Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication

Francesco Buscemi1, Kodai Kobayashi1, Shintaro Minagawa1, Paolo Perinotti2,3, and Alessandro Tosini2,3

1Department of Mathematical Informatics, Nagoya University, Furo-cho, Chikusa-ku, 464-8601 Nagoya, Japan
2QUIT Group, Department of Physics, University of Pavia, via Bassi 6, 27100 Pavia, Italy
3INFN Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

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While there is general consensus on the definition of incompatible POVMs, moving up to the level of instruments one finds a much less clear situation, with mathematically different and logically independent definitions of incompatibility. Here we close this gap by introducing the notion of $q-compatibility$, which unifies different notions of POVMs, channels, and instruments incompatibility into one hierarchy of resource theories of communication between separated parties. The resource theories that we obtain are $complete$, in the sense that they contain complete families of free operations and monotones providing necessary and sufficient conditions for the existence of a transformation. Furthermore, our framework is fully $operational$, in the sense that free transformations are characterized explicitly, in terms of local operations aided by causally-constrained directed classical communication, and all monotones possess a game-theoretic interpretation making them experimentally measurable in principle. We are thus able to pinpoint exactly what each notion of incompatibility consists of, in terms of information-theoretic resources.

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[3] Leevi Leppäjärvi and Michal Sedlák, "Incompatibility of quantum instruments", Quantum 8, 1246 (2024).

[4] Francesco Buscemi, Kodai Kobayashi, and Shintaro Minagawa, "A complete and operational resource theory of measurement sharpness", Quantum 8, 1235 (2024).

[5] Kaiyuan Ji and Eric Chitambar, "Incompatibility as a Resource for Programmable Quantum Instruments", PRX Quantum 5 1, 010340 (2024).

[6] Ning Gao, Dantong Li, Anchit Mishra, Junchen Yan, Kyrylo Simonov, and Giulio Chiribella, "Measuring Incompatibility and Clustering Quantum Observables with a Quantum Switch", Physical Review Letters 130 17, 170201 (2023).

[7] Stan Gudder, "Multi-Observables and Multi-Instruments", arXiv:2307.11223, (2023).

[8] Stanley Gudder, "A Theory of Quantum Instruments", arXiv:2305.17584, (2023).

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

[10] Stan Gudder, "Quantum Channel Conditioning and Measurement Models", arXiv:2403.08126, (2024).

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