Order preserving maps on quantum measurements

Teiko Heinosaari; Maria Anastasia Jivulescu; Ion Nechita

doi:10.22331/q-2022-11-10-853

Order preserving maps on quantum measurements

Teiko Heinosaari^1,2, Maria Anastasia Jivulescu³, and Ion Nechita⁴

¹Quantum algorithms and software, VTT Technical Research Centre of Finland Ltd
²Department of Physics and Astronomy, University of Turku, Finland
³Department of Mathematics, Politehnica University of Timişoara, Romania
⁴Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France

Published:	2022-11-10, volume 6, page 853
Eprint:	arXiv:2202.00725v2
Doi:	https://doi.org/10.22331/q-2022-11-10-853
Citation:	Quantum 6, 853 (2022).

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Abstract

We study the partially ordered set of equivalence classes of quantum measurements endowed with the post-processing partial order. The post-processing order is fundamental as it enables to compare measurements by their intrinsic noise and it gives grounds to define the important concept of quantum incompatibility. Our approach is based on mapping this set into a simpler partially ordered set using an order preserving map and investigating the resulting image. The aim is to ignore unnecessary details while keeping the essential structure, thereby simplifying e.g. detection of incompatibility. One possible choice is the map based on Fisher information introduced by Huangjun Zhu, known to be an order morphism taking values in the cone of positive semidefinite matrices. We explore the properties of that construction and improve Zhu's incompatibility criterion by adding a constraint depending on the number of measurement outcomes. We generalize this type of construction to other ordered vector spaces and we show that this map is optimal among all quadratic maps.

Featured image: A partially order set is mapped to another partially order set by an order preserving map. In the image side the incompatibility of the red and blue elements is vident, hence they are incompatible also on the domain side.

► BibTeX data

@article{Heinosaari2022orderpreservingmaps,
  doi = {10.22331/q-2022-11-10-853},
  url = {https://doi.org/10.22331/q-2022-11-10-853},
  title = {Order preserving maps on quantum measurements},
  author = {Heinosaari, Teiko and Jivulescu, Maria Anastasia and Nechita, Ion},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {853},
  month = nov,
  year = {2022}
}

► References

[1] S.T. Ali, C. Carmeli, T. Heinosaari, and A. Toigo. Commutative POVMs and fuzzy observables. Found. Phys., 39:593–612, 2009.
https://doi.org/10.1007/s10701-009-9292-y

[2] N. Andrejic and R. Kunjwal. Joint measurability structures realizable with qubit measurements: Incompatibility via marginal surgery. Phys. Rev. Research, 2:043147, 2020.
https://doi.org/10.1103/PhysRevResearch.2.043147

[3] F. Buscemi, G.M. D'Ariano, M. Keyl, P. Perinotti, and R.F. Werner. Clean positive operator valued measures. J. Math. Phys., 46:082109, 2005.
https://doi.org/10.1063/1.2008996

[4] R. Bhatia. Matrix analysis, volume 169 of Graduate Texts in Mathematics. Springer, 1997.
https://doi.org/10.1007/978-1-4612-0653-8

[5] P. Busch, T. Heinosaari, J. Schultz, and N. Stevens. Comparing the degrees of incompatibility inherent in probabilistic physical theories. EPL, 103:10002, 2013.
https://doi.org/10.1209/0295-5075/103/10002

[6] A. Bluhm and I. Nechita. Joint measurability of quantum effects and the matrix diamond. J. Math. Phys., 59:112202, 2018.
https://doi.org/10.1063/1.5049125

[7] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, Cambridge, 2004.
https://doi.org/10.1017/CBO9780511804441

[8] I. Bengtsson and K. Życzkowski. Geometry of quantum states. Cambridge University Press, Cambridge, 2006.
https://doi.org/10.1017/CBO9780511535048

[9] P. Busch. Unsharp reality and joint measurements for spin observables. Phys. Rev. D, 33:2253–2261, 1986.
https://doi.org/10.1103/PhysRevD.33.2253

[10] C. Carmeli, T. Heinosaari, and A. Toigo. Informationally complete joint measurements on finite quantum systems. Phys. Rev. A, 85:012109, 2012.
https://doi.org/10.1103/PhysRevA.85.012109

[11] C. Carmeli, T. Heinosaari, and T. Toigo. Quantum incompatibility witnesses. Phys. Rev. Lett., 122:130402, 2019.
https://doi.org/10.1103/PhysRevLett.122.130402

[12] S. Designolle, M. Farkas, and J. Kaniewski. Incompatibility robustness of quantum measurements: a unified framework. New J. Phys., 21:113053, 2019.
https://doi.org/10.1088/1367-2630/ab5020

[13] R.D. Gill and S. Massar. State estimation for large ensembles. Phys. Rev. A, 61:042312, 2000.
https://doi.org/10.1103/PhysRevA.61.042312

[14] T. Guff, N.A. McMahon, Y.R. Sanders, and A. Gilchrist. A resource theory of quantum measurements. J. Phys. A: Math. Theor., 54:225301, 2021.
https://doi.org/10.1088/1751-8121/abed67

[15] T. Heinonen. Optimal measurements in quantum mechanics. Phys. Lett. A, 346:77–86, 2005.
https://doi.org/10.1016/j.physleta.2005.08.003

[16] E. Haapasalo, T. Heinosaari, and J.-P. Pellonpää. Quantum measurements on finite dimensional systems: relabeling and mixing. Quantum Inf. Process., 11:1751–1763, 2012.
https://doi.org/10.1007/s11128-011-0330-2

[17] T. Heinosaari, M.A. Jivulescu, and I. Nechita. Random positive operator valued measures. J. Math. Phys., 61:042202, 2020.
https://doi.org/10.1063/1.5131028

[18] E. Haapasalo and J.-P. Pellonpää. Optimal quantum observables. J. Math. Phys., 58:122104, 2017.
https://doi.org/10.1063/1.4996809

[19] A. Harrow. The church of the symmetric subspace. arXiv:1308.6595.
arXiv:1308.6595

[20] T. Heinosaari, D. Reitzner, and P. Stano. Notes on joint measurability of quantum observables. Found. Phys., 38:1133–1147, 2008.
https://doi.org/10.1007/s10701-008-9256-7

[21] P. Hrubeš. On families of anticommuting matrices. Linear Algebra and its Applications, 493:494–507, 2016.
https://doi.org/10.1016/j.laa.2015.12.015

[22] P. Hausladen and W.K. Wootters. A `pretty good' measurement for distinguishing quantum states. J. Mod. Opt., 41:2385–2390, 1994.
https://doi.org/10.1080/09500349414552221

[23] A. Jenčová and S. Pulmannová. How sharp are PV measures? Rep. Math. Phys., 59:257–266, 2007.
https://doi.org/10.1016/S0034-4877(07)80038-3

[24] A. Jenčová, S. Pulmannová, and E. Vinceková. Sharp and fuzzy observables on effect algebras. Int. J. Theor. Phys., 47:125–148, 2008.
https://doi.org/10.1007/s10773-007-9396-0

[25] R.V. Kadison. Order properties of bounded self-adjoint operators. Proceedings of the American Mathematical Society, 2:505–510, 1951.
https://doi.org/10.2307/2031784

[26] Y. Kuramochi. Construction of the least informative observable conserved by a given quantum instrument. J. Math. Phys., 56:092202, 2015.
https://doi.org/10.1063/1.4931625

[27] Y. Kuramochi. Minimal sufficient positive-operator valued measure on a separable Hilbert space. J. Math. Phys., 56:102205, 2015.
https://doi.org/10.1063/1.4934235

[28] H. Martens and W.M. de Muynck. Nonideal quantum measurements. Found. Phys., 20:255–281, 1990.
https://doi.org/10.1007/BF00731693

[29] M. Newman. Note on an algebraic theorem of Eddington. J. London Math. Soc., 1:93–99, 1932.
https://doi.org/10.1112/jlms/s1-7.2.93

[30] A.J. Scott. Tight informationally complete quantum measurements. J. Phys. A: Math. Gen., 39:13507, 2006.
https://doi.org/10.1088/0305-4470/39/43/009

[31] P. Skrzypczyk, M.J. Hoban, A. B. Sainz, and N. Linden. The complexity of compatible measurements. Phys. Rev. Research, 2:023292, 2020.
https://doi.org/10.1103/PhysRevResearch.2.023292

[32] R. Uola, K. Luoma, T. Moroder, and T. Heinosaari. Adaptive strategy for joint measurements. Phys. Rev. A, 94:022109, 2016.
https://doi.org/10.1103/PhysRevA.94.022109

[33] S. Yu and C.H. Oh. Quantum contextuality and joint measurement of three observables of a qubit. arXiv: 1312.6470.
arXiv:1312.6470

[34] H. Zhu and B.-G. Englert. Quantum state tomography with fully symmetric measurements and product measurements. Phys. Rev. A, 84:022327, 2011.
https://doi.org/10.1103/PhysRevA.84.022327

[35] H. Zhu, M. Hayashi, and L. Chen. Universal steering criteria. Phys. Rev. Lett., 116:070403, 2016.
https://doi.org/10.1103/PhysRevLett.116.070403

[36] H. Zhu. Information complementarity: a new paradigm for decoding quantum incompatibility. Sci. Rep., 5:14317, 2015.
https://doi.org/10.1038/srep14317

Cited by

[1] Yui Kuramochi, "Infinite dimensionality of the post-processing order of measurements on a general state space", Journal of Physics A: Mathematical and Theoretical 55 43, 435301 (2022).

[2] Faedi Loulidi, Ion Nechita, and Clément Pellegrini, "A physical noise model for quantum measurements", arXiv:2305.19766, (2023).

[3] Huangjun Zhu, "Quantum Measurements in the Light of Quantum State Estimation", PRX Quantum 3 3, 030306 (2022).

[4] Qing-Hua Zhang and Ion Nechita, "A Fisher Information-Based Incompatibility Criterion for Quantum Channels", Entropy 24 6, 805 (2022).

[5] Faedi Loulidi and Ion Nechita, "Measurement Incompatibility versus Bell Nonlocality: An Approach via Tensor Norms", PRX Quantum 3 4, 040325 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-09-22 09:58:29) and SAO/NASA ADS (last updated successfully 2023-09-22 09:58:30). The list may be incomplete as not all publishers provide suitable and complete citation data.

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