A deliciously accessible introduction to quantum mechanics

Metascience, Apr 2017

Benjamin Eva

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A deliciously accessible introduction to quantum mechanics

A deliciously accessible introduction to quantum mechanics Benjamin Eva 0 1 0 Munich Center for Mathematical Philosophy , Ludwig Str 31, 80539 Munich , Germany 1 Jeffrey Bub: Bananaworld: quantum mechanics for primates. Oxford University Press , 2016, 304 pp, $44.95HB Perhaps the single most important development in the recent history of quantum foundations has been the emerging dominance of the quantum information theoretic paradigm. In particular, the idea that the fundamental defining difference between classical and quantum theories can be characterized purely in terms of what kinds of correlations they allow, has led to a truly remarkable generalization and broadening of the subject's purview. Jeffrey Bub's new book Bananaworld: Quantum Mechanics for Primates provides an exceptionally clear, insightful and complete overview of the key conceptual innovations of the information theoretic revolution in quantum foundations. In addition, the book manages to articulate and explain the famously difficult subtleties of quantum correlations with an absolutely minimal amount of mathematical machinery. This accessibility is not bought at the cost of rigor. The exposition is always thorough and precise and Bub never resorts to metaphor or hand waving. Indeed, although the book is intended for a general audience, there is much here to excite the experts. Bananaworld begins by tracing the history of the information theoretic approach to quantum theory all the way back to Heisenberg's operator theoretic formulation. The crucial idea here is that representing physical quantities by possibly noncommuting operators introduces the possibility of intrinsically random events obeying new kinds of non-local probabilistic correlations. As Bub puts it, ''Quantum Mechanics is fundamentally a theory about the structure of information, insofar as a theory of information in the physical sense is essentially a theory of probabilistic correlation'' (7). The central task of the book then is to explicate and analyze the kinds of probabilistic correlations that are characteristic of quantum systems ''without the distraction of the mathematical formalism of quantum mechanics.'' This is done via the introduction of the eponymous conceit of Bananaworld. - The idea is this. Imagine an island covered with banana trees. Bananas on these trees grow in bunches. And there are two, and only two, ways of peeling these bananas: from the stem (S) or the top (T). Once a banana has been peeled it cannot be peeled again, and it will taste either ??ordinary?? (o) or ??incredible?? (i). This setting, although simple and amusing, shares a deep structural analogy with quantum measurement scenarios. Each tree can be thought of as corresponding to a particular preparation procedure. The bunches of bananas on the tree correspond to collections of physical systems in the states prepared by the tree. The two ways of peeling the bananas correspond to a pair of non-commuting observables, e.g. photon polarization or spin in orthogonal directions, and the two possible tastes of the bananas correspond to possible measurement outcomes for those observables, e.g. spin/polarization up or down in the measured direction. We can imagine two people, Alice and Bob, taking one banana each from the same bunch, going to different spots on the island and then testing the tastes of the bananas with different peelings. This gives us a way of thinking about non-local correlations. After introducing this extremely general and intuitive framework, Bub goes on to provide a concise and enjoyable introduction to special relativity, culminating in a discussion of the Lorentz transformations and the distinction between timelike and spacelike separated events. This introduction, although short, is helpful in aiding the reader to understand the significance of the no-signaling assumptions that play a prominent role in later discussions of quantum correlations. Bub also uses this section to illustrate a fundamental interpretational distinction between quantum mechanics and special relativity. Special relativity, although unintuitive and deeply surprising to those who are unacquainted with the theory, follows naturally from fundamental physical principles: that the laws of the physics are the same in all inertial reference frames and that the speed of light in a vacuum is the same for all observers. But the same cannot be said of quantum theory. As Bub explains, ??we don?t see that quantum phenomena must be the way they are because of some basic physical principles?? (26). Chapter 2 provides a basic introduction to superposition, entanglement, complementarity and the Born rule. Everything is done in terms of polarized photons, which makes the introduction especially helpful for students without a strong physics background. But chapter 3 is where things really get interesting. It begins with a historical discussion of the EPR correlations, phrased in terms of photon polarization. Bub then goes on to translate the EPR correlations into the language of Bananaworld. Specifically, he imagines that there can be ??EPR banana trees?? on which the bunches always contain exactly two bananas. The peelings and tastes of the bananas in these bunches are correlated in a way that persists when they are separated by arbitrary distance. In particular, they are correlated so that (i) if the peelings are the same (SS or TT) then the tastes are the same with equal probability for 00 and 11, (ii) if the peelings are different (ST or TS), then the tastes are uncorrelated with equal probability for 00, 01, 10, 11, (iii) the marginal probabilities for the tastes 0 and 1 when a banana is peeled S or T is 1/2, regardless of what happens to the other banana. Bub also translates the no-signaling principle into this framework: ??the taste of a banana is independent of how a remote banana is peeled, or whether or not a remote banana is peeled?? (52). Bub then goes on to show that this kind of correlation can be simulated with local resources, and so can be thought of as resulting from a common cause structure. And since quantum mechanics does not say anything about such a common cause structure, it must, the story goes, be incomplete. This presentation provides an extremely accessible and perspicuous outline of the core structure of the EPR argument that will surely be of great use to students. The second part of chapter three provides a similar translation of Popescu? Rohrlich (PR) correlations into the Bananaworld framework. Bub then provides an elegant derivation of the essentially non-classical nature of these correlations. Unlike EPR correlations, PR correlations cannot be simulated with local resources and so cannot be given a common cause explanation. In fact, using local resources the best possible simulation of PR correlations that one can obtain has a rate of success of . But using quantum resources, e.g. shared entangled states of a common resource, one can obtain a simulation with a success rate of roughly 0.85. Thus, there exist essentially non-local quantum correlations that simply cannot be explained in a local way. This, in a nutshell, is Bell?s theorem. Again, this presentation of Bell?s theorem is novel, insightful and accessible. By the end of Chapter 3, the reader has been guided through two of the most profound and fundamental theoretical results in the foundations of quantum mechanics. Further, they have reached this point using only basic probability theory and the conceptual setting of Bananaworld. At this point, it is worth mentioning a couple of important stylistic features of the book. First, a couple of the chapters have extra subsections under the heading ??more??, which build on the themes of the preceding sections but can be skipped by those who want to focus on the book?s central arguments. For example, Section 3 ends with a brief review of some relevant trigonometry and an introduction to Boolean algebras. There is also a technical appendix at the end of the book covering some of the mathematical fundamentals of the usual Hilbert space formalism for quantum theory. These features allow the more technically minded reader to better understand the central analogies between the Bananaworld setting and fully formalized Hilbert space quantum theory. Another nice feature of the book is that each important argument is followed by a brief bullet-pointed summary recalling the key points. This feature will be much appreciated by anyone new to the field. Chapter 4 focuses on demonstrating the ??intrinsically random?? nature of PR correlations. It begins by introducing correlation arrays, which go on to play a useful simplifying role in the analysis of non-local correlations. Next is a very nice discussion of the no-cloning theorem and its relation to free choice and no-signaling principles. The demonstration of the intrinsic randomness of PR correlations is given in terms of the inability of an eavesdropper to predict in a way that is better than a random guess the outcomes of measurements on entangled photons using only information about events before their preparation. Chapter 5 provides a general survey of the relationship between classical, quantum and superquantum correlations. This chapter is particularly interesting for philosophers of physics who might not be well acquainted with the current literature on generalized probabilistic theories, an important strand of research in quantum foundations at the moment. Chapter 6 provides another accessible and insightful discussion of one of the most puzzling and mysterious features of the quantum world: contextuality. Bub focuses specifically on the GHZ state, the Mermin square and the Kochen-Specker theorem. Chapters 7 and 8 introduce central topics from quantum information theory. Chapter 7 focuses on quantum cryptography while chapter 8 provides an outline of some of the best known theoretical results in quantum computation: teleportation, superdense coding and Shor?s factoring algorithm. Chapter 9 focuses on the question of why nature does not appear to allow for superquantum correlations. Most of the discussion focuses on the relationship between the Tsirelson bound and general information theoretic principles. Chapter 10 is the most straightforwardly philosophical in the book. It introduces the Everettian and Bohmiam interpretations of quantum theory and also includes a discussion of decoherence and hidden variables. One particularly nice part of this chapter is the discussion of the PBR theorem, which has not yet been much discussed in the philosophical literature. In the second half of the chapter, Bub goes on to present his own version of an ??information theoretic?? interpretation of quantum theory, and provides a very interesting discussion of what this means for the measurement problem. However, this section is rather too short to constitute a convincing defense of the desired interpretation. In particular, it seems inevitable that those who favor a realist interpretation of the quantum state will be somewhat dissatisfied with Bub?s approach, and the analogy between quantum measurements and phase transitions is unlikely to be sufficient to dispel these concerns. But this, as I see it, is not a major problem. The aim of the book is not to defend a general interpretation of quantum theory, but rather to allow the reader, with a minimum of technical machinery, to see deeply into the heart of the conceptual problems of quantum correlations. And in this respect, Bananaworld is an unmitigated success. Even experts can gain new insights from the consistently novel and streamlined presentation of the fundamental ideas and results of the field. Bananaworld is fantastically good at mapping the boundaries between classical, quantum and superquantum theories. It guides the reader toward developing an intuitive sense of what is distinctive about quantum mechanics. In many ways, Bananaworld reminds me of Jordan?s (2006) wonderful book Quantum Mechanics in Simple Matrix Form. Like Bananaworld, that book uses extremely simple mathematics to elucidate the basic structure of quantum information. But one thing that sets Bananaworld apart from such illustrious predecessors is the sheer breadth of its scope. After reading Bananaworld, the reader will have an impressively complete overview of the information theoretic approach to quantum foundations, including recent developments like the PBR theorem and generalized probabilistic theories. Bananaworld is also especially well suited as a central text for self-contained graduate courses in philosophy of physics. The concise, conceptually focused and mathematically straightforward presentation of the material will be extremely helpful for philosophy students hoping to gain a serious understanding of quantum mechanics. I really have very few qualms with the book. But one thing that might have served to make the presentation even more comprehensive and accessible would be a short introduction to graphical models of causation. Probabilistic independence assumptions play an important role throughout the discussion of quantum correlations, and students in particular would probably benefit from being able to think of these conditions in graphical terms. There is also an important and growing strand of literature applying graphical causal models to the analysis of quantum information, and it would be nice for the reader to be able to engage with these developments. But given the extensive scope of Bananaworld, that is not really a fair complaint. In summary, Bananaworld represents an invaluable contribution to the quantum foundations literature that provides a serious, comprehensive and impressively accessible introduction to this flourishing field. Jordan , T.F. 2006 . Quantum mechanics in simple matrix form . New York : Dover.

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Benjamin Eva. A deliciously accessible introduction to quantum mechanics, Metascience, 2017, 251-255, DOI: 10.1007/s11016-017-0183-0