Researchers answer a question inspired by a century old problem in quantum mechanics
Chris Heunen and Andre Kornell (Department of Computer Science, Tulane University) made a breakthrough in the foundations of quantum informatics by proposing a solution to a problem open for nearly a century: what is mathematically special about Hilbert spaces and their operators? Their result is remarkable because it characterises an analytic theory in a purely algebraic way, linking two fields that typically resist each other.
With an answer to this question, scientists can begin to understand quantum mechanics in a more operational way. They can also begin to design better programming languages for quantum computers, and ultimately write programs and protocols for quantum technology that are more clever and larger.
Quantum mechanics is the most precisely tested theory in the history of science. However, its mathematical rules are a bit of a mystery – they do not have a conceptual explanation from prior principles humans are intuitively familiar with. They work remarkably well, but nobody "really" knows "why".
Hilbert spaces and their operators are the mathematical foundation of quantum mechanics. They’re a category that’s made up of the rules used to describe quantum systems and the relationships between them.
Heunen and Kornell’s research shows just what distinguishes the category of Hilbert spaces from all other categories. It demonstrates how to recognize when any given category is in fact that of Hilbert spaces.
Knowing that is a first step in "understanding" the mathematical rules of quantum theory. It allows researchers to work with descriptions of quantum systems in a different way, especially if the characteristic properties of the category of Hilbert spaces could be physically justified or interpreted. In particular, the properties are suited to constructing programming languages for quantum protocols.
The characteristic properties found in this research are remarkable because they are entirely algebraic: they explain "continuous" things like probabilities and complex numbers, in terms of purely "non-continuous" things like combining two objects in an independent way.
The full article is published in the PNAS.