Operational relevance of resource theories of quantum measurements

Michał Oszmaniec; Tanmoy Biswas

doi:10.22331/q-2019-04-26-133

Operational relevance of resource theories of quantum measurements

Michał Oszmaniec and Tanmoy Biswas

Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, Wita Stwosza 57, 80-308 Gdańsk, Poland

Published:	2019-04-26, volume 3, page 133
Eprint:	arXiv:1901.08566v3
Doi:	https://doi.org/10.22331/q-2019-04-26-133
Citation:	Quantum 3, 133 (2019).

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Abstract

For any resource theory it is essential to identify tasks for which resource objects offer advantage over free objects. We show that this identification can always be accomplished for resource theories of quantum measurements in which free objects form a convex subset of measurements on a given Hilbert space. To this aim we prove that every resourceful measurement offers advantage for some quantum state discrimination task. Moreover, we give an operational interpretation of robustness, which quantifies the minimal amount of noise that must be added to a measurement to make it free. Specifically, we show that this geometric quantity is related to the maximal relative advantage that a resourceful measurement offers in a class of minimal-error state discrimination (MESD) problems. Finally, we apply our results to two classes of free measurements: incoherent measurements (measurements that are diagonal in the fixed basis) and separable measurements (measurements whose effects are separable operators). For both of these scenarios we find, in the asymptotic setting in which the dimension or the number of particles increase to infinity, the maximal relative advantage that resourceful measurements offer for state discrimination tasks.

► BibTeX data

@article{Oszmaniec2019operational,
  doi = {10.22331/q-2019-04-26-133},
  url = {https://doi.org/10.22331/q-2019-04-26-133},
  title = {Operational relevance of resource theories of quantum measurements},
  author = {Oszmaniec, Micha{\l{}} and Biswas, Tanmoy},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {133},
  month = apr,
  year = {2019}
}

► References

[1] E. Chitambar and G. Gour, arXiv:1806.06107 (2018).
arXiv:1806.06107

[2] B. Coecke, T. Fritz, and R. W. Spekkens, Information and Computation 250, 59 (2016).
https://doi.org/10.1016/j.ic.2016.02.008

[3] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Phys. Rev. Lett. 111, 250404 (2013).
https://doi.org/10.1103/PhysRevLett.111.250404

[4] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016).
https://doi.org/10.1088/1751-8113/49/14/143001

[5] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997).
https://doi.org/10.1103/PhysRevLett.78.2275

[6] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[7] A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, P. Joshi, W. Kłobus, and A. Wójcik, Phys. Rev. Lett. 112, 120401 (2014).
https://doi.org/10.1103/PhysRevLett.112.120401

[8] J. Barrett, Phys. Rev. A 75, 032304 (2007).
https://doi.org/10.1103/PhysRevA.75.032304

[9] R. Gallego and L. Aolita, Phys. Rev. X 5, 041008 (2015).
https://doi.org/10.1103/PhysRevX.5.041008

[10] V. Veitch, C. Ferrie, D. Gross, and J. Emerson, New Journal of Physics 14, 113011 (2012).
http://stacks.iop.org/1367-2630/14/i=11/a=113011

[11] S. M. Barnett and S. Croke, Adv. Opt. Photon. 1, 238 (2009).
https://www.osapublishing.org/aop/abstract.cfm?uri=aop-1-2-238

[12] J. Bae and L.-C. Kwek, J. Phys. A: Math. Gen. 48, 083001 (2015).
http://stacks.iop.org/1751-8121/48/i=8/a=083001

[13] N. Brunner, M. Navascués, and T. Vértesi, Phys. Rev. Lett. 110, 150501 (2013).
https://doi.org/10.1103/PhysRevLett.110.150501

[14] I. Marvian and R. W. Spekkens, Phys. Rev. A 94, 052324 (2016).
https://doi.org/10.1103/PhysRevA.94.052324

[15] J. Bae, W.-Y. Hwang, and Y.-D. Han, Phys. Rev. Lett. 107, 170403 (2011).
https://doi.org/10.1103/PhysRevLett.107.170403

[16] C. Lomont, arxiv:0411037 (2005).
arXiv:quant-ph/0411037

[17] P. Skrzypczyk and N. Linden, Phys. Rev. Lett. 122, 140403 (2019).
https://doi.org/10.1103/PhysRevLett.122.140403

[18] R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Phys. Rev. Lett. 122, 140402 (2019).
https://doi.org/10.1103/PhysRevLett.122.140402

[19] G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999).
https://doi.org/10.1103/PhysRevA.72.022310

[20] F. G. S. L. Brandão and G. Gour, Phys. Rev. Lett. 115, 070503 (2015).
https://doi.org/10.1103/PhysRevLett.115.070503

[21] B. Regula, Journal of Physics A: Mathematical and Theoretical 51, 045303 (2017).
https://doi.org/10.1088/1751-8121/aa9100

[22] J. Aberg, arXiv:quant-ph/0612146 (2006).
arXiv:quant-ph/0612146

[23] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
https://doi.org/10.1103/PhysRevLett.113.140401

[24] A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017).
https://doi.org/10.1103/RevModPhys.89.041003

[25] S. Bandyopadhyay and M. Nathanson, Phys. Rev. A 88, 052313 (2013).
https://doi.org/10.1103/PhysRevA.88.052313

[26] J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018).
https://doi.org/10.1017/9781316848142

[27] E. Bagan, M. A. Ballester, R. Muñoz Tapia, and O. Romero-Isart, Phys. Rev. Lett. 95, 110504 (2005).
https://doi.org/10.1103/PhysRevLett.95.110504

[28] V. Kargin, The Annals of Statistics 33, 959 (2004).
https://doi.org/10.1214/009053604000001219

[29] M. Nathanson, J. Math. Phys. 46, 141 (2005).
https://doi.org/10.1063/1.1914731

[30] E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter, Communications in Mathematical Physics 328, 303 (2014).
https://doi.org/10.1007/s00220-014-1953-9

[31] G. M. D'Ariano, P. Lo Presti, and M. G. A. Paris, Phys. Rev. Lett. 87, 270404 (2001).
https://doi.org/10.1103/PhysRevLett.87.270404

[32] M. Horodecki and J. Oppenheim, Nat. Commun. 4 (2013), 10.1038/ncomms3059.
https://doi.org/10.1038/ncomms3059

[33] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009).
https://doi.org/10.1103/PhysRevLett.102.010401

[34] D. Cavalcanti, P. Skrzypczyk, and I. Šupić, Phys. Rev. Lett. 119, 110501 (2017).
https://doi.org/10.1103/PhysRevLett.119.110501

[35] A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
https://doi.org/10.1103/PhysRevLett.116.120404

[36] T. Biswas, M. G. Diaz, and A. Winter, Proc. R. Soc. A 473 (2017), 10.1098/rspa.2017.0170.
https://doi.org/10.1098/rspa.2017.0170

[37] M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. Johnston, and G. Adesso, Phys. Rev. A 93, 042107 (2016a).
https://doi.org/10.1103/PhysRevA.93.042107

[38] C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, and G. Adesso, Phys. Rev. Lett. 116, 150502 (2016).
https://doi.org/10.1103/PhysRevLett.116.150502

[39] M. Piani and J. Watrous, Phys. Rev. Lett. 114, 060404 (2015).
https://doi.org/10.1103/PhysRevLett.114.060404

[40] M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. Johnston, and G. Adesso, Phys. Rev. A 93, 042107 (2016b).
https://doi.org/10.1103/PhysRevA.93.042107

[41] T. Heinosaari, J. Kiukas, and D. Reitzner, Phys. Rev. A 92, 022115 (2015).
https://doi.org/10.1103/PhysRevA.92.022115

[42] L. Guerini, J. Bavaresco, M. Terra Cunha, and A. Acín, J. Math. Phys. 58, 092102 (2017).
https://doi.org/10.1063/1.4994303

[43] M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, Phys. Rev. Lett. 119, 190501 (2017).
https://doi.org/10.1103/PhysRevLett.119.190501

[44] M. Oszmaniec, F. B. Maciejewski, and Z. Puchała, arXiv:1807.08449 (2018).
arXiv:1807.08449

[45] M. Kleinmann and A. Cabello, Phys. Rev. Lett. 117, 150401 (2016).
https://doi.org/10.1103/PhysRevLett.117.150401

[46] 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

[47] F. G. Brandao, Phys. Rev. A 72, 022310 (2005).
https://doi.org/10.1103/PhysRevA.72.022310

[48] M. Hillery, Phys. Rev. A 93, 012111 (2016).
https://doi.org/10.1103/PhysRevA.93.012111

[49] F. Bischof, H. Kampermann, and D. Bruss, arXiv:1812.00018 (2018).
arXiv:1812.00018

[50] T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio., arXiv:1806.07332 (2018).
arXiv:1806.07332

[51] M. Bera, arXiv:1809.02578 (2018).
arXiv:1809.02578

[52] L. Vandenberghe and S. Boyd, SIAM Review 38, 49 (1996).
https://epubs.siam.org/doi/abs/10.1137/1038003

[53] K. Korzekwa, S. Czachorski, Z. Puchała, and K. Zyczkowski, arXiv:1812.09083 (2018).
arXiv:1812.09083

[54] L. Gurvits and H. Barnum, Phys. Rev. A 68, 042312 (2003).
https://doi.org/10.1103/PhysRevA.68.042312

[55] A. Montanaro, Commun Math Phys 273, 619 (2007).
https://doi.org/10.1007/s00220-007-0221-7

[56] P. Hayden, D. Leung, P. W. Shor, and A. Winter, Communications in Mathematical Physics 250, 371 (2004).
https://doi.org/10.1007/s00220-004-1087-6

[57] W. Matthews, S. Wehner, and A. Winter, Communications in Mathematical Physics 291, 813 (2009).
https://doi.org/10.1007/s00220-009-0890-5

[58] A. Tavakoli, M. Smania, T. Vértesi, D. Rosset, and N. Brunner, arXiv:1811.12712 (2018).
arXiv:1811.12712

[59] P. Mironowicz and M. Pawłowski, arXiv:1811.12872 (2018).
arXiv:1811.12872

[60] R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Guhne, Phys. Rev. Lett. 122, 130404 (2019).
https://doi.org/10.1103/PhysRevLett.122.130404

[61] R. Takagi. and B. Regula, arXiv:1901.08127 (2019).
arXiv:1901.08127

[62] C. Carmeli, T. Heinosaari, and A. Toigo, Phys. Rev. Lett. 122, 13040 (2019).
https://doi.org/10.1103/PhysRevLett.122.130402

[63] P. Skrzypczyk, I. Šupić, and D. Cavalcanti, Phys. Rev. Lett. 122, 130403 (2019).
https://doi.org/10.1103/PhysRevLett.122.130403

[64] D. Cavalcanti and P. Skrzypczyk, Reports on Progress in Physics 80, 024001 (2016).
https://doi.org/10.1088/1361-6633/80/2/024001

[65] D. Gross, S. T. Flammia, and J. Eisert, Phys. Rev. Lett. 102, 190501 (2009).
https://doi.org/10.1103/PhysRevLett.102.190501

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[10] Stanisław Kurdziałek and Rafał Demkowicz-Dobrzański, "Measurement Noise Susceptibility in Quantum Estimation", Physical Review Letters 130 16, 160802 (2023).

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[16] Hüseyin Talha Şenyaşa and Gökhan Torun, "Golden states in resource theory of superposition", Physical Review A 105 4, 042410 (2022).

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