The uncertainty principle determines the non-locality of quantum mechanics

Abstract: Two central concepts of quantum mechanics are Heisenberg's uncertainty principle, and a subtle form of non-locality that Einstein famously called spooky action at a distance''. These two fundamental features have thus far been distinct concepts. Here we show that they are inextricably and quantitatively linked. Quantum mechanics cannot be more non-local with measurements that respect the uncertainty principle. In fact, the link between uncertainty and non-locality holds for all physical theories.More specifically, the degree of non-locality of any theory is determined by two factors -- the strength of the uncertainty principle, and the strength of a property calledsteering'', which determines which states can be prepared at one location given a measurement at another.

• The paper demonstrates that stronger uncertainty relations cap quantum non-locality by formalizing the role of steering.
• It employs entropic measures and Bell inequalities to rigorously quantify quantum measurement limitations and non-local effects.
• The findings pave the way for advanced quantum protocols by unifying core quantum mechanical concepts into actionable insights.

The Uncertainty Principle and Non-Locality in Quantum Mechanics

The paper "The uncertainty principle determines the non-locality of quantum mechanics" authored by Jonathan Oppenheim and Stephanie Wehner, addresses a nuanced relationship between two pivotal quantum mechanical concepts: the Heisenberg uncertainty principle and quantum non-locality, commonly referred to as "spooky action at a distance." The authors propose a formal link between these concepts, suggesting not only their interconnectedness but also a framework where the degree of non-locality in any physical theory is governed by the uncertainty principle and a property known as "steering."

Key Contributions

1. Uncertainty Principle and Quantum Measurements: The authors explore Heisenberg's uncertain relations, discussing how quantum mechanics intrinsically limits our ability to predict measurement outcomes with certainty, especially when dealing with non-commuting observables, such as position and momentum. The modern approach frames these restrictions using entropic measures like Shannon entropy to quantify the extent of uncertainty over various outcomes.
2. Non-Locality and Bell Inequalities: The principle of non-locality allows for correlations between outcomes of measurements on distinct quantum systems, surpassing what would be feasible under classical statistics. The study revisits Bell inequalities, rigorously discussing conditions under which quantum theories can exhibit non-local correlations without violating the no-signalling principle.
3. Steering: This concept is discussed in the context of Einstein-Podolsky-Rosen (EPR) paradox, where steering refers to the ability to influence the state of a distant part of the system via local measurements. The paper posits that steering is a key determinant in the extent to which quantum mechanics exhibits non-locality.

Relation Between Uncertainty, Steering, and Non-Locality

A salient feature of the paper is the derivation that non-locality in quantum mechanics is bounded by the uncertainty principle guided by the steerability of states. It provides a two-fold insight:

• The authors mathematically formalize that stronger uncertainty relations correlate with a higher limit on possible non-locality. This is quantified in fine-grained terms rather than merely relying on the coarse measures of traditional entropic uncertainty relations.
• For specific non-local games like CHSH, the degree of non-locality that can be achieved quantum mechanically is succinctly tied to the steerability to maximally certain states. This establishes a direct bridge linking the robustness of quantum correlations to inherent quantum uncertainty.

Implications and Future Perspectives

This investigation sets the stage for further exploration of quantum mechanical frameworks beyond classical interpretations. The relationship established might steer efforts in formulating new protocols in quantum information science, where exploiting non-local correlations is key—ranging from quantum cryptography to teleportation scenarios.

From a theoretical viewpoint, this work encourages revisiting other fundamental quantum phenomena under the lens of uncertainty and non-locality, potentially leading to more unified theories in quantum mechanics that can delineate the reach and limits of quantum correlations.

As quantum mechanics continues to underpin advancements in technology and fundamental science, the clarity and precision brought about by works such as this, that cleanly links foundational principles in new ways, is invaluable. The paper not only provides comprehensive theoretical support for the bound between non-locality and uncertainty but also invites future inquiry into whether such bounds are universally applicable across all proposed quantum theories and how they might be impacted by other physical constraints.