Additivity of entropic uncertainty relations

René Schwonnek

Institut für Theoretische Physik, Leibniz Universität Hannover, Germany

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We consider the uncertainty between two pairs of local projective measurements performed on a multipartite system. We show that the optimal bound in any linear uncertainty relation, formulated in terms of the Shannon entropy, is additive. This directly implies, against naive intuition, that the minimal entropic uncertainty can always be realized by fully separable states. Hence, in contradiction to proposals by other authors, no entanglement witness can be constructed solely by comparing the attainable uncertainties of entangled and separable states. However, our result gives rise to a huge simplification for computing global uncertainty bounds as they now can be deduced from local ones.
Furthermore, we provide the natural generalization of the Maassen and Uffink inequality for linear uncertainty relations with arbitrary positive coefficients.

For pairs of local measurements performed on multipartite systems, it is shown that the optimal bound in any linear uncertainty relation, formulated in terms of Shannon entropies, is additive. This directly implies that minimal uncertainty can always be realized by fully separable states. Hence, we have an example for a task which is not improved by the use of entanglement. On the other hand, this gives rise to a huge simplification to the structure of global uncertainty bounds as they now can be deduced from local ones by a single letter formula. Furthermore, an extension of the Maassen and Uffink bound for arbitrary linear relations, with positive coefficients, is provided.

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[2] Qing-Hua Zhang and Shao-Ming Fei, "Entropic uncertainty relations with quantum memory in a multipartite scenario", Physical Review A 108 1, 012211 (2023).

[3] Bo-Fu Xie, Fei Ming, Dong Wang, Liu Ye, and Jing-Ling Chen, "Optimized entropic uncertainty relations for multiple measurements", Physical Review A 104 6, 062204 (2021).

[4] Yuan-Yuan Zhao, Guo-Yong Xiang, Xiao-Min Hu, Bi-Heng Liu, Chuan-Feng Li, Guang-Can Guo, René Schwonnek, and Ramona Wolf, "Entanglement Detection by Violations of Noisy Uncertainty Relations: A Proof of Principle", Physical Review Letters 122 22, 220401 (2019).

[5] René Schwonnek, Koon Tong Goh, Ignatius W. Primaatmaja, Ernest Y.-Z. Tan, Ramona Wolf, Valerio Scarani, and Charles C.-W. Lim, "Device-independent quantum key distribution with random key basis", Nature Communications 12 1, 2880 (2021).

[6] Alberto Riccardi, Giovanni Chesi, Chiara Macchiavello, and Lorenzo Maccone, "Tight Bounds from Multiple‐Observable Entropic Uncertainty Relations", Annalen der Physik 2400020 (2024).

[7] Ana Costa, Roope Uola, and Otfried Gühne, "Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems", Entropy 20 10, 763 (2018).

[8] Timo Simnacher, Nikolai Wyderka, René Schwonnek, and Otfried Gühne, "Entanglement detection with scrambled data", Physical Review A 99 6, 062339 (2019).

[9] Mario Berta, David Sutter, and Michael Walter, "Quantum Brascamp–Lieb Dualities", Communications in Mathematical Physics 401 2, 1807 (2023).

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