No-Cloning In Categorical Quantum Mechanics

TL;DR

This paper explores the categorical formulation of quantum mechanics to reveal a surprising connection between the no-cloning theorem, logic, and the foundations of mathematics, extending known results in categorical logic.

Contribution

It introduces a novel perspective linking no-cloning in quantum mechanics to fundamental issues in logic and mathematics, strengthening prior categorical logic results.

Findings

Establishes a connection between no-cloning and logical foundations.

Shows that certain categorical assumptions lead to trivialization similar to classical logic.

Reveals a heretofore unnoticed link between proof theory limitations and quantum no-go theorems.

Abstract

Recently, the author and Bob Coecke have introduced a categorical formulation of Quantum Mechanics. In the present paper, we shall use it to open up a novel perspective on No-Cloning. What we shall find, quite unexpectedly, is a link to some fundamental issues in logic, computation, and the foundations of mathematics. A striking feature of our results is that they are visibly in the same genre as a well-known result by Joyal in categorical logic showing that a `Boolean cartesian closed category' trivializes, which provides a major road-block to the computational interpretation of classical logic. In fact, they strengthen Joyal's result, insofar as the assumption of a full categorical product (both diagonals and projections) in the presence of a classical duality is weakened. This shows a heretofore unsuspected connection between limitative results in proof theory and No-Go theorems in quantum mechanics.