Extremum-entropy-based Heisenberg-like uncertainty relations

TL;DR

This paper develops extremum-entropy-based uncertainty relations for quantum systems, incorporating spin and spatial degrees of freedom, and demonstrates improved bounds over existing relations.

Contribution

It introduces a variational extremization approach using information-theoretic measures to derive sharper uncertainty-like relations for multi-dimensional fermionic systems.

Findings

Derived new uncertainty relations that include spin effects.

Achieved tighter bounds compared to previous uncertainty relations.

Applied the method to $d$-dimensional quantum systems with fermions.

Abstract

In this work we use the extremization method of various information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy) for $d$-dimensional quantum systems, which complementary describe the spreading of the quantum states of natural systems. Under some given constraints, usually one or two radial expectation values, this variational method allows us to determine an extremum-entropy distribution, which is the \textit{least-biased} one to characterize the state among all those compatible with the known data. Then we use it, together with the spin-dependent uncertainty-like relations of Daubechies-Thakkar type, as a tool to obtain relationships between the position and momentum radial expectation values of the type $\langle r^{\alpha}\rangle^{\frac{k}{\alpha}}\langle p^k\rangle\geq f(k,\alpha,q,N), q=2s+1$, for $d$-dimensional systems of $N$ fermions with spin $s$. The resulting uncertainty-like products, which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system, are shown to often improve the best corresponding relationships existing in the literature.