Quantisation and Prediction

Quantisation and Prediction Another look at the aim and structure of quantum theory Jean-Michel Del hotel London School of Economics and Political Science Department of Philosophy, Logic and Scientific Method Thesis submitted for the Degree of Doctor of Philosophy University of London 2004 UMI Number: U615622 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U615622 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 OF poLrrtCAL ^ AMD \ 1 H f c S £ S F ZZ 7 Z < 7 7 Our present quantum mechanical formalism is a peculiar mixture describing in part laws of Nature, in part incomplete human information about Nature - all scrambled up together by Bohr into an omelette that nobody has seen how to unscramble. Yet we think the unscrambling is a prerequisite for any further advance in basic physical theory... E.T. Jaynes - Erwin with his psi can do Calculations quite a few. But one thing has not been seen Just what psi really mean. Felix Bloch There exists a body of exact mathematical laws, but these cannot be interpreted as expressing simple relationships between objects existing in space and time. Werner Heisenberg ...if quantum theory were not successful pragmatically, we would have no interest in its interpretation. It is precisely because of the enormous success of the QM mathematical formalism that it becomes crucially important to learn what that mathematics means. To find a rational physical interpretation of the QM formalism ought to be considered the top priority research problem of theoretical physics; until this is accomplished, all other theoretical results can only be provisional and temporary. E.T. Jaynes ABSTRACT It is argued (Part A) that quantum mechanics can be derived as a principle-based dynamical framework, the basic equation of which is an alternative form of the Hamilton-Jacobi equation. Schrodinger’s equation obtains as a result of linearising that equation, and so-called wave functions can be given no straightforward physical interpretation. It is suggested, partly in relation to a theorem by Gromov, that a finite action quantum would make it practically inevitable, for purposes of prediction, to resort to a probabilistic formulation. The structure of the space of square-integrable solutions of the Schrodinger equation happens to lend itself to the introduction of the appropriate kind of predictive scheme. Investigating the nature and scope of such a scheme is the subject of Part B. It is shown that basic features of the formalism of quantum theory, like composition rules for ‘amplitudes’ or the ‘Born’ probability rule, can be derived independently of any physical assumptions. A generalisation of the basic formalism using tensor product composition appears to be required if all correlations are to be extracted from locally accessed data. A detailed discussion of quantum teleportation leads to the conclusion that a ‘one-shot’ account leads to a distorted picture of what is actually achieved. An analogy with classical cryptography is made and the statistical significance of the ‘transfer’, which does not require introducing any novel form of ‘quantum information’, is emphasised. Results obtained over the last decade using the extended formalism of positive operator-valued measures are reviewed and discussed. These lend further support to the idea that the set of basic ‘quantum’ rules functions as a general kind of probabilistic scheme for prediction, the structural features of which are not constrained in any direct way by the underlying physics. On the other hand, the very existence of such a predictive framework hinges on selecting a particular class of solutions of the Schrodinger equation, which selection has been incorrectly interpreted as reflecting a physical necessity. Contents Introduction 1 Part A 1 The emergence of quantum mechanics - a historical outline 6 2 On 'states’ and physical quantities in quantum mechanics 34 2.1 Quantum mechanics without 'states’? 34 2.2 The representation of physical quantities 45 3 Quantum mechanics as a principle theory 52 3.1 The Special Theory of Relativity: a prime instance of a principle theory.. 3.2 Some steps toward a principle-based account of quantum mechanics 57 3.3 Quantum mechanics from a single principle 67 Part B 4 Quantum rules and objectification 80 4.1 Quantum theory, systems and preparations.......80 4.2 Basic rules of quantum theory 82 4.3 The SAQM is incompatible with a simple-minded ensemble view 87 4.4 Objectification and quantum theory 88 5 Deriving rules for amplitudes and 'quantum’ probabilities 92 5.1 The composition rules for amplitudes 92 5.2 From amplitudes to probabilities 97 5.3 Destouches’ derivation of the ‘quantum’ probability rule .......99 6 Quantum theory as a linear scheme for prediction 110 6.1 Basic features 110 6.2 Angular and statistical distance in quantum theory 117 6.3 Evolving, measuring and predicting 125 7 Hilbert space composition and quantum theory 130 7.1 n-ary preparations 130 7.2 Correlations and entanglement 134 7.3 Binary case objectification 140 8 Quantum teleportation - a case study 143 8.1 ‘Teleporting a quantum state’ 143 8.2 The fastest way from A to B? 150 8.3 From classical cryptography to quantum teleportation 159 8.4 Dense coding, entanglement swapping 165 9 Further developments 169 9.1 Gleason’s theorem and positive operator-valued measures 169 9.2 ‘Quantum’ Bayesian update, complex numbers and 0-composition. 9.3 A simple axiom set for quantum theory 175 9.4 Can information-theoretical concepts and methods illuminate the foundations of quantum theory?... 184 * 10 Finale 196 Bibliography 208 Introduction Three quarters of a century have now gone by since Bohr’s Como lecture (1927). All the same, there is as yet no consensus regarding how the most basic features of the quantum-mechanical formalism should properly be understood. Half seriously, one could say that quantum physics has been a tremendously successful answer to a question that we have yet to ask and formulate clearly. Up to the present day, the dominant trend has consisted in defending one’s favourite interpretation against other major competitors or in the face of a number of ‘no go’ theorems. The feeling is also widespread that the most counterintuitive aspects of quantum theory point to an inherent weirdness of the physical world, involving uncontrollable influences at a distance, a blurring of the distinction between being and non-being or some inscrutable variety of holism. Indeed, the popularity of baffling interpretations like Many Worlds and its variants testifies to the fact that (too) many physicists or philosophers readily surrender to the idea of a sweeping ‘non-classicality’ of quantum mechanics. By contrast, comparatively rare attempts to ground the theory on a small set of well motivated, and perhaps more mundane principles having clear physical significance, tend to be considered futile or regressive (‘hidden variables’). Whilst ‘no go’ proofs can be technically impressive, they all boil down to the acknowledgment of two well-known and related basic features of the quantum-mechanical formalism: a metric vector space structure and a non- commutative algebra of linear operators. These underlie both the ‘non- classicality’ and the efficiency of the algorithm whereby expectation values and probabilities are routinely calculated in the laboratory. The rise of Quantum Information Technology (QIT) has recently prompted a revival of interest in the logico-algebraic structure of quantum theory, which had once been the rather exclusive concern of a small band of quantum logicians and philosophers. New ‘hard’ results, admittedly of unequal conceptual importance, have been obtained over the last ten years. Technical breakthroughs like Shor’s algorithm for factoring large numbers into primes or technological prospects like that of manufacturing a quantum computing network in the not-too-distant future have bred optimism. Almost inevitably, this has led some to hope that a change of focus toward information- theoretical concepts and methods might hold a (the?) key to a final, if belated understanding of the aim and structure of quantum theory. In all fairness, nothing in what has been achieved in QIT so far, or what can be reasonably 1 foreseen, really leads one to suspect that any such revelation (revolution?) is under way. On the other hand, the rejuvenation of old concerns, the fresh outlook of new participants in the debate and - last but not least - the sharpening and more extensive use of some relatively new tools like positive operator-valued measures might all contribute in a beneficial way to our understanding of quantum theory. Thus, Chapter 9 of this thesis is devoted to some recent and outstanding contributions by theorists who all participate to some degree in the QIT enterprise. The upshot is a re-evaluation of the nature and purpose of the basic formalism of quantum theory: this is seen to function as a linear probabilistic scheme for prediction, the mathematical structure of which is essentially independent of any assumption regarding the nature or behaviour of physical objects. It is one of the aims of this dissertation to convince the reader that the whole of quantum physics can justifiably be established as following from the satisfaction of a fundamental structural principle, in a similar sense as the refoundation of mechanics prompted by the Special Theory of Relativity (STR) can be shown to follow from assuming the universal validity of a single1 relativity principle.

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