Modern Quantum Challenges: Theory

Maciej Lewenstein, Dagmar Bruß, J. I. Cirac +5 more

Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like quantum gravity, string theories, etc. concern quantum theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: (a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement; and (b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system). Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of quantum information theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.

Behram N. Kurșunoǧlu, Neal Moore

Charles Paul Enz

Tanner Leighton

Abstract This article advocates for a pragmatist view on quantum theory, offering a response to David Wallace’s recent criticisms of Richard Healey’s quantum pragmatism. In particular, I challenge Wallace’s general claim that quantum pragmatists—and antirepresentationalists more broadly—lack the resources to make sense of the novel “quantum” language used throughout modern physics in applications of quantum theory. I conclude by posing a challenge to quantum representationalists.

Charles Paul Enz

Haoxiang Li

Quantum mechanics has a history more than 100 years, and there are still many challenging open-ended questions. The problems encountered in modern physics such as matrix mechanics and quantum information often involve both quantum theory and open-ended problems in modern mathematics such as linear algebra. Distillability problem is an important open-ended problem in the field of quantum information and linear algebra. This article recalls the development history and application of linear algebra, summarizes the background and progress of distillability conjecture, and attempts to explore this problem by summarizing and using the knowledge of linear algebra and matrix theory learned in the course.