Resource Marginal Problems

Chung-Yun Hsieh1,2, Gelo Noel M. Tabia3,4,5,6, Yu-Chun Yin7,4, and Yeong-Cherng Liang4,5

1H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK
2ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain
3Foxconn Quantum Computing Research Center, Taipei 114, Taiwan
4Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
5Physics Division, National Center for Theoretical Sciences, Taipei 106319, Taiwan
6Center for Quantum Technology, National Tsing Hua University, Hsinchu 300, Taiwan
7Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan

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We introduce the $\textit{resource marginal problems}$, which concern the possibility of having a resource-free target subsystem compatible with a $given$ collection of marginal density matrices. By identifying an appropriate choice of resource R and target subsystem T, our problems reduce, respectively, to the well-known $\textit{marginal problems}$ for quantum states and the problem of determining if a given quantum system is a resource. More generally, we say that a set of marginal states is $\textit{resource-free incompatible}$ with a target subsystem T if all global states compatible with this set must result in a resourceful state in T of type R. We show that this incompatibility $induces$ a resource theory that can be quantified by a monotone and obtain necessary and sufficient conditions for this monotone to be computable as a conic program with finite optimum. We further show, via the corresponding witnesses, that (1) resource-free incompatibility is equivalent to an operational advantage in some channel-discrimination tasks, and (2) some specific cases of such tasks fully characterize the convertibility between marginal density matrices exhibiting resource-free incompatibility. Through our framework, one sees a clear connection between any marginal problem – which implicitly involves some notion of incompatibility – for quantum states and a resource theory for quantum states. We also establish a close connection between the physical relevance of resource marginal problems and the ground state properties of certain many-body Hamiltonians. In terms of application, the universality of our framework leads, for example, to a further quantitative understanding of the incompatibility associated with the recently-proposed entanglement marginal problems and entanglement transitivity problems.

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Cited by

[1] Huan-Yu Ku, Chung-Yun Hsieh, and Costantino Budroni, "Measurement incompatibility cannot be stochastically distilled", arXiv:2308.02252, (2023).

[2] Chung-Yun Hsieh, Matteo Lostaglio, and Antonio Acín, "Quantum channel marginal problem", Physical Review Research 4 1, 013249 (2022).

[3] Chung-Yun Hsieh, Huan-Yu Ku, and Costantino Budroni, "Characterisation and fundamental limitations of irreversible stochastic steering distillation", arXiv:2309.06191, (2023).

[4] Chung-Yun Hsieh and Shin-Liang Chen, "A thermodynamic approach to quantifying incompatible instruments", arXiv:2402.13080, (2024).

[5] Chung-Yun Hsieh and Manuel Gessner, "General quantum resources provide advantages in work extraction tasks", arXiv:2403.18753, (2024).

[6] Gelo Noel M. Tabia, Kai-Siang Chen, Chung-Yun Hsieh, Yu-Chun Yin, and Yeong-Cherng Liang, "Entanglement transitivity problems", npj Quantum Information 8, 98 (2022).

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