv:1302.5311
[76] L. Weihua and W. Junde, J. Phys. A Math. Theor. 43, 395206 (2010).
https://doi.org/10.1088/1751-8113/43/39/395206
[77] B. Prunaru, J. Phys. A Math. Theor. 44, 185203 (2011).
https://doi.org/10.1088/1751-8113/44/18/185203
[78] A. Arias, A. Gheondea, and S. Gudder, J. Math. Phys. 43, 5872 (2002).
https://doi.org/10.1063/1.1519669
[79] L. Weihua and W. Junde, J. Math. Phys. 50, 103531 (2009).
https://doi.org/10.1063/1.3253574
[80] G. M. D’Ariano, P. Perinotti, and M. Sedlák, J. Math. Phys. 52, 082202 (2011).
https://doi.org/10.1063/1.3610676
[81] M. H. Mohammady, Phys. Rev. A 103, 042214 (2021b).
https://doi.org/10.1103/PhysRevA.103.042214
[82] V. Pata, Fixed Point Theorems and Applications, UNITEXT, Vol. 116 (Springer International Publishing, Cham, 2019).
https://doi.org/10.1007/978-3-030-19670-7
[83] G. Pisier, Introduction to Operator Space Theory (Cambridge University Press, 2003).
https://doi.org/10.1017/CBO9781107360235
[84] Y. Kuramochi and H. Tajima, (2022), arXiv:2208.13494.
arXiv:2208.13494
[85] R. V. Kadison, Ann. Math. 56, 494 (1952).
https://doi.org/10.2307/1969657
[86] M.-D. Choi, Illinois J. Math. 18, 565 (1974).
https://doi.org/10.1215/ijm/1256051007
[87] W. F. Stinespring, Proc. Am. Math. Soc. 6, 211 (1955).
https://doi.org/10.2307/2032342
[88] T. Miyadera and H. Imai, Phys. Rev. A 78, 052119 (2008).
https://doi.org/10.1103/PhysRevA.78.052119
[89] T. Miyadera, L. Loveridge, and P. Busch, J. Phys. A Math. Theor. 49, 185301 (2016).
https://doi.org/10.1088/1751-8113/49/18/185301
[90] K. Kraus, States, Effects, and Operations Fundamental Notions of Quantum Theory, edited by K. Kraus, A. Böhm, J. D. Dollard, and W. H. Wootters, Lecture Notes in Physics, Vol. 190 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1983).
https://doi.org/10.1007/3-540-12732-1
[91] P. Lahti, Int. J. Theor. Phys. 42, 893 (2003).
https://doi.org/10.1023/A:1025406103210
[92] J.-P. Pellonpää, J. Phys. A Math. Theor. 47, 052002 (2014).
https://doi.org/10.1088/1751-8113/47/5/052002
[93] S. Luo and Q. Zhang, Theor. Math. Phys. 151, 529 (2007).
https://doi.org/10.1007/s11232-007-0039-7
[94] G. M. D’Ariano, P. L. Presti, and P. Perinotti, J. Phys. A. Math. Gen. 38, 5979 (2005).
https://doi.org/10.1088/0305-4470/38/26/010
[95] C. A. Fuchs and C. M. Caves, Open Syst. Inf. Dyn. 3, 345 (1995).
https://doi.org/10.1007/BF02228997
[96] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Phys. Rev. Lett. 76, 2818 (1996).
https://doi.org/10.1103/PhysRevLett.76.2818
Cited by
[1] Yui Kuramochi and Hiroyasu Tajima, “Wigner-Araki-Yanase theorem for continuous and unbounded conserved observables”, arXiv:2208.13494, (2022).
[2] M. Hamed Mohammady and Takayuki Miyadera, “Quantum measurements constrained by the third law of thermodynamics”, arXiv:2209.06024, (2022).
[3] M. Hamed Mohammady, “Thermodynamically free quantum measurements”, arXiv:2205.10847, (2022).
[4] Lauritz van Luijk, Reinhard F. Werner, and Henrik Wilming, “Covariant catalysis requires correlations and good quantum reference frames degrade little”, arXiv:2301.09877, (2023).
[5] M. Hamed Mohammady, “Thermodynamically free quantum measurements”, Journal of Physics A Mathematical General 55 50, 505304 (2022).
[6] M. Hamed Mohammady and Takayuki Miyadera, “Quantum measurements constrained by the third law of thermodynamics”, Physical Review A 107 2, 022406 (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-06-05 13:40:12). 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 2023-06-05 13:40:10: Could not fetch cited-by data for 10.22331/q-2023-06-05-1033 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.
- SEO Powered Content & PR Distribution. Get Amplified Today.
- PlatoAiStream. Web3 Data Intelligence. Knowledge Amplified. Access Here.
- Minting the Future w Adryenn Ashley. Access Here.
- Buy and Sell Shares in PRE-IPO Companies with PREIPO®. Access Here.
- Source: https://quantum-journal.org/papers/q-2023-06-05-1033/