General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures

Kohdai Kuroiwa1,2 and Hayata Yamasaki3,4

1Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
2Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
3Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
4Institute for Quantum Optics and Quantum Information — IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria

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Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover $\textit{catalytic replication}$ of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce $\textit{consistent resource measures}$ that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.

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[2] Ludovico Lami, Bartosz Regula, Ryuji Takagi, and Giovanni Ferrari, "Framework for resource quantification in infinite-dimensional general probabilistic theories", Physical Review A 103 3, 032424 (2021).

[3] Zi-Wen Liu and Andreas Winter, "Many-body quantum magic", arXiv:2010.13817.

[4] Hayata Yamasaki, Madhav Krishnan Vijayan, and Min-Hsiu Hsieh, "Hierarchy of quantum operations in manipulating coherence and entanglement", arXiv:1912.11049.

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