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.
 Coles, Berta, Tomamichel, and Wehner, ``Entropic Uncertainty Relations and Their Applications'' Reviews of Modern Physics 89, 015002 (2017).
 Arthursand Kelly ``On the Simultaneous Measurement'' Bell System Technical Journal 44, 725-729 (1965).
 Davies ``Quantum Theory of Open Systems'' Academic Press (1976).
 Leonhardt, Böhmer, and Paul, ``Uncertainty Relations for Realistic Joint Measurements of Position and Momentum in Quantum Optics'' Optics Communications 119, 296-300 (1995).
 Werner ``The Uncertainty Relation for Joint Measurement of Position and Momentum'' Quantum Information and Computation 4, 546-562 (2004).
 Martensand W. M. Muynck ``Disturbance, Conservation Laws and the Uncertainty Principle'' Journal of Physics A: Mathematical and General 25, 4887 (1992).
 Branciard ``Error-Tradeoff and Error-Disturbance Relations for Incompatible Quantum Measurements'' Proceedings of the National Academy of Sciences 110, 6742-6747 (2013).
 Busch, Lahti, and Werner, ``Quantum Root-Mean-Square Error and Measurement Uncertainty Relations'' Reviews of Modern Physics 86, 1261-1281 (2014).
 Wolf ``Quantum Channels'' (2012).
 Paulsen ``Completely Bounded Maps'' Cambridge University Press (2003).
 Gilchrist, Langford, and Nielsen, ``Distance Measures to Compare Real and Ideal Quantum Processes'' Physical Review A 71, 062310 (2005).
 Devetak ``The Private Classical Capacity and Quantum Capacity of a Quantum Channel'' IEEE Transactions on Information Theory 51, 44-55 (2005).
 Renes ``Duality of Privacy Amplification against Quantum Adversaries and Data Compression with Quantum Side Information'' Proceedings of the Royal Society A 467, 1604-1623 (2011).
 Korzekwa, Jennings, and Rudolph, ``Operational Constraints on State-Dependent Formulations of Quantum Error-Disturbance Trade-off Relations'' Physical Review A 89, 052108 (2014).
 Sacchi ``Entanglement Can Enhance the Distinguishability of Entanglement-Breaking Channels'' Physical Review A 72, 014305 (2005).
 Rene Schwonnek, "Additivity of entropic uncertainty relations", Quantum 2, 59 arXiv:1801.04602 (2018).
 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).
 Anna-Lena K. Hashagen and Michael M. Wolf, "Universality and Optimality in the Information–Disturbance Tradeoff", Annales Henri Poincaré 20 1, 219 (2019).
 Hubert de Guise, Lorenzo Maccone, Barry C. Sanders, and Namrata Shukla, "State-independent uncertainty relations", Physical Review A 98 4, 042121 (2018).
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