Heisenberg's uncertainty principle was originally formulated in 1927 as a quantitative relation between the "mean error" of a measurement of one observable and the disturbance thereby caused on another observable. Heisenberg derived this famous relation under an additional assumption on quantum measurements that has been abandoned in the modern theory, and its universal validity was questioned in a debate on the sensitivity limit to gravitational-wave detectors in 1980s. A universally valid form of the error-disturbance relation was shown to be derived in the modern framework of general quantum measurements in 2003. We have experienced a considerable progress in theoretical and experimental study of error-disturbance relations in the last decade. In 2013 Branciard showed a new stronger form of universally valid error-disturbance relations, one of which is proved tight for spin measurements carried out in "pure" states. Nevertheless, a recent information-theoretical study of error-disturbance relations has suggested that Branciard relations can be considerably strengthened for measurements in mixed states. Here, we show a method for strengthening Branciard relations in mixed states and derive several new universally valid and stronger error-disturbance relations in mixed states. In particular, it is proved that one of them gives an ultimate error-disturbance relation for spin measurements, which is tight in any state. The new relations will play an important role in applications to state estimation problems including quantum cryptographic scenarios.
Heisenberg's uncertainty principle was originally formulated in 1927 as a quantitative relation between the "mean error" of a measurement of one observable and the disturbance thereby caused on another observable. Heisenberg derived this famous relation under an additional assumption on quantum measurements that has been abandoned in the modern theory, and its universal validity was questioned in a debate on the sensitivity limit to gravitational-wave detectors in 1980s. A universally valid form of the error-disturbance relation was shown to be derived in the modern framework of general quantum measurements in 2003. We have experienced a considerable progress in theoretical and experimental study of error-disturbance relations in the last decade. In 2013 Branciard showed a new stronger form of universally valid error-disturbance relations, one of which is proved tight for spin measurements carried out in "pure" states. Nevertheless, a recent information-theoretical study of error-disturbance relations has suggested that Branciard relations can be considerably strengthened for measurements in mixed states. Here, we show a method for strengthening Branciard relations in mixed states and derive several new universally valid and stronger error-disturbance relations in mixed states. In particular, it is proved that one of them gives an ultimate error-disturbance relation for spin measurements, which is tight in any state. The new relations will play an important role in applications to state estimation problems including quantum cryptographic scenarios.
The indeterminacy inherent in quantum measurements is an outstanding character of quantum theory, which manifests itself typically in the uncertainty principle. In the last decade, several universally valid forms of error-disturbance uncertainty relations were derived for completely general quantum measurements for arbitrary states. Subsequently, Branciard established a form that is optimal for spin measurements for some pure states. However, the bound in his inequality is not stringent for mixed states. One of the present authors recently derived a new bound tight in the corresponding mixed state case. Here, a neutron-optical experiment is carried out to investigate this new relation: it is tested whether error and disturbance of quantum measurements disappear or persist in mixing up the measured ensemble. The attainability of the new bound is experimentally observed, falsifying the tightness of Branciard's bound for mixed spin states.
Heisenberg’s uncertainty principle in a formulation of uncertainties, intrinsic to any quantum system, is rigorously proven and demonstrated in various quantum systems. Nevertheless, Heisenberg’s original formulation of the uncertainty principle was given in terms of a reciprocal relation between the error of a position measurement and the thereby induced disturbance on a subsequent momentum measurement. However, a naive generalization of a Heisenberg-type error-disturbance relation for arbitrary observables is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid, Ozawa’s relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance under certain conditions. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg’s original EDUR is violated, and Ozawa’s and Branciard’s EDURs are valid in a wide range of experimental parameters, as well as the tightness of Branciard’s relation.
The quantification of the ``measurement uncertainty''aspect of Heisenberg's uncertainty principle---that is, the study of trade-offs between accuracy and disturbance, or between accuracies in an approximate joint measurement on two incompatible observables---has regained a lot of interest recently. Several approaches have been proposed and debated. In this paper we consider Ozawa's definitions for inaccuracies (as root-mean-square errors) in approximate joint measurements, and study how these are constrained in different cases, whether one specifies certain properties of the approximations---namely their standard deviations and/or their bias---or not. Extending our previous work [C. Branciard, Proc. Natl. Acad. Sci. USA 110, 6742 (2013)], we derive error-trade-off relations, which we prove to be tight for pure states. We show explicitly how all previously known relations for Ozawa's inaccuracies follow from ours. While our relations are in general not tight for mixed states, we show how these can be strengthened and how tight relations can still be obtained in that case.
Heisenberg's uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: Although Heisenberg's first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg's intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a tight relation characterizing the optimal tradeoff between the error on one observable vs. the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario.
The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.
Heisenberg's uncertainty principle is probably the most famous statement of quantum physics and its essential aspects are well described by a formulations in terms of standard deviations. However, a naive Heisenberg-type error-disturbance relation is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid Ozawa's relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg's original EDUR is violated, and the Ozawa's and Branciard's EDURs are valid in a wide range of experimental parameters, applying a new measurement procedure referred to as two-state method.