AI 中文总结
本文严格推导了基于后验可观测量和酉变换的误差-扰动关系，证明其在单个测量层面成立，并验证了传统平均形式的有效性，为理解测量基本权衡提供了理论基础。
AI 中文摘要
海森堡最初在测量后状态必须与肯纳德-罗伯逊不确定关系保持一致的假设下设想了误差-扰动关系。这意味着测量误差必须通过后验可观测量$\hat{x}_t$来表述，而不是先验可观测量$\hat{x}_0$，因为后验测量误差决定了电子的测量后不确定性。基于arXiv:1504.0379中报告的初步概念基础，本文从可观测量的酉变换方程出发，对误差-扰动原理进行了严格的推导。后验探针可观测量$\hat{X}_t$的每个读数$X$完成一次单个测量事件，产生特定的条件对象状态。肯纳德-罗伯逊不确定关系必须对这些状态成立。具体而言，我们证明了误差$\epsilon_{X}(\hat{x}_t)$与扰动$\eta_{X}(\hat{p}_0)$之间的不确定关系在单个测量层面上严格成立，其特征由特定读数$X$决定。我们还验证了传统的误差-扰动关系（其中误差通过对无条件状态进行平均来评估）成立。我们的结果为理解单个测量结果中的基本权衡提供了精炼的理论基础。
英文摘要
Heisenberg originally envisioned the error-disturbance relation under the premise that the state after measurement must remain consistent with the Kennard-Robertson uncertainty relation. This implies that the measurement error must be formulated via a posterior observable $\hat{x}_t$, rather than a prior observable $\hat{x}_0$, because the posterior measurement error determines the post-measurement uncertainty of the electron. Building upon the preliminary conceptual foundation reported in arXiv:1504.03779, this paper presents a rigorous derivation of the error-disturbance principle formulated from the unitary transformation equations of observables. Each readout $X$ of a posterior probe observable $\hat{X}_t$ completes a single measurement event, producing a specific conditional object state. The Kennard-Robertson uncertainty relation must hold for these states. Specifically, we show that the uncertainty relation between the error $ε_{X}(\hat{x}_t)$ and the disturbance $η_{X}(\hat{p}_0)$ holds strictly at the level of individual measurements, characterized by the specific readout $X$. We also verify that conventional error-disturbance relations, where the errors are evaluated by averaging over the unconditioned state, hold true. Our results provide a refined theoretical basis for understanding the fundamental trade-off in individual measurement outcomes.
CommentsThis paper builds upon the preliminary framework presented in arXiv:1504.03779
