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  • Open Access

Uncertainty relations via conditional expectation and variance

Shiting Ma, Peng Zhou, Haoyu Chen, and Dong Wang*

  • *Contact author: dwang@ahu.edu.cn
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APS Open Sci. 1, 000158 – Published 1 October, 2026

DOI: https://doi.org/10.1103/959j-v71v

Abstract

The uncertainty principle constitutes a cornerstone of quantum mechanics. However, its conventional formulations do not adequately describe uncertainty in scenarios involving conditional measurement. In this work, we propose uncertainty relations based on conditional variance and conditional expectation. We define the conditional expectation operator E(Â|B̂), its dual operator E(B̂|Â), and a corresponding conditional covariance term Rcorr. We demonstrate that even for noncommuting observables  and B̂, their conditional expectation operators can possess a nonvanishing commutator. This leads to a strengthened uncertainty relation applicable to both pure and mixed states. In addition to the standard term arising from the noncommutativity of the observables themselves, our relation incorporates three distinct contributions: one from the noncommutativity of the conditional expectations, one from quantum coherence, and one from the conditional covariance. These contributions collectively yield a significantly tighter lower bound. Numerical simulations verify the generality and enhanced tightness of the proposed relation. Our results provide a refined theoretical tool for the joint estimation of multiple observables in composite quantum systems, with potential applications in quantum precision metrology and secure quantum key distribution.

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APS Open Sci. 1, 000158– Published 1 October, 2026
Vol. 1
  • Received 13 April 2026
  • Accepted 22 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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