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arXiv:2302.06002

Unified Derivation of Uncertainty Relations and Their Saturation Conditions

Chi-Kwong Li, Mikio Nakahara

Published Feb 12, 2023• 0 citations
arXivHTMLPDF

Abstract

We analyze uncertainty relations due to Kennard, Robertson, Schrödinger, Maccone and Pati in a unified way from matrix theory point of view. Short proofs are given to these uncertainty relations and characterizations of the saturation conditions are given.
DOI
10.48550/arxiv.2302.06002

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The existence of incompatible observables constitutes one of the most prominent characteristics of quantum mechanics (QM) and can be revealed and formalized through uncertainty relations. The Heisenberg-Robertson-Schrödinger uncertainty relation (HRSUR) was proved at the dawn of quantum formalism and is ever-present in the teaching and research on QM. Notwithstanding, the HRSUR possess the so called triviality problem. That is to say, the HRSUR yields no information about the possible incompatibility between two observables if the system was prepared in a state which is an eigenvector of one of them. After about 85 years of existence of the HRSUR, this problem was solved recently by Lorenzo Maccone and Arun K. Pati. In this article, we start doing a brief discussion of general aspects of the uncertainty principle in QM and recapitulating the proof of HRSUR. Afterwards we present in simple terms the proof of the Maccone-Pati uncertainty relation, which can be obtained basically via the application of the parallelogram law and Cauchy-Schwarz inequality.
Abstract The Heisenberg uncertainty relation, together with Robertson’s generalisation, serves as a fundamental concept in quantum mechanics, showing that noncommutative pairs of observables cannot be measured precisely. In this study, we explore the Robertson-type uncertainty relations to demonstrate their effectiveness in establishing a series of thermodynamic uncertainty relations and quantum speed limits in open quantum dynamics. The derivation utilises a scaled continuous matrix product state representation that maps the time evolution of the quantum continuous measurement to the time evolution of the system and field. Specifically, we consider the Maccone–Pati uncertainty relation, a refinement of the Robertson uncertainty relation, to derive thermodynamic uncertainty relations and quantum speed limits. These newly derived relations, which use a state orthogonal to the initial state, yield bounds that are tighter than previously known bounds. Moreover, we consider the Robertson–Schrödinger uncertainty, which extends the Robertson uncertainty relation. Our findings not only reinforce the significance of the Robertson-type uncertainty relations, but also expand its applicability in identifying uncertainty relations in open quantum dynamics.