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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Abstract

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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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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