Abstract
Nonlocality, which is the key feature of quantum theory, has been linked with the uncertainty principle by fine-grained uncertainty relations, by considering combinations of outcomes for different measurements. However, this approach assumes that information about the system to be fine-grained is local, and does not present an explicitly computable bound. Here, we generalize above approach to general quasi-fine-grained uncertainty relations (QFGURs) which applies in the presence of quantum memory and provides conspicuously computable bounds to quantitatively link the uncertainty to entanglement and Einstein–Podolsky–Rosen (EPR) steering, respectively. Moreover, our QFGURs provide a framework to unify three important forms of uncertainty relations, i.e., universal uncertainty relations, the uncertainty principle in the presence of quantum memory, and fine-grained uncertainty relation. This result gives a direct significance to uncertainty principle, and allows us to determine whether a quantum measurement exhibits typical quantum correlations, meanwhile, it reveals a fundamental connection between basic elements of quantum theory, specifically, uncertainty measures, combined outcomes for different measurements, quantum memory, entanglement and EPR steering.
Highlights
The uncertainty principle, articulated in 1927 by Heisenberg [1], plays a crucial role in highlighting the non-classical nature of quantum probabilities
The challenge is, in order to acquire strong criteria of EPR steering and entanglement which are comparable with other approaches, we need to consider the optimal bound of quasi-fine-grained uncertainty relations (QFGURs) which we would leave for future work
Based on the notion of local probability relations from a measured system and a quantum memory, we prove a special format, namely a quasi-fine-grained uncertainty relation, which unifies universal uncertainty relations (UURs), UPQM, and FGUR
Summary
The uncertainty principle, articulated in 1927 by Heisenberg [1], plays a crucial role in highlighting the non-classical nature of quantum probabilities. Even though FGUR nicely presents a close connection between a quantum game in terms of Bell inequalities and uncertainty relation based on the winning condition (a particular choice of measurement outcomes), one would be curious can different aspects of quantum correlations, such as Einstein-Podolsky-Rosen (EPR) steering [45, 46] and entanglement be linked with uncertainty from a single framework with clearly computable bounds of FGURs. The question, naturally arises: can all these uncertainty relations be unified into a general form? For illustrative purposes of the general framework, we provide a numerical example and show that our approach gives a lower bound to test steering and entanglement than previous coarse-graining entropic functions
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