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Efficient Characterisation and Optimal Control of Open Quantum Systems

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Efficient Characterisation and Optimal Control of Open Quantum Systems - Mathematical Foundations and Physical Applications DISSERTATION zur Erlangung des akademischen Grades DOKTOR DER NATURWISSENSCHAFTEN (Dr. rer. nat.) im Fach Physik eingereicht am Fachbereich Mathematik und Naturwissenschaften Universitat¨ Kassel von Daniel Maximilian Reich Betreuerin: Prof. Dr. Christiane Koch Zweitgutachter: Prof. Dr. Steffen Glaser Eingereicht am: 18. Dezember 2014 Disputation am: 30. April 2015 Zusammenfassung Kein physikalisches System kann komplett von seiner Umgebung isoliert werden. Daher ist die Untersuchung offener Quantensysteme von enormer Wichtigkeit um eine verl¨assliche und akkurate Kontrolle komplexer Quantensysteme zu gew¨ahrleisten.In der Praxis muss die Zuverl¨assigkeit eines Kontrollfeldes durch Zertifizierung der gew¨unschten Zeitentwicklung des Systems ¨uberpr¨uftwerden, w¨ahrendeine hohe Genauigkeit durch pr¨aziseKontrollstrategien in der Gegenwart von Dekoh¨arenz- effekten garantiert werden muss. Der erste Teil dieser Arbeit pr¨asentiert einen algebraischer Rahmen zur Bestimmung der min- imalen Vorraussetzungen zur eindeutigen Charakterisierung beliebiger unit¨arerGatter in offenen Quantensystemen, unabh¨angigvon der speziellen physikalischen Implementierung eines Quanten- bauelements. In diesem Rahmen werden Theoreme hergeleitet, mit Hilfe derer entschieden werden kann, ob ein gegebener Satz an Eingangszust¨andenf¨ursolch ein Quantenelement hinreichend zur Beurteilung der erfolgreichen Implementation ist. Dies erm¨oglicht insbesondere die Herleitung der minimale Zahl an notwendigen Eingangszust¨andenmit dem Ergebnis, dass drei solcher Zust¨ande gen¨ugen- unabh¨angigvon der Gr¨oßedes physikalischen Systems. Diese Resultate erm¨oglichen es, die fundamentalen Grenzen in Bezug auf Zertifizierung und Tomographie offener Quantensysteme herauszustellen. Eine Kombination dieser Ergebnisse mit modernen Monte Carlo Zertifizierung- stechniken erlaubt desweiteren eine signifikante Verbesserung der Skalierung in Bezug auf die Sys- temgr¨oßebei der Zertifizierung allgemeiner unit¨arerQuantengatter. Diese Verbesserung ist nicht nur auf Quanteninformationstr¨agermit zwei Zust¨anden,genannt Qubits, beschr¨anktsondern kann auch f¨urelementare Informationstr¨agerbeliebiger Dimension, den sogenannten Qudits, verallgemeinert werden. Der zweite Teil dieser Arbeit besch¨aftigtsich zun¨achst mit der Anwendung dieser neuen Erkennt- nisse auf die Theorie der optimalen Kontrolle (OCT). OCT f¨urQuantensysteme verwendet Konzepte aus den Ingenieurwissenschaften wie R¨uckkopplung und Optimierung, um konstruktive und destruk- tive Interferenzen auf solch eine Art und Weise zu erzeugen, dass man mit ihnen physikalische Prozesse in eine bestimmte Richtung steuern kann. Es zeigt sich, dass die obigen mathematischen Resultate es erm¨oglichen, neue Optimierungsfunktionale aufzustellen, welche signifkante Einsparungen f¨urnu- merische Kontrollalgorithmen mit sich bringen. Dies gilt sowohl in Bezug auf Prozessorlaufzeit als auch in Bezug auf den notwendigen Speicher. Zum Abschluss dieser Arbeit werden zwei zentrale Probleme der Quanteninformationsverarbeitung aus der Perspektive der optimalen Kontrolle diskutiert - die Erzeugung reiner Zust¨andesowie die Im- plementierung unit¨arerGatter in offenen Quantensysteme. In beiden F¨allenwerden dabei spezifische physikalische Beispiele behandelt: Im ersten Fall handelt es sich um das K¨uhlender Vibrationsfrei- heitsgrade eines Molek¨ulsmit Hilfe optischen Pumpens, w¨ahrendim zweiten Fall ein supraleitendes Phasenqudit betrachtet wird. Dabei wird insbesondere demonstriert, wie spezielle Eigenschaften der Umgebung dazu verwendet werden k¨onnen,gewisse Kontrollziele zu erreichen. Abstract Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control - the preparation of pure states and the imple- mentation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets. Acknowledgements First of all I would like to thank Prof. Christiane Koch for the supervision of this thesis. It was her encouragement, commitment and patience that allowed me to obtain the results that enabled this work in the first place. I am also grateful to all the people I had the chance to work with. In particular I would like to thank Michael Goerz for teaching me almost everything I know about programming and Giulia Gualdi for the many engaging discussions and for making our office a small beacon of mathematical physics during her stay. My thanks also go to all present and past group members, Martin Berglund, MichalTomza, Esteban Goetz, and Wojciech Skomorowski, for creating a very pleasant and friendly working atmosphere. My thanks also go to Anne M¨ullerwho helped to ensure that I did not try to only survive by theorems, formulas, and Fortran code. Finally, I would like to thank my parents and the rest of my family for their unconditional and limitless support. The popular stereotype of the researcher Is that of a skeptic and a pessimist. Nothing could be further from the truth! Scientists must be optimists at heart, In order to block out the incessant chorus of those who say 'It cannot be done.' Academician Prokhor Zakharov, University Commencement (from \Sid Meier's Alpha Centauri") Contents 1 Introduction 1 2 Quantum States, Observables and Dynamical Maps 3 2.1 Breakdown of Local Realism and the Copenhagen Interpretation ............ 4 2.2 Hilbert Space ......................................... 5 2.3 Observables and Measurements ............................... 7 2.4 Liouville Space ........................................ 10 2.5 Open Quantum Systems ................................... 11 2.6 Dynamical Maps ....................................... 13 2.7 The Choi-Jamio lkowski Isomorphism ............................ 15 2.8 Unitary Dynamical Maps (UDMs) ............................. 16 3 Minimal Characterisation and Identification of UDMs 19 3.1 Exponential Scaling ..................................... 20 3.2 Unitary Characterisation in Liouville Space ........................ 21 3.3 Minimal Characterisation of UDMs ............................. 24 3.4 Efficient Identification of Unitarity for Dynamical Maps . 27 4 Tomography and Process Certification 29 4.1 Process Tomography of General Dynamical Maps ..................... 30 4.2 State Tomography of General Density Matrices ...................... 31 4.3 Qubit Systems and Pauli Matrices - Stabilisers and the Clifford Group . 33 4.4 Distance Measures and Fidelities .............................. 35 4.5 Minimal Fidelities involving Pure States for Certification of UDMs . 39 4.6 Classical Fidelities ...................................... 43 4.7 Statistical Approaches to Tomography and Certification . 44 4.8 Monte Carlo Process Certification with State Fidelities . 47 4.9 Qupit Systems and Generalised Pauli Operators ...................... 52 4.10 Optimal Choice of Measurement Basis ........................... 53 5 Dynamics and Control of Open Quantum System 55 5.1 Isolated Quantum System Dynamics ...........................
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