The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.
 J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1, 195 (1964).
 S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Advances in quantum cryptography, Adv. Opt. Photon. 12, 1012 (2020).
 D. Cavalcanti and P. Skrzypczyk, Quantum steering: a review with focus on semidefinite programming, Reports on Progress in Physics 80, 024001 (2016a).
 J. I. de Vicente, On nonlocality as a resource theory and nonlocality measures, Journal of Physics A: Mathematical and Theoretical 47, 424017 (2014).
 D. Cavalcanti and P. Skrzypczyk, Quantitative relations between measurement incompatibility, quantum steering, and nonlocality, Phys. Rev. A 93, 052112 (2016b).
 S.-L. Chen, C. Budroni, Y.-C. Liang, and Y.-N. Chen, Natural framework for device-independent quantification of quantum steerability, measurement incompatibility, and self-testing, Phys. Rev. Lett. 116, 240401 (2016).
 L. Tendick, H. Kampermann, and D. Bruß, Quantifying necessary quantum resources for nonlocality, Phys. Rev. Research 4, L012002 (2022).
 A. Bera, T. Das, D. Sadhukhan, S. S. Roy, A. Sen(De), and U. Sen, Quantum discord and its allies: A review of recent progress, Reports on Progress in Physics 81, 024001 (2017).
 K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Operational resource theory of imaginarity, Phys. Rev. Lett. 126, 090401 (2021).
 M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, Simulating positive-operator-valued measures with projective measurements, Phys. Rev. Lett. 119, 190501 (2017).
 L. Guerini, J. Bavaresco, M. T. Cunha, and A. Acín, Operational framework for quantum measurement simulability, Journal of Mathematical Physics 58, 092102 (2017).
 P. Skrzypczyk and N. Linden, Robustness of measurement, discrimination games, and accessible information, Phys. Rev. Lett. 122, 140403 (2019).
 R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Gühne, Quantifying quantum resources with conic programming, Phys. Rev. Lett. 122, 130404 (2019).
 S. Designolle, R. Uola, K. Luoma, and N. Brunner, Set coherence: Basis-independent quantification of quantum coherence, Phys. Rev. Lett. 126, 220404 (2021).
 R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks, Phys. Rev. X 9, 031053 (2019).
 A. F. Ducuara and P. Skrzypczyk, Operational interpretation of weight-based resource quantifiers in convex quantum resource theories, Phys. Rev. Lett. 125, 110401 (2020).
 R. Uola, C. Budroni, O. Gühne, and J.-P. Pellonpää, One-to-one mapping between steering and joint measurability problems, Phys. Rev. Lett. 115, 230402 (2015).
 M. Piani and J. Watrous, Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 114, 060404 (2015).
 S. Designolle, M. Farkas, and J. Kaniewski, Incompatibility robustness of quantum measurements: a unified framework, New J. Phys. 21, 113053 (2019a).
 P. Skrzypczyk, M. Navascués, and D. Cavalcanti, Quantifying Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 112, 180404 (2014).
 T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying Coherence, Phys. Rev. Lett. 113, 140401 (2014).
 R. Uola, T. Bullock, T. Kraft, J.-P. Pellonpää, and N. Brunner, All quantum resources provide an advantage in exclusion tasks, Phys. Rev. Lett. 125, 110402 (2020b).
 T.-C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68, 042307 (2003).
 Y. Liu and X. Yuan, Operational resource theory of quantum channels, Phys. Rev. Research 2, 012035 (2020).
 B. Dakić, V. Vedral, and C. Brukner, Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett. 105, 190502 (2010).
 R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Operational advantage of quantum resources in subchannel discrimination, Phys. Rev. Lett. 122, 140402 (2019).
 H.-Y. Ku, S.-L. Chen, C. Budroni, A. Miranowicz, Y.-N. Chen, and F. Nori, Einstein-Podolsky-Rosen steering: Its geometric quantification and witness, Phys. Rev. A 97, 022338 (2018).
 Z. Puchała, L. Pawela, A. Krawiec, and R. Kukulski, Strategies for optimal single-shot discrimination of quantum measurements, Phys. Rev. A 98, 042103 (2018).
 P. Skrzypczyk, I. Šupić, and D. Cavalcanti, All sets of incompatible measurements give an advantage in quantum state discrimination, Phys. Rev. Lett. 122, 130403 (2019).
 C. Carmeli, T. Heinosaari, and A. Toigo, State discrimination with postmeasurement information and incompatibility of quantum measurements, Phys. Rev. A 98, 012126 (2018).
 J. Bae, D. Chruściński, and M. Piani, More entanglement implies higher performance in channel discrimination tasks, Phys. Rev. Lett. 122, 140404 (2019).
 C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, and G. Adesso, Robustness of coherence: An operational and observable measure of quantum coherence, Phys. Rev. Lett. 116, 150502 (2016).
 Y. Kuramochi, Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements (2020), arXiv:2002.03504.
 R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Operational framework for nonlocality, Phys. Rev. Lett. 109, 070401 (2012).
 T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, Journal of Physics A: Mathematical and Theoretical 49, 123001 (2016).
 S. Designolle, P. Skrzypczyk, F. Fröwis, and N. Brunner, Quantifying measurement incompatibility of mutually unbiased bases, Phys. Rev. Lett. 122, 050402 (2019b).
 R. Cleve, P. Hoyer, B. Toner, and J. Watrous, Consequences and limits of nonlocal strategies, in Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. (IEEE, 2004).
 T. Heinosaari, J. Kiukas, D. Reitzner, and J. Schultz, Incompatibility breaking quantum channels, Journal of Physics A: Mathematical and Theoretical 48, 435301 (2015b).
 D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Bell inequalities for arbitrarily high-dimensional systems, Phys. Rev. Lett. 88, 040404 (2002).
 M. Grant and S. Boyd, in Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, edited by V. Blondel, S. Boyd, and H. Kimura (Springer-Verlag Limited, 2008) pp. 95–110.
 K. Toh, M. Todd, and R. Tutuncu, Sdpt3 — a Matlab software package for semidefinite programming, Optimization Methods and Software (1999).
 D. Popovici and Z. Sebestyén, Norm estimations for finite sums of positive operators, Journal of Operator Theory 56, 3 (2006).
 J. Bavaresco, M. T. Quintino, L. Guerini, T. O. Maciel, D. Cavalcanti, and M. T. Cunha, Most incompatible measurements for robust steering tests, Phys. Rev. A 96, 022110 (2017).
 A. Klappenecker and M. Rötteler, Constructions of mutually unbiased bases, in Finite Fields and Applications, edited by G. L. Mullen, A. Poli, and H. Stichtenoth (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004) pp. 137–144.
 S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, A new proof for the existence of mutually unbiased bases, Algorithmica 34, 512 (2002).
 D. A. Levin, Y. Peres, and E. L. Wilmer, Markov chains and mixing times (American Mathematical Society, Providence, RI, 2009).
 A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization (Society for Industrial and Applied Mathematics, 2001).
 T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, Quantifying operations with an application to coherence, Phys. Rev. Lett. 122, 190405 (2019).
 Lucas Tendick, Hermann Kampermann, and Dagmar Bruß, "Distributed Quantum Incompatibility", Physical Review Letters 131 12, 120202 (2023).
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