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Quantum Markov Chain Mixing and Dissipative Engineering

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Quantum Markov Chain Mixing and Dissipative Engineering Quantum Markov Chain Mixing and Dissipative Engineering Kastoryano, Michael James Publication date: 2012 Document version Early version, also known as pre-print Citation for published version (APA): Kastoryano, M. J. (2012). Quantum Markov Chain Mixing and Dissipative Engineering. Download date: 27. Sep. 2021 FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN Quantum Markov Chain Mixing and Dissipative Engineering PhD thesis Michael James Kastoryano The PhD School of Science Quantum Information Group QUANTOP – Danish National Research Foundation Centre for Quantum Optics Niels Bohr Institute and Niels Bohr International Academy Academic supervisors: Michael M. Wolf and Anders S. Sørensen December, 2011 Quantum Markov Chain Mixing and Dissipative Engineering Michael James Kastoryano This thesis is submitted in partial fulfilment of the requirements for the Ph.D. degree at the University of Copenhagen. Copyright c 2011 Michael James Kastoryano Abstract This thesis is the fruit of investigations on the extension of ideas of Markov chain mixing to the quantum setting, and its application to problems of dissipative engineering. A Markov chain describes a statistical process where the probability of future events depends only on the state of the system at the present point in time, but not on the history of events. Very many important processes in nature are of this type, therefore a good understanding of their behavior has turned out to be very fruitful for science. Markov chains always have a non-empty set of limiting distributions (stationary states). The aim of Markov chain mixing is to obtain (upper and/or lower) bounds on the number of steps it takes for the Markov chain to reach a stationary state. The natural quantum extensions of these notions are density matrices and quantum channels. We set out to develop a general mathematical framework for studying quantum Markov chain mixing. We introduce two new distance measures into the quantum setting; the quantum c2- divergence and Hilbert’s projective metric. The quantum c2 divergence allows us to extend the class of functional techniques called Poincaré inequalities to the quantum set- ting. In the process, we identify the appropriate framework for discussing notions such as detailed balance and c2-mixing: monotone Riemannian metrics on matrix spaces. This insight allows us to characterize the connection between spectral properties of a (primi- tive) quantum channel and its mixing time. Within the same framework we also derive a restricted quantum version of the celebrated conductance bound (Cheeger’s inequal- ity). We then consider Hilbert’s projective metric - a well known tool in Perron-Frobenius theory - in the context of quantum information theory. As a classic tool for analyzing convergence of positive maps on cones, Hilbert’s projective metric is especially good at providing existence proofs of maps between two spaces (or cones). We relate this measure to distinguishability measures on restricted cones of operators, and analyze the observable loss of information after the application of a quantum channel under a restrict set of mea- surements. Various notions of contractivity of quantum channels are revealed through Hilbert’s metric. Still on the topic of quantum Markov chains, we introduce the notion of cutoff phe- nomenon to the quantum setting. The cutoff phenomenon describes the situation when a Markov chain does not converge for a potentially long time, and then at a specific point in time abruptly converges to equilibrium. In the thermodynamic limit, the convergence profile will then look like a step function. We show that this type of convergence behavior occurs, as measured in trace-norm, when a system is subject to a product channel, and the noise on each product is modeled by a one-parameter semigroup of quantum channels. Finally, we consider three independent tasks of dissipative engineering. The first task, which is aimed at experimental realization in the near future, consists in dissipatively iii iv ABSTRACT preparing a maximally entangled state of two atoms trapped in an optical cavity. We show that this is indeed possible with very hight fidelity, and in a short amount of time, using present day technology. We also show that the scaling of the fidelity with the quality factor of the cavity (the cooperativity) scales quadratically better in our dissipative setup than in any known coherent unitary setup; indicating that dissipative state preparation can lead to fundamental improvements over closed system protocols. The second task that we consider is dissipative preparation of graph states, where we show the this class of states can be prepared in a time scaling as log n in the number of stabilizer elements. We also show that this process exhibits a cutoff. As a third task of dissipative engineering, we revisit the dissipative quantum computing construction of [VWC09]; we rigorously prove that it is as efficient as the circuit model. Dansk resumé Denne Ph.D.-afhandling omhandler videreudvikling af ideer angående konvergensen af Markov-kæder i kvantemekanikken samt dens anvendelse inden for dissipativ manipu- lering. Markov-kæder beskriver en diskret tidslig proces for hvilken sandsynligheden af fremtidige begivenheder kun afhænger af systemets nuværende tilstand og ikke dens his- torik. Mange centrale processer i naturen er af denne type, hvorved en god forståelse af Markov-kædernes opførsel, har haft stor betydning for videnskaben. Markov-kæder har altid et ikke tomt sæt af stationære tilstande. Formålet med studier af Markov-kæders konvergens er at fastslå øvre og nedre grænser for antallet af skridt det tager for Markov- kæden at nå en stationær tilstand. Den naturlige kvantemekaniske udvidelse af disse be- greber er tæthedsmatricer samt kvantekanaler. I denne afhandling ønsker vi at udvikle en general ramme for analyser af Markov-kæders konvergens. To nye afstandsmål bliver introducerede, den kvantemekaniske c2-divergens samt Hilberts projektive metrik. Den kvantemekaniske c2-divergens muliggøre udvidelsen af funk- tionelle metoder kaldet Poincaré uligheder til kvantemekanikken. I den forbindelse iden- tificere vi de nødvendige værktøjer for at diskutere begreber som detaljerede balance eller c2-blanding: monotone Riemannske metrikker på matrix run. Dette muliggøre en karakteristik af forbindelsen imellem de spektrale egenskaber af en primitiv kvantekanal og dens konvergenstid. Inden for samme ramme udledes en begrænset version af den berømte grænse for ledningsevnen, Cheeger’s ulighed. Efterfølgende studeres Hilberts projektive metrik - der er et velkendt koncept i Perron-Frobenius teori - i sammenhæng med kvanteinformationsteori. Som et af de klassiske værktøjer til at analysere konver- gensen af positive funktioner på kegler, er Hilberts projektive metrik i særdeleshed god i forbindelse med eksistens beviser for afbildninger imellem to rum (eller kegler). Dette af- standsmål relateres til forskellighedsmål på begrænsede kegler af operatorer, og analyser af observablens tab af information efter anvendelse af en kvantekanal under et begrænset sæt af målinger. Forskellige forståelser af sammentrækningen af kvantekanaler afsløres igennem Hilberts metrik Stadig i forbindelse med kvantemekaniske Markov-kæder introducere vi begreber af afbrydelses- fænomener i rammerne af kvanteteorien. Afbrydelses fænomener beskriver situationen hvor en Markov-kæde ikke konvergere for en potentiel lang tid, hvorefter den for en speci- fik tid brat vil konvergere mod sin stationære tilstand. I den termodynamiske grænse, vil konvergens profilen ligne en trappe funktion. Det vises at denne type af konvergens op- står, målt af spor normen, når systemet udsættes for en kvanteproduktkanal samt at støjen til ethvert punkt kan modellers som en en-parameter semigruppe af kvantekanaler. Til sidst beskæftiger vi os med tre uafhængige opgaver i forbindelse med dissipativ ma- nipulering. Først opgave, med henblik på snarlig eksperimentel virkeliggørelse, består v vi DANSK RESUMÉ i en dissipativ forberedelse af en maksimalt sammenfilteret tilstand imellem to atomer fanget i en optisk cavity. Vi viser at det med nutidens teknologi er muligt at frembringe en sådan tilstand på kort tid og med høj nøjagtighed. Det vises også at nøjagtigheden skalere kvadratisk bedre i det dissipative tilfælde sammenlignet med kendte kohærente unitære systemer. Dette er en indikation på at dissipativ tilstands forberedelse kan lede til fundamentale forbedringen i forhold til lukkede systemer. Den anden opgave, som vi antager, er dissipativ forberedelse af graftilstander, her vises det at disse tilstand kan forberedes med en tids skalering som log n i antallet af stabilisator elementer. Det vises også at denne processer har en afgrænsning. For den tredje opgave i dissipativ manipulering genovervejer vi den dissipative kvante computer af [VWC09]; det bevises stringent at denne model er ligeså effektiv som den almindelige kvantecomputere. Acknowledgments This thesis concludes three years of Ph.D. studies carried out jointly at the Niels Bohr Institute, University of Copenhagen, and at the Niels Bohr International Academy. I gratefully acknowledge the generous support by the faculty of the Niels Bohr Institute and the Niels Bohr International academy, and of the European projects QUEVADIS and COQUIT. These have been three wonderful years in a highly stimulating , as well as gratifying, envi- ronment. Naturally, the personal and professional educational process which culminates with this thesis would not have been possible
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