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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

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. 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 in 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” [

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 [

The idea that Quantum Theory is in its backbone a new theory of information became very concrete with the rise 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 [

Recently, a new derivation of Quantum Theory from purely information-theoretic principles has been presented in [

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 [

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 (

Alice’s laboratory. Alice has at disposal many devices, each of them having an input system and an output system (represented by different wires) and possibly a set of outcomes labelling different processes that can take place. The devices can be connected in series and in parallel to form circuits. A circuit with no input and no output wires represents an experiment starting from the preparation of a state with a given source and ending with some measurement(s). Specifying a theory for Alice’s laboratory means specifying which are the allowed devices and specifying a rule to predict the probability of outcomes in such experiments.

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 [

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

The second way to associate a circuit to an experiment is to represent the instance of the experiment corresponding to a particular sequence of outcomes. For example, the circuit

In summary, our basic framework to treat general theories of information is based on the combination of the graphical language of circuits with elementary probability theory. Such a combination of circuits and probabilities, originally introduced in [

The features of the probability distributions arising in Alice’s experiments depend on the particular physical theory describing her laboratory: At this basic level, the theory could be classical or quantum, or any other fictional theory that we may be able to invent. We now start restricting the circle of possible theories: first of all, we make sure that Alice’s laboratory is not in a fictional Wonderland, but in a standard world enjoying some elementary properties common to Classical and Quantum Theory. The first property is:

The word

Causality is implicit in the framework in most works in the tradition of generalized probabilistic theories [

Let us set more requirements on the processes taking place in Alice’s laboratory. For every random process, there is also a

Our second principle is:

This principle establishes that

If Alice describes the system as being in a pure state, then this means that she has maximal knowledge about the system’s preparation. Instead, if Alice describes the system as being in a mixed state, then she is ignoring (or choosing to ignore) some information about the preparation. When Alice describes the preparation of her system with a mixed state

In other words,

Suppose that Alice wants to transfer to another experimenter Bob all the information she possesses about a system. If the system’s state

Our fourth principle guarantees the possibility of such an ideal compression:

Due to the Ideal Compression principle, Alice can transfer information without transferring the particular physical system in which information is embodied. In the example of the roll of the die, Ideal Compression principle can be illustrated as follows: if our information about the outcome of the roll is described by a probability distribution

Compressing information. Alice encodes information (here represented by a pile of books) in a suitable system carrying the smallest possible amount of data (here a USB stick). The most advantageous situation is when the compression is

The next principle concludes our list of requirements that are satisfied both by Classical and Quantum Theory:

Local Tomography plays a crucial role in reducing the complexity of experimental setups needed to characterize the state of multi-partite systems, ensuring that all the information contained in a composite system is accessible to joint local measurements, as illustrated in

Local Tomography. Alice can reconstruct the state of compound systems using only local measurements on the components. A world where this property did not hold would contain global information that cannot be accessed with local experiments.

The five principles presented so far define a family of theories of information that can be regarded as a standard. If it were just for these principles, Alice’s experiments could still be described, for example, by Classical Theory. What is then special about Quantum Theory? What makes it different from any other theory of information satisfying the five basic principles presented so far? Our answer is the following: Quantum Theory is the only theory of information that is compatible with a description of physical processes only in terms of pure states and reversible interactions. In a sense, Quantum Theory is

Let us spell out our last principle precisely. In Quantum Theory, every random process can be simulated as a reversible interaction of the system with a pure environment (

Every random process can be simulated in an essentially unique way as a reversible interaction of the system with a pure environment.

The Purity and Reversibility principle is closely connected with the idea of

The Purity and Reversibility principle concludes our list. For finite systems (systems whose state is determined by a finite number of outcome probabilities) the six principles presented above describe Quantum Theory completely [

physical systems are associated to complex Hilbert spaces;

the maximum number of perfectly distinguishable states of the system is equal to the dimension of the corresponding Hilbert space;

the pure states of a system are described by the unit vectors in the corresponding Hilbert space (up to a global phase);

the reversible processes on a system are described by the unitary operators on the corresponding Hilbert space (up to a global phase);

the measurements on a system are described by resolutions of the identity in terms of positive operators

the mixed states of a system are described by density matrices on the corresponding Hilbert space;

the probabilities of outcomes in a measurement are given by the Born rule

the Hilbert space associated to a composite system is the tensor product of the Hilbert spaces associated to the components;

random processes are described by completely positive trace-preserving maps.

Although the derivation of [

We now illustrate two important messages of the Purity and Reversibility Principle. The first message is that irreversibility can be always modelled as loss of control over an environment. In other words, the principle states a law of

The second important message of the Purity and Reversibility Principle is that we can simulate every physical process using a

Purity and Reversibility can be expressed in an elegant way as

Remarkably, the compatibility of the ignorance about a part with the maximal knowledge about the whole is also the key idea in a recent proposal for the foundations of statistical mechanics [

Before concluding, some remarks are in order. First of all, it is important to stress that the principles in [

The difference between the information-theoretic syntax and physical semantics can be well exemplified by discussing how much of the Schrödinger equation can be reconstructed in the information-theoretic approach. As we already mentioned, from our principles we can derive that the reversible transformations of a system are described by unitary operators on the corresponding Hilbert space. As a consequence, a reversible time-evolution in continuous time will be described by a family of unitary transformations

It is important to note that also the very scope of the information-theoretic derivations focuses on the syntax, rather than on the semantics: Questions like “What is an observer?” or “What is a measurement?” are not addressed by the principles. Neither [

In conclusion, building on the results of [

Now that our portrait of Quantum Theory has been completed, a natural avenue of future research consists in exploring the alternative theories that are allowed if we relax some of the principles. Given the structure of our work, which highlights Purity and Reversibility as “the characteristic trait” of Quantum Theory, it becomes interesting to study theories in which one weakens some of the first five (standard) principles while keeping Purity and Reversibility. All these alternative theories could be rightfully called “quantum”, for they share with the standard Quantum Theory its distinctive feature. One natural weakening of the principles would be to relax Local Tomography, thus allowing Quantum Theory on real Hilbert spaces, an interesting toy theory which exhibits quite peculiar information-theoretic features [

GC acknowledges support from the National Basic Research Program of China (973) 2011CBA00300 (2011CBA00301) and from Perimeter Institute for Theoretical Physics in the initial stage of this work. Research at QUIT has been supported by the EC through the project COQUIT. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. We acknowledge the three anonymous referees of this paper for valuable comments that have been useful in improving the original manuscript.