Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Operational theories, ontological models and contextuality Contextuality for general probabilistic theories 3.1 GPT systems 3.2 Operational theory associated to a GPT system 3.3 Simulations of GPT systems 3.4 Properties of univalent simulations Hierarchy of contextuality and 4.1 Motivation and the resource theory 4.2 Contextuality of composite systems 4.3 Quantifying contextuality via the classical excess 4.4 Parity oblivious multiplexing success probability with free classical resources as a measure of contextuality Discussion 5.1 Contextuality and information erasure 5.2 Relation with previous works on contextuality and GPTs Conclusion, Acknowledgments, and References Operational theories, ontological models and contextuality Operational theories, ontological models and contextuality Operational theories, ontological models and contextuality Contextuality for general probabilistic theories 3.1 GPT systems 3.2 Operational theory associated to a GPT system 3.3 Simulations of GPT systems 3.4 Properties of univalent simulations Contextuality for general probabilistic theories Contextuality for general probabilistic theories 3.1 GPT systems 3.1 GPT systems 3.2 Operational theory associated to a GPT system 3.2 Operational theory associated to a GPT system 3.3 Simulations of GPT systems 3.3 Simulations of GPT systems 3.4 Properties of univalent simulations 3.4 Properties of univalent simulations Hierarchy of contextuality and 4.1 Motivation and the resource theory 4.2 Contextuality of composite systems 4.3 Quantifying contextuality via the classical excess 4.4 Parity oblivious multiplexing success probability with free classical resources as a measure of contextuality Hierarchy of contextuality and 4.1 Motivation and the resource theory Hierarchy of contextuality and 4.1 Motivation and the resource theory 4.2 Contextuality of composite systems 4.2 Contextuality of composite systems 4.3 Quantifying contextuality via the classical excess 4.3 Quantifying contextuality via the classical excess 4.4 Parity oblivious multiplexing success probability with free classical resources as a measure of contextuality 4.4 Parity oblivious multiplexing success probability with free classical resources as a measure of contextuality Discussion 5.1 Contextuality and information erasure 5.2 Relation with previous works on contextuality and GPTs Discussion 5.1 Contextuality and information erasure 5.1 Contextuality and information erasure 5.2 Relation with previous works on contextuality and GPTs 5.2 Relation with previous works on contextuality and GPTs Conclusion, Acknowledgments, and References Conclusion, Acknowledgments, and References Conclusion, Acknowledgments, and References A Physicality of the Holevo projection A Physicality of the Holevo projection 6 Conclusion Generalized contextuality, a leading notion of nonclassicality, is of crucial importance both in the foundations of quantum theory and in quantum information processing. Despite this, there is no complete characterisation of generalised contextuality as a resource. In this work we address this shortcoming. Based on recent developments formulating noncontextuality of GPT systems via simplex embeddability, we have defined a resource theory of contextuality of GPT systems in prepare-and-measure scenarios. The free resources are the noncontextual systems and the free operations are univalent simulations with free access to classical systems. Using these notions, we motivate a hierarchy of contextuality for GPT systems. A new contextuality monotone arises naturally from our considerations— the classical excess, which expresses the minimum error of a univalent simulation by the countably-infinite classical system. We have also shown how a standard witness of contextuality, in the form of the POM success probability, can be used to define a contextuality monotone. We have also discussed how GPT simulations could describe a physical process of information erasure that would explain the fine-tuning associated with contextuality[8] in a similar way to how the quantum equilibration process proposed by Valentini explains the fine-tuning associated with nonlocality in Bohmian mechanics. We have argued that the information erasure in this case would be different than a simple coarse-graining. An interesting avenue for future research would be to further characterize this kind of information erasure and propose a test to detect heat dissipation in experiments manifesting contextuality. Even though such a proposal is undoubtedly radical, we believe that it could explain the fine-tuning associated with contextuality without abandoning the ontological models framework. Another direction for further investigation is to develop a solid interpretation and establish the use-cases of the resource theory of GPT-contextuality that we provide. It is known that a resource theory of a given notion need not be unique. We have examples of this fact, like the resource theories of entanglement based on LOCC [81] and LOSR [82, 83] free operations, respectively. This does not mean that only one of them is “the correct resource theory of entanglement”. Rather, these resource theories may be applicable in different contexts or for different purposes. In the case of entanglement, one can say that LOCC operations are relevant for communication tasks in the context of quantum internet and LOSR operations are relevant for the study of entanglement in Bell scenarios. In this respect, we have discussed the relation of our work with other studies of generalized contextuality for GPTs in section 5.2. One that we did not mention is [72]. It would be interesting to find a relationship between our classical excess and the simulation cost of contextuality defined therein. Finally, it would be also interesting to extend the methods developed here to the realm of resource theories of more general fine-tunings [4], such as violations of time symmetry [84] and bounded ontological distinctness [85]. Acknowledgments The authors thank the participants of the PIMan workshop – Orange (CA), March 2019 – where the idea of this project originated. In particular, Luke Burns and Justin Dressel, who were part of the initial discussions on the project. The authors further thank Rafael Wagner for insightful explanations regarding the approach of Duarte and Amaral [44]. LC thanks Farid Shahandeh and TGa thanks Markus M¨uller for helpful discussions. This project started when LC was supported by the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust and by the Army Research Office (ARO) (Grant No. W911NF-18-1-0178). LC also acknowledges funding from the Einstein Research Unit “Perspectives of a Quantum Digital Transformation” and from the Horizon Europe project FoQaCiA, GA no.101070558. TGa acknowledges support from the Austrian Science Fund (FWF) via project P 33730-N. TGo acknowledges support from the Austrian Science Fund. This research was funded in whole or in part by the Austrian Science Fund (FWF) via the START Prize Y1261-N. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Colleges and Universities. For open access purposes, the authors have applied a CC BY public copyright license to any accepted manuscript version arising from this submission. References [1] David Schmid. Generalized noncontextuality. 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Galley, Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria and Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Vienna, Austria (thomas.galley@oeaw.ac.at); (3) Tomas Gonda, Institute for Theoretical Physics, University of Innsbruck, Austria (tomas.gonda@uibk.ac.at). Authors: Authors: (1) Lorenzo Catani, International Iberian Nanotechnology Laboratory, Av. Mestre Jose Veiga s/n, 4715-330 Braga, Portugal (lorenzo.catani4@gmail.com); (2) Thomas D. Galley, Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria and Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Vienna, Austria (thomas.galley@oeaw.ac.at); (3) Tomas Gonda, Institute for Theoretical Physics, University of Innsbruck, Austria (tomas.gonda@uibk.ac.at). This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv [8] A recent work that connects preparation contextuality and information erasure is [80], where the authors show that any ontological model reproducing the statistics of a sequential protocol involving incompatible projective measurements involves more information erasure than what operational quantum theory predicts. This fact is strictly related to the presence of preparation contextuality, as the same final quantum state in the protocol is represented by two different ontic distributions. This result can be seen as an example of fine-tuning of information erasure (i.e. more erasure at the ontological level than at the operational level), which, despite being interesting, differs from our idea of viewing contextuality as arising from a process of information erasure that explains the operational equivalences of distinct ontological representations.
