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arXiv:1710.08305 · Journal of Physics Conference Series

Quantum information aspects of noncommutative quantum mechanics

Orfeu Bertolami A, Alex E. Bernardini, Pedro Leal

Published Jan 1, 2018• 3 citations
arXivHTMLPDF

Abstract

Some fundamental aspects related with the construction of Robertson-Schrödinger-like uncertainty-principle inequalities are reported in order to provide an overall description of quantumness, separability and nonlocality of quantum systems in the noncommutative phase-space. Some consequences of the deformed noncommutative algebra are also considered in physical systems of interest.

DOI
10.1088/1742-6596/952/1/012016

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Some fundamental aspects related with the construction of Robertson-Schrödinger-like uncertainty-principle inequalities are reported in order to provide an overall description of quantumness, separability and nonlocality of quantum systems in the noncommutative phase-space. Some consequences of the deformed noncommutative algebra are also considered in physical systems of interest.
A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.
We show that a possible violation of the Robertson-Schr\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase space (even if it violates the Robertson-Schr\"odinger uncertainty principle) will always be a quantum state of an appropriate noncommutative extension of quantum mechanics. Conversely, all canonical noncommutative extensions of quantum mechanics display states that violate the Robertson-Schr\"odinger uncertainty principle.
Abstract We report a universal improvement to the standard Robertson–Schrödinger uncertainty relation. Our result shows that the Robertson–Schrödinger lower bound can be supplemented by a new noncommutativity-induced term. This term represents a previously overlooked quantum contribution and becomes more pronounced as the state becomes more mixed. Moreover, it is expressed as the expectation value of a positive observable, namely the squared modulus of the commutator, and therefore preserves the direct, experimentally accessible character of the Robertson–Schrödinger relation. For two-level quantum systems, our relation becomes an exact equality for any state and any pair of observables, thereby ensuring the tightness of the bound in the strongest possible sense. The relation also yields, as a corollary, a complete proof of a general uncertainty bound that had previously been supported only by numerical evidence.
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.