Selected for a Viewpoint in Physics PHYSICAL REVIEW A 84, 012311 (2011) Informational derivation of quantum theory Giulio Chiribella ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Ontario, Canada N2L 2Y5 † Giacomo Mauro D’Ariano ‡ and Paolo Perinotti § QUIT Group, Dipartimento di Fisica “A. Volta” and INFN Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy ‖ (Received 29 November 2010; published 11 July 2011) We derive quantum theory from purely informational principles. Five elementary axioms—causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning—define a broad class of theories of information processing that can be regarded as standard. One postulate—purification—singles out quantum theory within this class. DOI: 10.1103/PhysRevA.84.012311 PACS number(s): 03.67.Ac, 03.65.Ta I. INTRODUCTION More than 80 years after its formulation, quantum theory is still mysterious. The theory has a solid mathematical foun- dation, addressed by Hilbert, von Neumann, and Nordheim in 1928 [1] and brought to completion in the monumental work by von Neumann [2]. However, this formulation is based on the abstract framework of Hilbert spaces and self-adjoint operators, which, to say the least, are far from having an intuitive physical meaning. For example, the postulate stating that the pure states of a physical system are represented by unit vectors in a suitable Hilbert space appears as rather artificial: which are the physical laws that lead to this very specific choice of mathematical representation? The problem with the standard textbook formulations of quantum theory is that the postulates therein impose particular mathematical structures without providing any fundamental reason for this choice: the mathematics of Hilbert spaces is adopted without further questioning as a prescription that “works well” when used as a black box to produce experimental predictions. In a satisfactory axiomatization of quantum theory, instead, the mathematical structures of Hilbert spaces (or C* algebras) should emerge as consequences of physically meaningful postulates, that is, postulates formulated exclusively in the language of physics: this language refers to notions like physical system, experiment, or physical process and not to notions like Hilbert space, self-adjoint operator, or unitary operator. Note that any serious axiomatization has to be based on postulates that can be precisely translated in mathematical terms. However, the point with the present status of quantum theory is that there are postulates that have a precise mathe- matical statement, but cannot be translated back into language of physics. Those are the postulates that one would like to avoid. The need for a deeper understanding of quantum the- ory in terms of fundamental principles was clear since * gchiribella@perimeterinstitute.ca † http://www.perimeterinstitute.ca ‡ dariano@unipv.it § paolo.perinotti@unipv.it ‖ http://www.qubit.it the very beginning. Von Neumann himself expressed his dissatisfaction with his mathematical formulation of quan- tum theory with the surprising words “I don’t believe in Hilbert space anymore,” reported by Birkhoff in [3]. Re- alizing the physical relevance of the axiomatization prob- lem, Birkhoff and von Neumann made an attempt to un- derstand quantum theory as a new form of logic [4]: the key idea was that propositions about the physical world must be treated in a suitable logical framework, different from classical logics, where the operations AND and OR are no longer distributive. This work inaugurated the tradition of quantum logics, which led to several attempts to axiomatize quantum theory, notably by Mackey [5] and Jauch and Piron [6] (see Ref. [7] for a review on the more recent progresses of quantum logics). In general, a certain degree of technicality, mainly related to the emphasis on infinite-dimensional systems, makes these results far from providing a clear-cut description of quantum theory in terms of fundamental principles. Later Ludwig initiated an axiomatization program [8] adopting an operational approach, where the basic notions are those of preparation devices and measuring devices and the postulates specify how preparations and measurements combine to give the probabilities of experimental outcomes. However, despite the original intent, Ludwig’s axiomatization did not succeed in deriving Hilbert spaces from purely operational notions, as some of the postulates still contained mathematical notions with no operational interpretation. More recently, the rise of quantum information science moved the emphasis from logics to information processing. The new field clearly showed that the mathematical principles of quantum theory imply an enormous amount of information- theoretic consequences, such as the no-cloning theorem [9,10], the possibility of teleportation [11], secure key distribution [12–14], or of factoring numbers in polynomial time [15]. The natural question is whether the implication can be reversed: is it possible to retrieve quantum theory from a set of purely informational principles? Another contribution of quantum information has been to shift the emphasis to finite dimensional systems, which allow for a simpler treatment but still possess all the remarkable quantum features. In a sense, the study of finite dimensional systems allows one to decouple the 012311-1 1050-2947/2011/84(1)/012311(39) ©2011 American Physical Society