A complete and operational resource theory of measurement sharpness

Francesco Buscemi, Kodai Kobayashi, and Shintaro Minagawa

Department of Mathematical Informatics, Nagoya University, Furo-cho, Chikusa-ku, 464-8601 Nagoya, Japan

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We construct a resource theory of $sharpness$ for finite-dimensional positive operator-valued measures (POVMs), where the $sharpness-non-increasing$ operations are given by quantum preprocessing channels and convex mixtures with POVMs whose elements are all proportional to the identity operator. As required for a sound resource theory of sharpness, we show that our theory has maximal (i.e., sharp) elements, which are all equivalent, and coincide with the set of POVMs that admit a repeatable measurement. Among the maximal elements, conventional non-degenerate observables are characterized as the canonical ones. More generally, we quantify sharpness in terms of a class of monotones, expressed as the EPR–Ozawa correlations between the given POVM and an arbitrary reference POVM. We show that one POVM can be transformed into another by means of a sharpness-non-increasing operation if and only if the former is sharper than the latter with respect to all monotones. Thus, our resource theory of sharpness is $complete$, in the sense that the comparison of all monotones provides a necessary and sufficient condition for the existence of a sharpness-non-increasing operation between two POVMs, and $operational$, in the sense that all monotones are in principle experimentally accessible.

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[1] John von Neumann. Mathematical foundations of quantum mechanics. Princeton University Press, 1955.

[2] Jaroslav Řeháček Matteo Paris, editor. Quantum State Estimation, volume 649 of Lecture Notes in Physics. Springer Berlin, Heidelberg, 2004. doi:10.1007/​b98673.

[3] János A. Bergou. Discrimination of quantum states. Journal of Modern Optics, 57(3):160–180, 2010. arXiv:https:/​/​doi.org/​10.1080/​09500340903477756, doi:10.1080/​09500340903477756.

[4] Michele Dall'Arno, Francesco Buscemi, and Takeshi Koshiba. Guesswork of a quantum ensemble. IEEE Transactions on Information Theory, 68(5):3139–3143, 2022. doi:10.1109/​TIT.2022.3146463.

[5] E. B. Davies and J. T. Lewis. An operational approach to quantum probability. Communications in Mathematical Physics, 17(3):239–260, 1970. doi:10.1007/​BF01647093.

[6] Masanao Ozawa. Optimal measurements for general quantum systems. Reports on Mathematical Physics, 18(1):11–28, 1980. URL: https:/​/​www.sciencedirect.com/​science/​article/​pii/​0034487780900361, doi:10.1016/​0034-4877(80)90036-1.

[7] Paul Busch, Pekka J. Lahti, and Peter Mittelstaedt. The Quantum Theory of Measurement. Springer Berlin Heidelberg, 1996. doi:10.1007/​978-3-540-37205-9.

[8] Claudio Carmeli, Teiko Heinonen, and Alessandro Toigo. Intrinsic unsharpness and approximate repeatability of quantum measurements. Journal of Physics A: Mathematical and Theoretical, 40(6):1303, jan 2007. URL: https:/​/​dx.doi.org/​10.1088/​1751-8113/​40/​6/​008, doi:10.1088/​1751-8113/​40/​6/​008.

[9] Serge Massar. Uncertainty relations for positive-operator-valued measures. Phys. Rev. A, 76:042114, Oct 2007. URL: https:/​/​doi.org/​10.1103/​PhysRevA.76.042114, doi:10.1103/​PhysRevA.76.042114.

[10] Paul Busch. On the sharpness and bias of quantum effects. Foundations of Physics, 39(7):712–730, 2009. doi:10.1007/​s10701-009-9287-8.

[11] Kyunghyun Baek and Wonmin Son. Unsharpness of generalized measurement and its effects in entropic uncertainty relations. Scientific Reports, 6(1):30228, 2016. doi:10.1038/​srep30228.

[12] Yizhou Liu and Shunlong Luo. Quantifying unsharpness of measurements via uncertainty. Phys. Rev. A, 104:052227, Nov 2021. URL: https:/​/​doi.org/​10.1103/​PhysRevA.104.052227, doi:10.1103/​PhysRevA.104.052227.

[13] Michał Oszmaniec, Leonardo Guerini, Peter Wittek, and Antonio Acín. Simulating positive-operator-valued measures with projective measurements. Phys. Rev. Lett., 119:190501, Nov 2017. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.119.190501, doi:10.1103/​PhysRevLett.119.190501.

[14] Michał Oszmaniec, Filip B. Maciejewski, and Zbigniew Puchała. Simulating all quantum measurements using only projective measurements and postselection. Phys. Rev. A, 100:012351, Jul 2019. URL: https:/​/​doi.org/​10.1103/​PhysRevA.100.012351, doi:10.1103/​PhysRevA.100.012351.

[15] Masanao Ozawa. Heisenberg's original derivation of the uncertainty principle and its universally valid reformulations. Current Science, 109(11):2006–2016, 2015. URL: http:/​/​www.jstor.org/​stable/​24906690.

[16] Masanao Ozawa. Quantum measuring processes of continuous observables. Journal of Mathematical Physics, 25:79–87, 1984. URL: https:/​/​aip.scitation.org/​doi/​10.1063/​1.526000, doi:10.1063/​1.526000.

[17] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91:025001, Apr 2019. URL: https:/​/​doi.org/​10.1103/​RevModPhys.91.025001, doi:10.1103/​RevModPhys.91.025001.

[18] Arindam Mitra. Quantifying unsharpness of observables in an outcome-independent way. International Journal of Theoretical Physics, 61(9):236, 2022. doi:10.1007/​s10773-022-05219-2.

[19] Masanao Ozawa. Perfect correlations between noncommuting observables. Physics Letters A, 335(1):11–19, 2005. URL: https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960104016986, doi:10.1016/​j.physleta.2004.12.003.

[20] Masanao Ozawa. Quantum perfect correlations. Annals of Physics, 321(3):744–769, 2006. URL: https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491605001399, doi:10.1016/​j.aop.2005.08.007.

[21] Francesco Buscemi, Eric Chitambar, and Wenbin Zhou. Complete resource theory of quantum incompatibility as quantum programmability. Phys. Rev. Lett., 124:120401, Mar 2020. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.124.120401, doi:10.1103/​PhysRevLett.124.120401.

[22] Kaiyuan Ji and Eric Chitambar. Incompatibility as a resource for programmable quantum instruments. arXiv:2112.03717, 2021. URL: https:/​/​arxiv.org/​abs/​2112.03717.

[23] Francesco Buscemi, Kodai Kobayashi, Shintaro Minagawa, Paolo Perinotti, and Alessandro Tosini. Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication. Quantum, 7:1035, June 2023. doi:10.22331/​q-2023-06-07-1035.

[24] David Blackwell. Equivalent Comparisons of Experiments. The Annals of Mathematical Statistics, 24(2):265–272, 1953. URL: http:/​/​www.jstor.org/​stable/​2236332, doi:10.1214/​aoms/​1177729032.

[25] Francesco Buscemi. Comparison of quantum statistical models: Equivalent conditions for sufficiency. Communications in Mathematical Physics, 310(3):625–647, 2012. doi:10.1007/​s00220-012-1421-3.

[26] Francesco Buscemi, Michael Keyl, Giacomo Mauro D'Ariano, Paolo Perinotti, and Reinhard F. Werner. Clean positive operator valued measures. Journal of Mathematical Physics, 46(8):082109, 2005. arXiv:https:/​/​doi.org/​10.1063/​1.2008996, doi:10.1063/​1.2008996.

[27] Gerhart Lüders. Über die zustandsänderung durch den meßprozeß. Annalen der Physik (Leipzig), 8:322–328, 1951. URL: https:/​/​onlinelibrary.wiley.com/​doi/​10.1002/​andp.19504430510?__cf_chl_jschl_tk__=pmd_7hAcGnF999WRAeI9xOpY4b6DLNLqziEFL03Izd9rh_g-1635253796-0-gqNtZGzNAjujcnBszQu9, doi:10.1002/​andp.19504430510.

[28] J.P. Gordon and W.H. Louisell. Simultaneous measurements of noncommuting observables. In P.L. Kelley, B. Lax, and P.E. Tannenwald, editors, Physics of Quantum Electronics: Conference Proceedings, pages 833–840. McGraw-Hill, 1966.

[29] Paul Busch, Marian Grabowski, and Pekka J. Lahti. Operational Quantum Physics. Lecture Notes in Physics. Springer Berlin Heidelberg, 1995. URL: https:/​/​link.springer.com/​book/​10.1007/​978-3-540-49239-9.

[30] F. Buscemi, G. M. D'Ariano, and P. Perinotti. There exist nonorthogonal quantum measurements that are perfectly repeatable. Phys. Rev. Lett., 92:070403, Feb 2004. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.92.070403, doi:10.1103/​PhysRevLett.92.070403.

[31] Michele Dall'Arno, Giacomo Mauro D'Ariano, and Massimiliano F. Sacchi. Informational power of quantum measurements. Phys. Rev. A, 83:062304, Jun 2011. URL: https:/​/​doi.org/​10.1103/​PhysRevA.83.062304, doi:10.1103/​PhysRevA.83.062304.

[32] Michele Dall'Arno, Francesco Buscemi, and Masanao Ozawa. Tight bounds on accessible information and informational power. Journal of Physics A: Mathematical and Theoretical, 2014.

[33] Francesco Buscemi and Gilad Gour. Quantum relative lorenz curves. Phys. Rev. A, 95:012110, Jan 2017. URL: https:/​/​doi.org/​10.1103/​PhysRevA.95.012110, doi:10.1103/​PhysRevA.95.012110.

[34] Michele Dall'Arno and Francesco Buscemi. Tight conic approximation of testing regions for quantum statistical models and measurements, 2023. URL: https:/​/​arxiv.org/​abs/​2309.16153, doi:10.48550/​arXiv.2309.16153.

[35] Hans Martens and Willem M. de Muynck. Nonideal quantum measurements. Foundations of Physics, 20(3):255–281, March 1990. doi:10.1007/​BF00731693.

[36] A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47(10):777–780, May 1935. doi:10.1103/​PhysRev.47.777.

[37] Francesco Buscemi, Nilanjana Datta, and Sergii Strelchuk. Game-theoretic characterization of antidegradable channels. Journal of Mathematical Physics, 55(9):092202, 2014. arXiv:https:/​/​doi.org/​10.1063/​1.4895918, doi:10.1063/​1.4895918.

[38] F. Buscemi. Degradable channels, less noisy channels, and quantum statistical morphisms: An equivalence relation. Problems of Information Transmission, 52(3):201–213, 2016. doi:10.1134/​S0032946016030017.

[39] Francesco Buscemi and Nilanjana Datta. Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes. Phys. Rev. A, 93:012101, Jan 2016. URL: https:/​/​doi.org/​10.1103/​PhysRevA.93.012101, doi:10.1103/​PhysRevA.93.012101.

[40] Paul Skrzypczyk and Noah Linden. Robustness of measurement, discrimination games, and accessible information. Phys. Rev. Lett., 122:140403, Apr 2019. doi:10.1103/​PhysRevLett.122.140403.

[41] Claudio Carmeli, Teiko Heinosaari, and Alessandro Toigo. Quantum incompatibility witnesses. Phys. Rev. Lett., 122:130402, Apr 2019. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.122.130402, doi:10.1103/​PhysRevLett.122.130402.

[42] Claudio Carmeli, Teiko Heinosaari, and Alessandro Toigo. Quantum guessing games with posterior information. Reports on Progress in Physics, 85(7):074001, jun 2022. URL: https:/​/​dx.doi.org/​10.1088/​1361-6633/​ac6f0e, doi:10.1088/​1361-6633/​ac6f0e.

[43] Charles H Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A Smolin, and William K Wootters. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett., 76(5):722–725, jan 1996. doi:10.1103/​PhysRevLett.76.722.

[44] Francesco Buscemi. All entangled quantum states are nonlocal. Phys. Rev. Lett., 108:200401, May 2012. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.108.200401, doi:10.1103/​PhysRevLett.108.200401.

[45] John Watrous. The theory of quantum information. Cambridge university press, 2018. doi:10.1017/​9781316848142.

[46] V. P. Belavkin. Optimal multiple quantum statistical hypothesis testing. Stochastics, 1(1-4):315–345, 1975. arXiv:https:/​/​doi.org/​10.1080/​17442507508833114, doi:10.1080/​17442507508833114.

[47] H. Barnum and E. Knill. Reversing quantum dynamics with near-optimal quantum and classical fidelity. Journal of Mathematical Physics, 43(5):2097–2106, 2002. doi:10.1063/​1.1459754.

[48] Roope Uola, Tristan Kraft, Jiangwei Shang, Xiao-Dong Yu, and Otfried Gühne. Quantifying quantum resources with conic programming. Phys. Rev. Lett., 122:130404, Apr 2019. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.122.130404, doi:10.1103/​PhysRevLett.122.130404.

[49] Michał Oszmaniec and Tanmoy Biswas. Operational relevance of resource theories of quantum measurements. Quantum, 3:133, April 2019. doi:10.22331/​q-2019-04-26-133.

[50] Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks. Phys. Rev. X, 9:031053, Sep 2019. URL: https:/​/​doi.org/​10.1103/​PhysRevX.9.031053, doi:10.1103/​PhysRevX.9.031053.

[51] Godfrey Harold Hardy, John Edensor Littlewood, and George Polya. Inequalities. Cambridge university press, 1952.

[52] Albert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: theory of majorization and its applications. Springer, 2010.

[53] Francesco Buscemi. Degradable channels, less noisy channels, and quantum statistical morphisms: An equivalence relation. Probl Inf Transm, 52:201–213, 2016. doi:10.1134/​S0032946016030017.

[54] Anna Jencova. Comparison of quantum channels and statistical experiments, 2015. URL: https:/​/​arxiv.org/​abs/​1512.07016, doi:10.48550/​ARXIV.1512.07016.

[55] Francesco Buscemi. Reverse data-processing theorems and computational second laws. In Masanao Ozawa, Jeremy Butterfield, Hans Halvorson, Miklós Rédei, Yuichiro Kitajima, and Francesco Buscemi, editors, Reality and Measurement in Algebraic Quantum Theory, pages 135–159, Singapore, 2018. Springer Singapore.

[56] Francesco Buscemi, David Sutter, and Marco Tomamichel. An information-theoretic treatment of quantum dichotomies. Quantum, 3:209, December 2019. doi:10.22331/​q-2019-12-09-209.

[57] Anna Jencova. A general theory of comparison of quantum channels (and beyond). IEEE Transactions on Information Theory, 67(6):3945–3964, 2021. doi:10.1109/​TIT.2021.3070120.

[58] David Schmid, Denis Rosset, and Francesco Buscemi. The type-independent resource theory of local operations and shared randomness. Quantum, 4:262, April 2020. doi:10.22331/​q-2020-04-30-262.

[59] Wenbin Zhou and Francesco Buscemi. General state transitions with exact resource morphisms: a unified resource-theoretic approach. Journal of Physics A: Mathematical and Theoretical, 53(44):445303, oct 2020. URL: https:/​/​dx.doi.org/​10.1088/​1751-8121/​abafe5, doi:10.1088/​1751-8121/​abafe5.

[60] Denis Rosset, David Schmid, and Francesco Buscemi. Type-independent characterization of spacelike separated resources. Phys. Rev. Lett., 125:210402, Nov 2020. URL: https:/​/​doi.org/​10.1103/​PhysRevLett.125.210402, doi:10.1103/​PhysRevLett.125.210402.

[61] Denis Rosset, Francesco Buscemi, and Yeong-Cherng Liang. Resource theory of quantum memories and their faithful verification with minimal assumptions. Phys. Rev. X, 8:021033, May 2018. URL: https:/​/​doi.org/​10.1103/​PhysRevX.8.021033, doi:10.1103/​PhysRevX.8.021033.

[62] Francesco Buscemi. Complete positivity, markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations. Phys. Rev. Lett., 113:140502, Oct 2014. doi:10.1103/​PhysRevLett.113.140502.

[63] Bartosz Regula, Varun Narasimhachar, Francesco Buscemi, and Mile Gu. Coherence manipulation with dephasing-covariant operations. Phys. Rev. Research, 2:013109, Jan 2020. URL: https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013109, doi:10.1103/​PhysRevResearch.2.013109.

[64] Francesco Buscemi. Fully quantum second-law–like statements from the theory of statistical comparisons, 2015. URL: https:/​/​arxiv.org/​abs/​1505.00535, doi:10.48550/​ARXIV.1505.00535.

[65] Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics. Nature Communications, 9(1):5352, 2018. doi:10.1038/​s41467-018-06261-7.

[66] Cyril Branciard, Denis Rosset, Yeong-Cherng Liang, and Nicolas Gisin. Measurement-Device-Independent Entanglement Witnesses for All Entangled Quantum States. Physical Review Letters, 110(6):060405, February 2013. doi:10.1103/​PhysRevLett.110.060405.

Cited by

[1] Francesco Buscemi, Kodai Kobayashi, Shintaro Minagawa, Paolo Perinotti, and Alessandro Tosini, "Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication", Quantum 7, 1035 (2023).

[2] Gennaro Zanfardino, Wojciech Roga, Masahiro Takeoka, and Fabrizio Illuminati, "Quantum resource theory of Bell nonlocality in Hilbert space", arXiv:2311.01941, (2023).

[3] Michele Dall'Arno and Francesco Buscemi, "Tight conic approximation of testing regions for quantum statistical models and measurements", arXiv:2309.16153, (2023).

[4] Ties-A. Ohst and Martin Plávala, "Symmetries and Wigner representations of operational theories", arXiv:2306.11519, (2023).

[5] Albert Rico and Karol Życzkowski, "Discrete dynamics in the set of quantum measurements", arXiv:2308.05835, (2023).

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