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Violating the Assumption of Measurement Independence in Quantum Foundations Violating the assumption of Measurement Independence in Quantum Foundations A Project Report submitted by INDRAJIT SEN in partial fulfilment of the requirements for the award of the degree of Master of Science arXiv:1705.02434v1 [quant-ph] 6 May 2017 DEPARTMENT OF Physics INDIAN INSTITUTE OF TECHNOLOGY MADRAS. April 2016 PROJECT CERTIFICATE This is to certify that the project report titled Violating the assumption of Measure- ment Independence in Quantum Foundations, submitted by Indrajit Sen, to the Indian Institute of Technology, Madras, in partial fulfilment of the requirements for the award of the degree of Master of Science, is a bona fide record of the research work done by him under our supervision. The contents of this report, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma. Sibasish Ghosh Prabha Mandyam Project Guide Project Guide Associate Professor Assistant Professor Dept. of Physics Dept. of Physics IMSc Chennai IIT Madras Place: Chennai Date: 25 April 2016 ACKNOWLEDGEMENTS I’d like to express my gratitude to Prof. Sibasish Ghosh, with whom I have discussed my project work almost every week since the past one year. I thank him for the enor- mous time he has given me, and for the ease with which I could discuss any doubt I had, no matter how small. I am thankful to him for the freedom with which I could pursue a controversial topic in Quantum foundations, the viability of which seemed dim at the beginning, and for guiding me throughout the project. I’d like to thank Prof. Prabha Mandyam for introducing me to the world of Quantum foundations when I knew little about the people and literature in this field. I want to thank her for giving me the opportunity to do my Master’s project in foundations, and for her considerable inputs in editing the thesis to its current form. I’d also like to express my gratitude to Prof. MJW Hall who was extremely gener- ous in entertaining all my doubts on measurement independence and related topics. He proof read and checked several results(right or wrong) that I had, pointed out connect- ing ideas that I had missed, and was in general very kind to give so much of his time to me. His revolutionary work in the past few years opened up the topic of measurement independence from apathy, and this thesis would not have been possible without his bold paper of 2010. I’d like to thank my parents and my friends at IIT Madras for their wonderful com- pany. -Indrajit Sen i ABSTRACT KEYWORDS: Quantum foundations, Measurement Independence Though Quantum Mechanics is one of the most successful theories in Physics, experimentally verified to a very high degree of accuracy, there is still little consen- sus among physicists on a number of fundamental conceptual problems that the the- ory is plagued with since inception. Richard Feynman once remarked, "I think I can safely say that nobody understands quantum mechanics". Some of the founding fathers themselves- Erwin Schroedinger, Louis de Brogelie and Albert Einstein - were not sat- isfied with the theory10. The field of Quantum foundations strives to resolve these issues by reformulating the old discussions mathematically; proposing hidden variable theo- ries that reproduce Quantum Mechanics; proposing experimental tests that can be used to decide between various standpoints; and by working on the deeper theory of Quan- tum Gravity from the perspective of foundations. Measurement Independence is an assumption that has been used in such founda- tional arguments since the time of EPR thought experiment in 1935, and assumed in an overwhelming majority of hidden variable models of Quantum Mechanics since then. And yet, the term "measurement independence" is very recent - introduced only in 2010. Before this, it was known vaguely as the "free will assumption" or "lack of retrocausal- ity". Compared to other assumptions in foundations, like locality or contextuality, it has received little attention. This is because most researchers still consider the assumption too natural to expect its violation. In the past few years however, a lot of work has 10Some of the problems are: 1. The Measurement problem: when and how does the collapse of the wavefunction occur? 2. The Quantum to Classical transition problem: exactly how many microscopic(quantum) objects must be collected together for the composite object to be macroscopic(classical)? 3. The problem of Reality: whether there exists an objective real world independent of our observations. 4. The problem of Determinism: whether Nature is fundamentally random. 5. The problem of Completeness: In its domain Quantum Mechanics is correct, but is it a complete, description of Nature? ii been done in analysing the assumption. It has been quantified, and several interesting consequences of its violation have been derived. In this thesis we study the various contexts in which the assumption is used, re- view theorems about hidden variable models in light of relaxing measurement inde- pendence, develop several new measurement dependent hidden variable models which have interesting properties, finding application in the foundational question of "reality of wavefunction", and in classical simulation of quantum channels. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS i ABSTRACT ii LIST OF FIGURES vi ABBREVIATIONS vii 1 Introduction 1 1.1 Does Quantum Foundations matter? . 1 1.2 What is the Measurement Independence assumption? . 2 1.3 Outline of the thesis . 3 2 An introduction to hidden variable theories 4 2.1 Formulation of hidden variable theories . 4 2.2 Formulation of Measurement Independence . 7 2.2.1 Measurement Independence in Bell’s 1964 Theorem . 8 2.2.2 Measurement Independence in Bell’s 1976 Theorem . 9 2.2.3 Measurement Independence in Ontological Model Framework 10 2.3 The Ontic/Epistemic distinction . 10 2.3.1 Einstein’s Incompleteness Argument[19] . 11 2.3.2 Formalization of Incompleteness in Ontological Models frame- work . 12 2.3.3 Quantifying Epistemicity in Ontological models[32] . 15 2.4 Contextuality . 16 3 Consequences of violating Measurement Independence 19 3.1 Validity of theorems for ontological models in Measurement Dependent case . 19 3.1.1 Validity of Maroney’s theorem[32] . 19 3.1.2 Validity of Barrett’s theorem[24] . 20 iv 3.1.3 Validity of Leifer-Maroney’s results[25] . 21 3.1.4 Validity of Ballentine’s result[30] . 25 3.1.5 Validity of Bandyopadhyay et al’s results[38] . 26 3.2 The Brans model[33] . 28 3.2.1 Correlation between the particles and measurement choices 29 3.3 A maximally epistemic model in dN - The generalised Brans Model 30 3.3.1 How epistemic is the model? . 31 3.3.2 Does the model satisfy Preparation Independence[23]? . 32 3.3.3 What is the randomness in the model? . 33 3.4 Simulating Quantum Channels using Measurement Dependent models 34 3.4.1 Modified Kochen Specker model II . 36 3.4.2 Protocol to simulate quantum channels using modified Kochen Specker model II . 36 3.4.3 Calculation of Communication cost . 37 3.4.4 Advantage of using Measurement Dependent models for simu- lation . 38 3.5 A Measurement Dependent model that cross correlates particles and measurement choices in EPR scenario . 39 3.5.1 Modified Hall Model . 39 3.5.2 Foundational implications of such a model . 40 3.6 Can a ontic ontological model be converted to epistemic by intro- ducing Measurement Dependence? . 42 3.6.1 Modified Bell Mermin model . 42 4 Some observations on Preparation Independence 44 4.1 Preparation Independence for product state measurements . 44 4.2 Weakening the Preparation Independence Postulate . 45 4.2.1 Compatibility . 46 4.2.2 Local Compatibility . 47 5 Conclusion 48 5.1 List of new results . 48 5.2 Future Directions . 49 LIST OF FIGURES 2.1 Local hidden variables in Bell scenario . 9 3.1 Condition of local-causality . 41 vi ABBREVIATIONS MI Measurement Independence MD Measurement Dependent PI Preparation Independence vii CHAPTER 1 Introduction 1.1 Does Quantum Foundations matter? "I am a Quantum Engineer, but on Sundays I have principles." - John Bell ( as quoted in [1]) Bohmian Mechanics [2] is a completely deterministic, non-local, contextual hidden variable reformulation of standard Quantum Mechanics, where particles have classical trajectories. It reproduces all known experimental predictions of Quantum Mechanics, and has also been generalised to Quantum Field Theories [3]. Thus, explanation of experimental phenomena does not require us to abandon our classical notions, say of determinism, to explain events on microscopic scale. Infact, there are several hidden variable theories[4] [5] [6] which exactly reproduce the predictions of Quantum Me- chanics, implying that no unique interpretation can be drawn from the experimental data. Recently, hidden variable theories have also predicted phenomena which contra- dict Quantum Mechanics in early universe and around black holes [7], obviating the criticism of hidden variable theories that they contain no new Physics. In light of this, it becomes important to ask whether the concepts used in standard Quantum Mechanics : randomness, wave-function collapse, lack of causal explanation, are inevitable or whether, as suggested by Einstein [8], are indicative only of Quantum Mechanics being an incomplete and provisional theory, or whether, as is commonly held, a question of Semantics or Philosophy than Science. The result of such investigations have relevance in Quantum Information, Quantum Gravity and Cosmology. Bell’s theorem has applications in Quantum Cryptography[9] and Quantum random number generators[10] Certain classes of hidden variable mod- els can be used to classically simulate quantum channels and define communication complexity[11]. The problem of defining time, and causal structures in Quantum Grav- ity have connections to foundations.
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