# Uncertainty relations: An operational approach to the error-disturbance tradeoff

Joseph M. Renes1, Volkher B. Scholz1,2, and Stefan Huber1,3

1Institute for Theoretical Physics, ETH Zürich, Switzerland
2Department of Physics, Ghent University, Belgium
3Department of Mathematics, Technische Universität München, Germany

### Abstract

The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.

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### Cited by

[1] Krzysztof Urbanowski, "Uncertainty Relations: Curiosities and Inconsistencies", arXiv:2010.08339, Symmetry 12 10, 1640 (2020).

[2] T Bullock and P Busch, "Measurement uncertainty relations: characterising optimal error bounds for qubits", Journal of Physics A: Mathematical and Theoretical 51 28, 283001 (2018).

[3] Anna-Lena K. Hashagen and Michael M. Wolf, "Universality and Optimality in the Information–Disturbance Tradeoff", Annales Henri Poincaré 20 1, 219 (2019).

[4] Hubert de Guise, Lorenzo Maccone, Barry C. Sanders, and Namrata Shukla, "State-independent uncertainty relations", Physical Review A 98 4, 042121 (2018).

[5] Felipe G. Lacerda, Joseph M. Renes, and Renato Renner, "Classical Leakage Resilience from Fault-Tolerant Quantum Computation", Journal of Cryptology 32 4, 1071 (2019).

[6] Ilya Kull, Philippe Allard Guérin, and Frank Verstraete, "Uncertainty and trade-offs in quantum multiparameter estimation", Journal of Physics A: Mathematical and Theoretical 53 24, 244001 (2020).

[7] René Schwonnek, "Additivity of entropic uncertainty relations", arXiv:1801.04602, Quantum 2, 59 (2018).

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