# Quantum Theory, Namely the Pure and Reversible Theory of Information

###### Abstract

After more than a century since its birth, Quantum Theory still eludes our understanding. If asked to describe it, we have to resort to abstract and ad hoc principles about complex Hilbert spaces. How is it possible that a fundamental physical theory cannot be described using the ordinary language of Physics? Here we offer a contribution to the problem from the angle of Quantum Information, providing a short non-technical presentation of a recent derivation of Quantum Theory from information-theoretic principles qmfrompuri. The broad picture emerging from the principles is that Quantum Theory is the only standard theory of information compatible with the purity and reversibility of physical processes.

## I Introduction

Quantum Theory is booming: It allows us to describe elementary particles and fundamental forces, to predict the colour of the light emitted by excited atoms and molecules, to explain the black body spectrum and the photoelectric effect, to determine the specific heat and the speed of sound in solids, to understand chemical and biochemical reactions, to construct lasers, transistors, and computers. This extraordinary experimental and technological success, however, is dimmed by huge conceptual difficulties. After more than hundred years from the birth of Quantum Theory, we still struggle to understand its puzzles and hotly debate on its interpretations. And even leaving aside the vexed issue of interpretations, there is a more basic (and embarrassing) problem: We cannot even tell what Quantum Theory is without resorting to the abstract language of Hilbert spaces! Compare quantum mechanics with the classical mechanics of Newton and Laplace: Intuitive notions, such as position and velocity of a particle, are now replaced by abstract ones, such as unit vector in a complex Hilbert space. Physical systems are now represented by Hilbert spaces, pure states by unit vectors, and physical quantities by self-adjoint operators. What does this mean? Why should Nature be described by this very special piece of mathematics?

It is hard not to suspect that, despite all our experimental and technological advancement, we are completely missing the big picture. The situation was vividly portrayed by John Wheeler in a popular article on the New York Times, where he tried to attract the attention of the general public to what he was considering “the greatest mystery in physics today” wheelerNYT: “Balancing the glory of quantum achievements, we have the shame of not knowing “how come.” Why does the quantum exist?”

The need for a more fundamental understanding was clear since the early days of Quantum Theory. The first to be dissatisfied with the Hilbert space formulation was its founder himself, John von Neumann redei. Few years after the completion of his monumental book von32, von Neumann tried to understand Quantum Theory as a new form of logics. His seminal work in collaboration with Birkhoff BirkVN36 originated the field of quantum logics, which however did not succeed in producing a clear-cut picture capable to cross the borders of a small community of specialists. More recently, a fresh perspective on the origin of the quantum came from Wheeler. In his programme It from Bit, Wheeler argued that information should be the fundamental notion in our understanding of the whole of physics, based on the premise that “all things physical are information-theoretic in origin” wheelerIt. If we accept this premise, then nothing is more natural then looking for an information-theoretic understanding of quantum physics. Indeed, one of the most noteworthy features of quantum theory is the peculiar way in which it describes the extraction of information through measurements. This remarkable feature and its foundational import were discussed in depth by Wootters in his PhD thesis woot. In different guises, the idea of information being the core of Quantum Theory has been explored by several authors, notably by Weizsacker weiz, Zeilinger zei, and Brukner bruzei).

The idea that Quantum Theory is, in its backbone, a new theory of information became very concrete with the raise of Quantum Information. This revolutionary discipline revealed that Quantum Theory is not just a theory of unavoidable indeterminacy, as emphasized by its founders, but also a theory of new exciting ways to process information, ways that were unimaginable in the old classical world of Newton and Laplace. Quantum Information unearthed a huge number of operational consequences of Quantum Theory: quantum states cannot be copied wootterszurek; dieks but they can be teleported tele, the quantum laws allow for secure key distribution bb84; e91, for fast database search grover, and for the factorization of large numbers in polynomial time shor. These facts are so impressive that one may be tempted to promote some of them to the role of fundamental principles, trying to derive the obscure mathematics of Quantum Theory from them. The idea that the new discoveries of Quantum Information could offer the key to the mystery of the quantum was enthusiastically championed by Fuchs fuchs and Brassard brassard and rapidly led to a feverish quest for new information-theoretic principles, like information causality infocau, and to reconstructions of quantum theory from various informational ideas, like those of Refs. Har01; maurolast; philip; DakBru09; Mas10; har11; masanew.

Recently, a new derivation of Quantum Theory from purely information-theoretic principles has been presented in Ref. qmfrompuri (see also game for a short introduction to the background). In this work, which marks a first step towards the realization of Wheeler’s dream, Quantum Information is shown to maintain its promise for the understanding of fundamental physics: indeed, the key principle that identifies Quantum Theory is the Purification Principle purification, which is directly inspired by the research in Quantum Information. Quantum Theory is now captured by a complete set of information-theoretic principles, which can be stated using only the elementary language of systems, processes, and probabilities. With respect to related reconstructive works, the new derivation of Ref. qmfrompuri has the advantage of offering a clear-cut picture that nails down in few simple words what is special about of Quantum Theory: Quantum Theory is, in the first place, a theory of information, which shares some basic features with classical information theory, but differs from it on a crucial point, the purity and reversibility of information processing. In a standard set of theories of information, Quantum Theory appears to be the only theory where the limited knowledge about the processes that we observe in nature is enough to reconstruct a picture of the physical world where all processes are pure and reversible.

More precisely, when we state that Quantum Theory is a theory of information, we mean that the mathematical framework of the theory can be expressed by using only concepts and statements that have an informational significance, such as the concept of signalling, of distinguishability of states, or of encoding/decoding. Here we refer to “information” and “informational significance” in a very basic, primitive sense: in this paper we will not rely on specific measures of information, such as the Shannon, Von Neumann, or Renyi entropies. In fact, the very possibility of defining such quantitative measures is based on the specific mathematical structure of classical and quantum theory (chiefly, on the fact that in these theories every mixed state is a probabilistic mixture of perfectly distinguishable states), which, for the quantum case, is exactly what we want to pin down with our principles.

The informational concepts used in this paper are connected to the more traditional language of physics by viewing the possible physical processes as information processing events. For example, a scattering process can be viewed as an event—the interaction—that transforms the input information encoded in the momenta of the incoming particles into the output information encoded in the momenta of the scattered particles. From this perspective, the properties of the particular theory of information that we adopt immediately translate into properties of our physical description of the world. The natural question that we address here is: which properties of a theory of information imply that the description of the world must be quantum?

The purpose of this paper is to give a short, non-technical answer to the question, providing an account of the informational principles of Quantum Theory presented in Ref. qmfrompuri and of the worldview emerging from them. Hence, we will focus on the broad picture and on the connection of the principles with other fundamental areas of theoretical physics, while referring the reader to the comprehensive work of Ref.qmfrompuri for the mathematical definitions and for the rigorous proofs of the claims.

## Ii A complete set of information-theoretic principles for Quantum Theory

To portray Quantum Theory we set up a scene where an experimenter, Alice, has many devices in her laboratory and can connect them in series and in parallel to build up circuits (Fig. 1). In Alice’s laboratory, any device can have an input and an output system, and possibly some outcomes that Alice can read out. Each outcome labels a different process transforming the input into the output: the device itself can be viewed as implementing a random process. Some devices have no input: they are preparations, which initialize the system in some state. Other devices have no output: they are measurements, which absorb the system and produce an outcome with some probability.

From a slightly more formal point of view, Alice’s circuits can be described with a graphical language where boxes represent different devices and wires represent physical systems travelling from one device to the next purification, in a way that is inspired by the picturalist framework by Coecke bob. These circuits are essentially the same circuits that are commonly used in Quantum Information nielsenchuang, except for the fact that here we do not specify from the beginning the mathematical representation of the devices: we do not specify that the possible states are described by density matrices on some complex Hilbert space, or that the possible reversible evolutions are described by unitary operators. Retrieving these specific mathematical prescriptions from operationally meaningful assumptions is indeed the main technical point of Ref. qmfrompuri and of the other quantum reconstructions Har01; maurolast; philip; DakBru09; Mas10; har11; masanew.

Since the devices in Alice’s laboratory can have different outcomes, there are two natural ways to associate circuits to an experiment. First, a circuit can represent the schematic of Alice’s experimental setup. For example, the circuit

\xy@@ix@ | (1) |