2024/03/28 by Kin-ya Oda, Oda, Kin-ya, Naoya Ogawa +1 · 1 voice · 1 citation
Decision Sciences · Mathematics · #Scientific Measurement and Uncertainty Evaluation #Statistical and numerical algorithms #gr-qc #hep-ph #hep-th #quant-ph
paper · pdf · doi:10.48550/arxiv.2403.19440
openalex publication_date 2024/03/28 · openalex created_date 2024/03/30 · openalex updated_date 2026/07/28
We point out that the Gaussian wave-packet formalism can serve as a concrete realization of the joint measurement of position and momentum, which is an essential element in understanding Heisenberg's original philosophy of the uncertainty principle, in line with the universal framework of error, disturbance, and their uncertainty relations developed by Lee and Tsutsui. We show that our joint measurement in the Gaussian phase space, being a Positive-Operator-Valued-Measure (POVM) measurement, smoothly interpolates between the projective measurements of position and momentum. We, for the first time, have obtained the Lee-Tsutsui (LT) error and the refined Lee error for the position-momentum measurement. We find that the LT uncertainty relation becomes trivial, 0 = 0, in the limiting case of projective measurement of either position or momentum. Remarkably, in contrast to the LT relation, the refined Lee uncertainty relation, which assesses errors for local representability, provides a constant lower bound unaffected by these limits and is invariably saturated, for a pure Gaussian initial state. The obtained lower bound is in agreement with Heisenberg's value.