A Quantum Mechanical Bound for Space-Energy Cost with Respect to the Von Neumann Entropy

TL;DR

This paper critically examines a proposed quantum mechanical lower bound on space-energy cost related to von Neumann entropy, providing mathematical analysis, examples, and numerical results that suggest possible alternative bounds.

Contribution

It offers a mathematical critique of existing bounds, presents numerical investigations, and explores new approaches to relate energy-space cost to information stored in quantum systems.

Findings

Identified limitations in the existing inequality.

Numerical results suggest a different form of the bound.

Proposed an ansatz for a bound depending on stored information.

Abstract

This thesis discusses the possibility of uncertainty relations for space and energy given a state of fixed entropy. In particular, it discusses the results in the paper of Dam/Nguyen. There, the authors propose a lower bound for the mixed cost in energy and space required for physically storing information in a quantum mechanical system. We first critically examine the justifications for the bound given in the paper.This is done from a mathematical point of view, in contrast to the more physically motivated original paper.Then we give two examples that illustrate the limitations of this inequality. We also present numerical results to find an alternative energy-space bound in the finite dimensional version of this problem. They indicate a slightly different version of the inequality. In the end we describe a promising ansatz to find a lower energy-space bound that depends on the amount stored information. Unfortunately that did not result in a satisfactory result.