Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a road to quantum deletion

TL;DR

This paper rectifies previous claims about Quantum Theory over finite fields, introduces formalism for time evolution and observables, and demonstrates the possibility of quantum deletion with near-unitary operators.

Contribution

It establishes a consistent formalism for Quantum $ ext{F}_{un}$, including inner products, unitary evolution, and no-deletion results, extending quantum theory over finite fields.

Findings

Quantum $ ext{F}_1$ has a well-defined inner product.

Time evolution operators and unitarity are formalized in Quantum $ ext{F}_{un}$.

Quantum deletion can be achieved with almost unitary operators, with probability approaching 1.

Abstract

In a recent paper [2], Chang et al. have proposed studying "Quantum $\mathbb{F}_{un}$": the $q \mapsto 1$ limit of Modal Quantum Theories over finite fields $\mathbb{F}_q$, motivated by the fact that such limit theories can be naturally interpreted in classical Quantum Theory. In this letter, we first make a number of rectifications of statements made in [2]. For instance, we show that Quantum Theory over $\mathbb{F}_1$ {\em does} have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what is claimed in [2]. Starting from that formalism, we introduce time evolution operators and observables in Quantum $\mathbb{F}_{un}$, and we determine the corresponding unitary group. Next, we obtain a typical no-cloning in the general realm of Quantum $\mathbb{F}_{un}$. Finally, we obtain a no-deletion result as well. Remarkably, we show that we {\em can} perform quantum deletion by {\em almost unitary operators}, with a probability tending to $1$. Although we develop the construction in Quantum $\mathbb{F}_{un}$, it is also valid in any other Quantum Theory (and thus also in classical Quantum Theory).