Generalisations of Heisenberg's inequality

Masao Nagasawa

Gao Mingzhe

Abdessamad Belfakir, A. Belhaj, Y. Hassouni +2 more

Abstract In this paper, we address the optimal control problem in molecular systems, focusing on transitions within coherent states characterised by complex coefficients. Employing Hölder’s inequality, we establish a mathematical relationship between the energy requirement and the distance separating the initial and the target coherent states. A key part of our study is the application of this framework to the H2O molecule, specifically examining the local OH bond. Here, we demonstrate how energy requirements for the state transitions are influenced by the distance between these states. Furthermore, we investigate the effects of a heat bath coupled to the system, by analysing its impact on transferring the molecular system to different final coherent states. These coherent states are defined as almost eigenvectors of the Generalised Heisenberg Algebra (GHA) annihilation operator. By using the Perolomov approach, another type of coherent states for the Morse potential associated with the GHA can be constructed. By leveraging the GHA structure, we revisit and analyse Morse coherent states previously established for certain diatomic molecules, offering a deeper insight into the dynamics of state transitions under various conditions.

Sirawit Kajonsombat, Pongwit Srisangyingcharoen

A generalisation of the Leggett-Garg inequality (LGI) and the Bell-CHSH inequality is proposed in this work, by replacing a classical probability notion with an extended probability notion. The extended probability serves as an underlying layer of reality to the classical probability. Within the consistent history framework, the underlying layer is so-called a non-settleable history. Hence, the bounds of the LGIs and Bell-CHSH inequalities are extended. The resulting generalised inequalities are satisfied without violation for arbitrary measurement settings. Furthermore, generalised Macro- realism and generalised Local realism are introduced and analysed within the extended probability framework without relying on an explicit measurement. This opens up further possibilities on one of the great pursuits in physics, understanding on how a quantum system can possess classical properties.

Mihai Octavian Dima, Marian Petre

The (seldomly quoted) generalised-Heisenberg uncertainty relations are an effect of the quantum correlation coefficient inequalities. The quantum correlation coefficient determines how much a state can be compacted and on what basis. It is shown that how this can be used to best compress a signal (such as a radio wave, or a 2D laser complex field at a focal plane) while at the same time encrypting the signal.

Luis de la Peña