Beyond Robertson-Schrödinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918504797569024 |
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| author | Kimura, Gen Mayumi, Aina Ohno, Hiromichi Lee, Jaeha Chruściński, Dariusz |
| author_facet | Kimura, Gen Mayumi, Aina Ohno, Hiromichi Lee, Jaeha Chruściński, Dariusz |
| contents | We report a universal improvement to the standard Robertson--Schrödinger uncertainty relation. Our result shows that the Robertson--Schrödinger 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--Schrödinger 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_20404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond Robertson-Schrödinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs Kimura, Gen Mayumi, Aina Ohno, Hiromichi Lee, Jaeha Chruściński, Dariusz Quantum Physics Statistical Mechanics High Energy Physics - Theory Mathematical Physics We report a universal improvement to the standard Robertson--Schrödinger uncertainty relation. Our result shows that the Robertson--Schrödinger 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--Schrödinger 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. |
| title | Beyond Robertson-Schrödinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs |
| topic | Quantum Physics Statistical Mechanics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2504.20404 |