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arXiv · 2504.20404

Beyond Robertson-Schr\"odinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs

Gen Kimura·Aina Mayumi·Hiromichi Ohno·Jaeha Lee·Dariusz Chruściński
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Abstract

We report a universal improvement to the standard Robertson--Schr\"odinger uncertainty relation. Our result shows that the Robertson--Schr\"odinger lower bound can be supplemented by a new noncommutativity-induced term. This term represents a previously overlooked quantum contribution and becomes more pronounced as the state becomes more mixed. Moreover, it is expressed as the expectation value of a positive observable, namely the squared modulus of the commutator, and therefore preserves the direct, experimentally accessible character of the Robertson--Schr\"odinger relation. For two-level quantum systems, our relation becomes an \emph{exact equality} for \emph{any} state and \emph{any} pair of observables, thereby ensuring the tightness of the bound in the strongest possible sense. The relation also yields, as a corollary, a complete proof of a general uncertainty bound that had previously been supported only by numerical evidence.

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Gen Kimura, Aina Mayumi, Hiromichi Ohno, Jaeha Lee, Dariusz Chruściński. 2025-04-29. Beyond Robertson-Schr\"odinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs. https://arxiv.org/abs/2504.20404

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