Probing Foundations of Quantum Mechanics: A Study into Nonlocality and Quantum Gravity Parth Girdhar A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy School of Physics University of Sydney 2020 2 Abstract This thesis is about probing aspects of the foundations of quantum mechanics. Firstly, two notions of quantum nonlocality are explored: EPR-steering, the abil- ity to control a remote quantum state, and Bell nonlocality, the inconsistency of a theory with local causality. A necessary and sufficient witness of Einstein-Podolsky- Rosen (EPR) steering is derived for a two qubit system employing only correlations between two arbitrary dichotomic measurements on each party. It is demonstrated that all states that are EPR-steerable with such correlations are also Bell nonlocal, a surprising equivalence between these two fundamental concepts of quantum mechan- ics. Next, testing modifications of the quantum mechanical canonical commutation relations is addressed. These are properties of some quantum gravity theories that involve an effective minimal length. It is shown that optomechanical probes of po- sition noise spectrum of macroscopic oscillators can produce constraints on these theories. A comparison with current and future realistic experiments reveals the po- tential to beat constraints from direct experiments on elementary particles. Finally, it is studied how such modifications of quantum mechanics manifest in the theory of general continuous quantum position measurements. Several behaviours are found that deviate strongly from that of standard commutation relations. Contents Abstract . .2 Statement of Originality . .5 Statement of Contribution and Attribution . .5 Acknowledgements . .6 1 Introduction 9 1.1 The Quantum Formalism . 10 1.2 Bell's Theorem . 13 1.3 Einstein-Podolsky-Rosen Steering . 16 1.4 Generalisations . 18 1.4.1 Complexity of Determining Bell Nonlocality . 21 1.5 Modified Canonical Commutation Relations . 24 2 EPR-Steering and Bell Nonlocality in the CHSH Scenario 29 2.1 Necessary and Sufficient Steering Inequality . 31 2.2 Equivalence Classes of Measurements . 34 2.3 States Steerable via CHSH-type measurements are Non-local . 35 2.4 Discussion . 36 2.5 Appendix A . 38 2.6 Appendix B . 40 3 Probing Modified Commutators with Quantum Noise 47 3.1 Introduction . 47 3.2 Background . 48 3.3 Modified Noise Spectrum . 52 3.3.1 Setup . 52 3.3.2 Noise spectrum . 54 3.4 Constraints on the Modified Commutator . 57 3.4.1 Estimating current constraints . 59 3.4.2 Future constraints . 62 3.4.3 General driven oscillator . 68 3.5 Discussion . 69 3.6 Appendix . 72 3.6.1 First term of (3.13) . 73 3.6.2 Second term of (3.13) . 76 3 4 CONTENTS 3.6.3 Perturbed spectrum . 79 3.6.4 aLIGO modelling . 81 3.6.5 Steady-state expectation values without the adiabatic approx- imation . 83 3.6.6 Standard spectrum . 84 4 Modified Commutators and Continuous Position Measurements 87 4.1 Quantum Trajectory Equation . 87 4.2 Differentials of general mean & covariance matrix . 92 4.3 Modified canonical commutation relations . 93 4.4 Discussion . 99 5 Conclusion 103 CONTENTS 5 Statement of Originality I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged. This thesis has not been submitted for any degree or other purposes. Parth Girdhar Statement of Contribution and Attribution There are three main chapters in this thesis. Appendices are presented immediately after the main contents of each chapter. • Chapter 2 of this thesis is published as: Girdhar, P. and Cavalcanti, E.G., 2016. All two-qubit states that are steerable via Clauser-Horne-Shimony-Holt-type correlations are Bell nonlocal. Physical Review A, 94(3), p.032317. In the journal the publication is in two-column format but it has been adjusted to single column for this thesis. I am the lead author of the publication and derived all the results, advised by Eric Cavalcanti on the research questions. The question connecting Bell nonlocality and EPR-steering was prompted by Curtis Broadbent, I independently discovered the proof of the main theorem and its implications. I wrote the main drafts and minor edits were provided by Eric Cavalcanti. • Chapters 3 with minor revision is published as: Girdhar, P. and Doherty, A.C., 2020. Testing generalised uncertainty princi- ples through quantum noise. New Journal of Physics, 22(9), p.093073. Chapter 4 is currently unpublished. I derived all the results, advised by Andrew Doherty who suggested research questions and some of the techniques. I wrote both of these chapters and edited them in response to comments by Andrew Doherty. Parth Girdhar 6 CONTENTS Acknowledgements Firstly I wish to thank my principal advisor Andrew C. Doherty and auxiliary advisor Eric G. Cavalcanti. I am grateful to have had the opportunity to discuss and learn from you about many strands of quantum mechanics (and what might be lurking behind it) and receive feedback about my work. I am also thankful to Curtis Broadbent, Cyril Branciard, Michael Hall and Howard Wiseman for discussions and feedback. I appreciate the support from Nicolas Brunner, Peter Graham, Achim Kempf, Antony Milne, Terry Rudolph and for being available to host my research visits, talks and the subsequent exchange of ideas. In particular I acknowledge Tim Ralph for hosting the presentation of my research at RQI-10. To my friends in the USyd quantum information theory group I offer my thanks for the conversations on walks through physics and life, the group meetings (\best time of the week") and seminars. Thank you to Stephen Bartlett, Andrew Doherty, Steven Flammia, Arne Grimsmo and Isaac Kim for leading the activities of the group that have broadened my vision of quantum information theory. I also appreciate the lessons on practical innovation from colleagues downstairs working on quantum ex- periments. I acknowledge administrative support from Geraldine Arriesgado, Alexis George, Lorraine Di Masi, Anthony Monger, Jeremy Platt, Leanne Price, Satpal Sahota, Fran Vega, Wicky West, Nicole Yang and Amy Zhu. I cherish the companionship of my friends in collectives outside of physics, es- pecially all the volunteers of the USyd Vegesoc. Theodore, a constant source of positivity on my days on campus, did not survive to see the completion of this the- sis: I will always value our friendship and your dedication to the great causes of the world. This work could not have happened without the support of my immediate family. I am especially grateful to my mother whose help is immeasurable. Finally, my first gurus of physics Isaac Asimov, Albert Einstein and Stephen Hawking have guided me all this way, their writings shall inspire students eternally. CONTENTS 7 Mr W: Quantum mechanics is a real Bohr. Mr X: How dare you say that! It just shows you are as thick as a Planck! Mr Y: Oh Shut Up! Stop Bragg-ing about your knowledge Mr X! Mr X: Hey, you step out of this. You're just jealous of the abilities I was Born with. Mr Y: Why you! (All three fight) Mr Z: Guys, cut it out! I'll call the Pauli-ce if you don't stop fighting! Mr A: Just being Curie-ous, aren't you a Pauli-ce officer? Mr Z: No no. It seems my uniform has made me a Feyn-man (Girl comes by, Mr A offers flower to girl) Girl: Fer-mi? Mr A: Yes, if you A-Laue me to give it to you. Mr B: Hey, I'm her fiance! You're just a Neumann. Mr A: We'll see who's the Neumann. Bring it on! (Mr A and B start fighting) (Mr W, X and Y start fighting with Mr A and B for no reason) (Mr Z joins in for no reason either) Random: To complement I shall blast this Bohm. Muhahaha! (All die) THE END A QM skit, composed at age 14 8 CONTENTS Chapter 1 Introduction Some may say quantum mechanics is the cherry on top of a creamy dessert called physics. Its predictive power has amazed us all, beginning with Planck's realisation that the quantisation of energy absorption and emission of a blackbody explains the blackbody radiation spectral density, to the prediction in quantum electrodynamics (QED) of the electron anomalous magnetic dipole moment to a precision of better than one part in a trillion. In more recent years the success of quantum mechanics has opened the field of quantum technology, where the distinct features of quantum mechanics are exploited to bring enormous advantage in technological tasks like computation and communication. But there have been two questions about quantum mechanics that have been lingering since the birth of the theory itself: whether it is a complete framework of physics, and what its relationship is to gravity. Most physicists have segregated the questions by relying on the argument that the foundations of quantum mechanics are independent of the features that make it operational for statistical predictions. In principle these features could be combined with an operational theory of gravity. But even at an operational level it is generally regarded that experimentally prob- ing the quantum mechanics-gravity relationship, the problem of quantum gravity phenomenology, is not currently within reach. In this thesis I follow an alternate philosophy. That is, I examine how the op- erational features of quantum mechanics can be used to constrain the possibility of a more complete framework of nature as well as the relationship to gravity. In this way the two questions above are amalgamated together. If quantum mechanics is taken to be a complete framework then the phenomenon of Einstein-Podolsky-Rosen (EPR) steering (the ability to control a remote quantum state via measurements) implies that the theory contains an apparent nonlocality, in the sense that systems arbitrary distances apart can affect each other instantaneously.