Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of non-local correlations. The uncertainty principle states that there exist pairs of non-compatible measurements on quantum systems such that their outcomes cannot be simultaneously predicted by any observer. Non-local correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in quantum mechanics in the presence of entanglement. Here, we show that these two essential properties of quantum mechanics are quantitatively related. Namely, we provide an entropic uncertainty relation that gives a lower bound on the uncertainty of the binary outcomes of two measurements in terms of the maximum Clauser-Horne-Shimony-Holt value that can be achieved using the same measurements. We discuss an application of this uncertainty relation to certify a quantum source using untrusted devices.
Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of non-local correlations. The uncertainty principle states that there exist pairs of non-compatible measurements on quantum systems such that their outcomes cannot be simultaneously predicted by any observer. Non-local correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in quantum mechanics in the presence of entanglement. Here, we show that these two essential properties of quantum mechanics are quantitatively related. Namely, we provide an entropic uncertainty relation that gives a lower bound on the uncertainty of the binary outcomes of two measurements in terms of the maximum Clauser-Horne-Shimony-Holt value that can be achieved using the same measurements. We discuss an application of this uncertainty relation to certify a quantum source using untrusted devices.
Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of nonlocal correlations. The uncertainty principle states that there exist pairs of incompatible measurements on quantum systems such that their outcomes cannot both be predicted. On the other hand, nonlocal correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in the presence of entanglement. Here, we show that these two fundamental quantum effects are quantitatively related. Namely, we provide an entropic uncertainty relation for the outcomes of two binary measurements, where the lower bound on the uncertainty is quantified in terms of the maximum Clauser-Horne-Shimony-Holt value that can be achieved with these measurements. We discuss applications of this uncertainty relation in quantum cryptography, in particular, to certify quantum sources using untrusted devices.
The notion of nonlocality implicitly implies there might be some kind of spooky action at a distance in nature, however, the validity of quantum mechanics has been well tested up to now. In this work it is argued that the notion of nonlocality is physically improper, the basic principle of locality in nature is well respected by quantum mechanics, namely, the uncertainty principle. We show that the quantum bound on the Clauser, Horne, Shimony, and Holt (CHSH) inequality can be recovered from the uncertainty relation in a multipartite setting. We further argue that the super-quantum correlation demonstrated by the nonlocal box is not physically comparable with the quantum one. The origin of the quantum structure of nature still remains to be explained, some post-quantum theory which is more complete in some sense than quantum mechanics is possible and might not necessarily be a hidden variable theory.
We propose an uncertainty-complementarity balance relation and build quantitative connections among non-locality, complementarity, and uncertainty. Our balance relation, which is formulated in a theory-independent manner, states that for two measurements performed sequentially, the complementarity demonstrated in the first measurement in terms of disturbance is no greater than the uncertainty of the first measurement. Quantum theory respects our balance relation, from which the Tsirelson bound can be derived, up to an inessential assumption. In the simplest Bell scenario, we show that the bound of Clauser-Horne-Shimony-Holt inequality for a general non-local theory can be expressed as a function of the balance strength, a constance for the given theory. As an application, we derive the balance strength as well as the nonlocal bound of Popescu-Rohrlich box. Our results shed light on quantitative connections among three fundamental concepts, i.e., uncertainty, complementarity and non-locality.
Abstract Bell nonlocality refers to the nonlocal correlation strength among quantum systems that cannot be duplicated using local hidden variables, and uncertainty relations set the lower bound on the sum or product of variances for incompatible observables within a quantum system. In this work, the study of linking these two fundamental quantum features in terms of statistical characteristics within Bell tests is initiated. Two statistics determine the state‐dependent quantum upper bounds of the Bell values: the expectation values of local observables, and the joint uncertainty that is the product of local uncertainty. Specifically, it is showed that the joint measurement uncertainty determines the upper bounds of covariant Bell inequalities in the quantum region, with more joint measurement uncertainty required for more covariant correlation strengths. Conversely, the covariant Bell value defines the lower bound of joint uncertainty. In addition, given the maximally entangled states, Tsirelson's bounds of Bell inequalities can be achieved when the joint uncertainty reaches the maximum.
The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a fundamental distinction between the complementarity of observables in terms of uncertainty and in terms of information.
In quantum mechanics, nonlocality (a violation of a Bell inequality) is intimately linked to complementarity, by which we mean that consistently assigning values to different observables at the same time is not possible. Nonlocality can only occur when some of the relevant observables do not commute, and this noncommutativity makes the observables complementary. Beyond quantum mechanics, the concept of complementarity can be formalized in several distinct ways. Here we describe some of these possible formalizations and ask how they relate to nonlocality. We partially answer this question by describing two toy theories which display nonlocality and obey the no-signaling principle, although each of them does not display a certain kind of complementarity. The first toy theory has the property that it maximally violates the Clauser, Horn, Shimony, and Holt (CHSH) inequality, although the corresponding local observables are pairwise jointly measurable. The second toy theory also maximally violates the CHSH inequality, although its state space is classical and all measurements are mutually nondisturbing: if a measurement sequence contains some measurement twice with any number of other measurements in between, then these two measurements give the same outcome with certainty.