## Review: Quantum Theory from First Principles, G. M. D’Ariano, G. Chiribella and P. Perinotti, Cambridge, 2017

Quantum mechanics (hereafter: QM) makes probabilistic predictions about experimental results. It is therefore natural to ask whether the formal structure of quantum theory can be arrived at from principles that can be stated in purely operational or probabilistic terms. Attempts to reconstruct QM on such a basis go all the way back to von

Neumann’s early papers on the subject, and take in the substantial body of literature associated with quantum logic. Unfortunately, in spite of considerable mathematical ingenuity, these efforts never quite succeeded, and the search for a logical, operational or probabilistic foundation for QM had lost most of its momentum by the early 1980s.

The situation changed, dramatically, around the turn of the present century, owing to the development of quantum information theory (QIT). This provided just enough of a shift of viewpoint to enable considerable progress. First, QIT considers finite-dimensional systems, making these a respectable target. Secondly, QIT makes essential use of properties of composite systems and entangled states, making it seem reasonable to treat certain properties of such states as postulates. The power of such an approach was first demonstrated by Lucien Hardy in a pioneering paper entitled “quantum theory from five simple axioms”. This was followed by a spate of papers by various authors, including those of the present book, in which finite-dimensional quantum mechanics is deduced from various (but similar) packages of more or less operational assumptions.

One key idea exploited by all of these papers is that systems having a the same “information content” should be physically equivalent. Here information content refers to the maximum number of states that can be sharply distinguished by a single trial of a single measurement. Another is “locality” (or tomographic locality), which is the doctrine that the state of a composite system should depend only on the joint probabilities it assigns to the results of measurements on the two subsystems. Most of the cited papers also rely on a symmetry assumption, requiring that any two pure states should be related by a reversible physical transformation. The authors of this book replace this with a stronger purification principle, requiring that any state of any system A should be representable as the marginal, or reduced, state of a pure bipartite state on a compound system AB consisting of the given system together with an ancilla, B, and that this should be unique up to a reversible transformation on B. A rather amazing amount of structure flows from this one assumption.

This is first book-length treatment of these ideas. In that sense, it is a research monograph. However, very agreeably, it is presented as a textbook: explanations and examples are developed in detail, and in an attractive and easy-going style, and there are even exercises! This is a very good thing, and should be more widely imitated. In particular, this manner of presentation makes the book accessible to graduate students, or even well-prepared undergraduates, while also making it comparatively easy reading for the physicist of mathematician interested in an introduction to this area, and even for the expert who wants to understand the details of the authors’ particular approach.

The book is divided in to four parts. Part I deals with finite-dimensional quantum mechanics, or perhaps better to say, quantum information theory, presented in a style that emphasizes the convex geometry of the state space (that is, the space of density operators) and properties of completely positive maps. Along the way, the authors identify several features of this theory that, expressible in purely operational/probabilistic language, can serve as postulates for a more abstract, Hilbert-space independent, information theory. In Parts II and III, the authors give a general account of probabilistic theories satisfying their assumptions, showing convincingly that many well-known quantum information-theoretic phenomena follow naturally (and, in their formulation, elegantly) from these: Teleportation protocols, ensemble steering and much more, receive treatment here. The heart of the book is Chapter 7, in which the authors present the most immediate consequences of the purification postulate. In a deft, almost effortless way, they show, for instance, that the purifying states must have a steering property, that measurement inevitably corresponds to state-disturbance, and that unknown states can be “teleported” through entangled states. Later chapters build on this to recover, not only further standard results from quantum information theory, but also a substantial slice of the spectral theorem, without ever mentioning Hilbert space.

Using this machinery, the authors undertake in Part IV to reconstruct the finite-dimensional version of the standard quantum-mechanical formalism. That is, they show that any theory satisfying the postulates of Section II can be represented in such a way that states are density operators on some finite-dimensional complex Hilbert space, and

all density operators correspond to states; that measurement-outcomes are positive self-adjoint operators bounded by the identity operator (and conversely); and that processes are represented by completely positive mappings (and conversely). Although it requires some work, the proof of this result is elementary, in the sense that it does not lean on any mathematical results not already developed. This is in contrast to other reconstruction theorems. For instance, the quantum-logic approach requires the fundamental theorem of projective geometry, while most of the other post-Hardian reconstructions lean on the classification of compact group actions on spheres.

A striking feature of the book, apparent on nearly every page, is the use of a notation that replaces equations involving tensor products with diagrams. These take up space, and may be a bit off-putting at first for those not familiar with them, but it well repays the small effort needed to master this formalism: once one has learned to trust it (say, by translating a few examples back into equational language), many proofs become so natural as to be almost automatic. The book is well-produced, with an attractive layout and, in view of the complexity of the diagrammatic typesetting remarkably few typos. All in all, the authors are to be congratulated on producing a delightful, inspiring, and persuasive introduction to their way of looking at finite-dimensional quantum theory.

UPDATE: In the first version of this article, the authors were listed in the wrong order. This has been changed to the order they are listed on the book cover.