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

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.

References in corpus (2)

• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Weak mutually unbiased bases

Cited by in corpus (2)

• The complete Heyting algebra of subsystems and contextuality
• Quantum mechanics on profinite groups and partial order