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Conclusive Exclusion of Quantum States and Aspects of Thermo-Majorization

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Conclusive Exclusion of Quantum States and Aspects of Thermo-Majorization Conclusive Exclusion of Quantum States and Aspects of Thermo-Majorization By Christopher David Perry A thesis submitted to University College London for the degree of Doctor of Philosophy Department of Physics and Astronomy University College London 30th December, 2015 I, Christopher David Perry, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Signed ............................................ Date ................ 1 Conclusive Exclusion of Quantum States and Aspects of Thermo-Majorization By Christopher David Perry Doctor of Philosophy of Physics University College London Prof. Jonathan Oppenheim, Supervisor Abstract Part 1: Why can we not distinguish between pure, non-orthogonal quantum states? Re- garding the quantum state as a state of knowledge rather than something physically real, has the potential to answer this question and explain other quantum properties but such interpre- tations have recently been undermined. This important no-go result, due to Pusey, Barrett and Rudolph, makes use of a specific example of a task we term state exclusion. Here, a system is prepared in a state chosen from a known set and the aim is to determine a preparation that has not taken place. We formulate state exclusion as a semidefinite program, using it to investigate when exclusion is conclusively possible and how it can be achieved. Based on state exclusion, we construct a communication task which exhibits drastic, `infi- nite', separations between a variety of classical and quantum information and communication complexity measures. This serves to requisition the aforementioned foundational result for use in information theoretic protocols. Part 2: What does thermodynamics look like in the absence of the thermodynamic limit? In recent years there has been a concerted effort to apply techniques from quantum information theory to study the laws of thermodynamics at the nano-scale. This has led to the resource theory of thermal operations for determining when single-copy transformations are possible. However, if a deterministic transition is forbidden, can it occur probabilistically? Here we compute and bound the maximum probability with which nano-scale thermodynamical trans- formations can occur. Thermal operations assume that one can precisely manipulate all of the degrees of freedom in a very large heat bath. While this enables the derivation of ultimate limits on nano-scale 2 thermodynamics, it does not make them feasible to perform in reality. We show how allowed transitions can be implemented whilst manipulating only a single bath-qubit, making thermal operations more experimentally palatable. 3 List of Publications and Preprints The majority of the work presented in this thesis contains materials from the following publi- cations: S. Bandyopadhyay, R. Jain, J. Oppenheim, and C. Perry, Conclusive exclusion of quantum states, Phys. Rev. A, 89, 022336 (2014). T.A. Brun, M.-H. Hsieh, and C. Perry, Compatibility of state assignments and pooling of infor- mation, Phys. Rev. A, 92, 012107 (2015). C. Perry, R. Jain, and J. Oppenheim, Communication tasks with infinite quantum-classical separation, Phys. Rev. Lett. 115, 030504 (2015). A.M.´ Alhambra, J. Oppenheim, and C. Perry, What is the probability of a thermodynamical transition?, Preprint arXiv:1504.00020. Z.-W. Liu, C. Perry, Y. Zhu, D.E. Koh, and S. Aaronson, Doubly infinite separation of quantum information and communication, Preprint arXiv:1507.03546. C. Perry, P. Cwikli´nski,J.´ Anders, M. Horodecki, and J. Oppenheim, A sufficient set of exper- imentally implementable thermal operations, Preprint arXiv:1511.06553. 4 Acknowledgments The three years of my PhD and the completion of this thesis would have been near impossible without the input of many people both from a physics and sanity-preservation perspective. On the physics front, the foremost contributor has been my supervisor Jonathan. I am grateful to him for plucking me from a career of Excel spreadsheets in Stevenage to undertake the research presented here, for teaching me a lot of physics and for finally convincing me to take the plunge and learn a little thermodynamics. I am less grateful for the amount of chalk dust sitting in my lungs from his office's blackboard-wall combination. I have been fortunate to have had the opportunity to accumulate a number of excellent collaborators. In particular, I thank Som for his encouragement - especially in the early years, Rahul for his patience in guiding me through the subtleties of communication complexity theory and Alvaro´ for his willingness to decorate every whiteboard in the department with scrawled Lorenz curves. I have also benefited from many conversations with people both at UCL and on my travels and I cannot do them all justice here. Special thanks however, goes to Neil, and Na¨ıriand Hussain for countless `discussions' (particularly to Hussain for undertaking the task of proofreading this thesis for me). My examiners, Toby Cubitt and Tony Short, also deserve thanks: firstly for taking the time to read this tomb and secondly for their insightful comments that have helped to improve it. Sanity-preservation duties have been ably shouldered by my original office-mates, Pete and Na¨ıri. K3 drinks, flippantry, Jamie Dalrymple's profligacy, extended afternoon strolls and much more besides have made PhD life a lot easier. It would also be remiss of me not to mention the 12:15 lunch crew. Arne, Alexandros, The Moderate Johnny, Rafa, Roberta, Sarah, Alvaro,´ Enrico, Luca, Pik, Brendan and the adopted Brummies, Ilhan_ and Michela, have provided a welcome daily escape from my desk. Thank you to my parents and brother for all of their love, support and encouragement 5 throughout my life. Without it, I definitely would not have made it to where I am today. Finally, my deepest thanks go to Marie. Her smiles, laughter and love are invaluable. 6 Contents Introduction 13 I Conclusive Exclusion of Quantum States 16 Guide to Part I 17 1 State Discrimination 19 1.1 The task of state discrimination . 19 1.2 Semidefinite programs . 21 1.2.1 The SDP formalism . 21 1.2.2 Properties of SDPs . 22 1.3 State discrimination as an SDP . 24 1.3.1 Formulation as an SDP . 25 1.3.2 Optimal measurements . 26 1.3.3 Bounds on the probability of success . 27 1.4 Summary . 31 2 State Exclusion 32 2.1 An explanation for indistinguishability . 32 2.1.1 Ontological models . 33 2.1.2 Spekkens' toy bit and indistinguishability . 35 2.1.3 The PBR argument . 36 2.1.4 Aside: The impossibility of maximally epistemic models . 39 2.2 The task of state exclusion . 40 2.2.1 Exclusion as discrimination and vice versa . 42 7 2.2.2 Geometric interpretation . 43 2.3 State exclusion as an SDP . 43 2.3.1 Optimal measurements . 45 2.3.2 A necessary condition for single-state conclusive exclusion . 46 2.3.3 Lower bounds on the probability of error . 49 2.4 Aside: Alternative formulations of state exclusion . 50 2.4.1 Unambiguous state exclusion . 50 2.4.2 Worst-case error state exclusion . 51 2.5 Applications of the state exclusion SDP . 52 2.5.1 Optimality of the PBR measurement . 52 2.5.2 Measures of state assignment compatibility . 55 2.6 Summary . 60 3 Communication Tasks with Infinite Quantum-Classical Separation 62 3.1 Communication tasks and protocols . 62 3.2 The exclusion game . 64 3.3 Preliminaries and notation for communication protocols . 64 3.3.1 Asymptotic complexity . 65 3.3.2 Information theory . 66 3.3.3 Complexity measures for communication protocols and tasks . 68 3.3.4 Properties of and relationships between complexity measures . 71 3.4 Separations from the exclusion game . 74 3.4.1 Quantum information cost vs classical information cost . 74 3.4.2 Quantum communication complexity vs quantum information cost . 81 3.4.3 Quantum communication complexity vs classical communication complexity 86 3.5 Summary . 96 II Beyond the Thermodynamic Limit 99 Guide to Part II 100 4 Thermodynamics as a Resource Theory 102 4.1 Thermodynamics without the thermodynamic limit . 102 8 4.2 Resource theories . 104 4.2.1 Noisy operations . 107 4.2.2 Thermal operations . 112 4.3 Beyond thermo-majorization . 125 4.3.1 Thermodynamics with catalysts . 126 4.3.2 Dealing with coherences . 127 4.3.3 Average energy conservation . 127 4.4 Summary . 128 5 Probabilistic Thermodynamical Transitions 129 5.1 Probabilistic transitions . 129 5.2 Noisy operations . 130 5.2.1 Non-deterministic transitions . 130 5.2.2 Quantifying the purity of transition . 133 5.2.3 Aside: Entanglement of transition under LOCC . 136 5.3 Thermal operations . 138 5.3.1 Non-deterministic transitions . 139 5.3.2 Quantifying the work of transition . 143 5.4 Summary . 146 6 Towards Experimentally Friendly Thermal Operations 148 6.1 Coarse operations . 148 6.1.1 Partial Level Thermalizations . 149 6.1.2 Level Transformations . 150 6.1.3 Partial Isothermal Reversible Processes and Points Flows . 153 6.2 Coarse operations as thermal operations . 159 6.2.1 Partial Level Thermalizations . 159 6.2.2 Level Transformations . 161 6.3 Implementing allowed transformations using coarse operations . 162 6.3.1 States with the same ordering . 163 6.3.2 States with different ordering . 166 6.4 Summary . 172 9 III Appendices 174 A Proofs for state exclusion 175 A.1 A proof of the necessary condition for conclusive state discrimination . 175 A.2 Optimality of projective measurements . 177 B SDP formulations 179 B.1 The unambiguous state exclusion SDP . 179 B.2 The worst-case error state exclusion SDP . 180 B.3 The measure of equal support compatibility SDP . 182 C The tradeoff between the probability and the purity of a transition 185 10 List of Figures 2.1 Classes of ontological models.
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