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arXiv:2003.07203

A revision for Heisenberg uncertainty relation based on environment variable in the QCPB theory

Gen Wang

Published Mar 13, 2020• 1 citation
arXivHTMLPDF

Abstract

The Heisenberg uncertainty principle and its extensions are all still inequalities form which hold the superior approximate estimations. Based on quantum covariant Poisson bracket theory, we propose quantum geomertainty relation to modify and explain the uncertainty relation to positively give a complete description of reality that enhances the outcome of each measurement with certainty. It demonstrates that entanglement term exists between the observable and the environment and nicely explains how the environment has an effect on the measurement which causes the unavoidable influences.

DOI
10.48550/arxiv.2003.07203

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The Heisenberg uncertainty principle and its extensions are all still inequalities form which hold the superior approximate estimations. Based on quantum covariant Poisson bracket theory, we propose quantum geomertainty relation to modify and explain the uncertainty relation to positively give a complete description of reality that enhances the outcome of each measurement with certainty. It demonstrates that entanglement term exists between the observable and the environment and nicely explains how the environment has an effect on the measurement which causes the unavoidable influences.
The EPR argued that quantum mechanics is an incomplete description of reality. So far, the Heisenberg uncertainty principle and its extensions are all still inequalities form which hold the superior approximate estimations, a precise estimation and environmental variables have never appeared in any formula. With the indeterminism of observable quantities constrained by the uncertainty principle. The question arises whether there might be some deeper reality hidden beneath quantum mechanics, to be described by a more fundamental theory that can always predict the outcome of each measurement with certainty, this paper by using the QCPB attempts to answer this question. As a result of the QCPB theory, we geometrically propose an equality called quantum geomertainty relation (QGR) to modify the uncertainty relation based on the fundamental theory QCPB to positively give a complete description of reality that predicts the outcome of each measurement with certainty, meanwhile, the uncertainty relation is just a derivation from this quantum geometric certainty equality. It deals with the measurement in quantum equality for different manifolds equipped with various mathematical or physical structure. Accordingly, the environment joins the physical process, by taking environment variable as a geometric structure function in the QCPB into consideration, it has naturally solved the environment problem for the measurements. We demonstrate that entanglement term exists between the observable and the environment. Actually, the QCPB nicely explains how the environment has an effect on the measurement which causes the unavoidable influences. Conversely, we state that quantum mechanics is incomplete assuredly. Doubtlessly, the QCPB is surely a new way for such complete description of reality.
Abstract One of the foundational principles of quantum mechanics is Heisenberg’s uncertainty relation. According to this relation, no one can measure both the position of a particle and how fast it is moving simultaneously no matter how precise the measuring instruments. Heisenberg’s uncertainty relation and the associated Bohr’s principle of complementarity are explained using simple examples. The idea that measurement disturbs the system is a source of the uncertainty. The philosophical question whether the uncertainty represents only our inability to measure the properties precisely or it has much deeper consequences related to the question of reality is also discussed.
The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a fundamental distinction between the complementarity of observables in terms of uncertainty and in terms of information.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.
The Heisenberg uncertainty principle has a more precise formulation in terms of inequalities involving quantum entropies. Currently known entropic uncertainty relations are presented; they capture and extend Heisenberg's idea of the unpredictability of the outcomes of incompatible measurements. Distinct results are obtained for finite- and infinite-dimensional Hilbert spaces. Applications are surveyed, including the formulation of entanglement witnesses, current ideas about wave-particle duality, and the analysis of quantum cryptography.
For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.