Partial order and a $T_0$-topology in a set of finite quantum systems

TL;DR

This paper develops a 'whole-part' theory for finite quantum systems using a partial order and a $T_0$-topology, exploring embeddings, ubiquity of quantities, and topological structures in quantum subsystem sets.

Contribution

It introduces a new framework combining partial orders and $T_0$-topologies to analyze finite quantum systems and their attributes.

Findings

Entropic quantities are shown to be ubiquitous.

Sets of quantities form $T_0$-topological spaces with divisor topology.

These sets can be extended to dcpo by adding top elements.

Abstract

A `whole-part' theory is developed for a set of finite quantum systems $\Sigma (n)$ with variables in ${\mathbb Z}(n)$. The partial order `subsystem' is defined, by embedding various attributes of the system $\Sigma (m)$ (quantum states, density matrices, etc) into their counterparts in the supersystem $\Sigma (n)$ (for $m|n$). The compatibility of these embeddings is studied. The concept of ubiquity is introduced for quantities which fit with this structure. It is shown that various entropic quantities are ubiquitous. The sets of various quantities become $T_0$-topological spaces with the divisor topology, which encapsulates fundamental physical properties. These sets can be converted into directed-complete partial orders (dcpo), by adding `top elements'. The continuity of various maps among these sets is studied.