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Maximal Entanglement Maximal Entanglement Applications in Quantum Information and Particle Physics Alba Cervera Lierta Aquesta tesi doctoral està subjecta a la llicència Reconeixement 4.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution 4.0. Spain License. Tesi doctoral Maximal Entanglement Applications in Quantum Information and Particle Physics Alba Cervera Lierta Director: Dr. José Ignacio Latorre Sentís Maximal Entanglement Applications in Quantum Information and Particle Physics Programa de doctorat en física Autora: Alba Cervera Lierta Director: Dr. José Ignacio Latorre Sentís Tutor: Dr. Domènec Espriu Climent Copyright c 2019 Alba Cervera Lierta (CC-by) Chapter 2 image art and design by Carlos Villafranca. Quantum circuits have been designed with Q-circuit package. Source can be found at https://cquic.unm.edu/resources/. Feynman diagrams have been designed with TikZ-Feynman package. Documentation can be found in Joshua P. Ellis, Computer Physics Communications 210, 103-123 (2017). LATEX template “The Legrand Orange Book” v. 2.4 by Mathias Legrand with modifications. Unless explicitly stated in the caption, I have produced all the figures that appear in this work, which will often closely follow those in the corresponding publications. First printing, April 2019. A les meves àvies i avis. Agraïments Aquesta tesi no hauria estat possible sense el suport i l’ajut de moltes persones. Algunes d’aquestes han format part de la meva vida abans de començar aquest camí, altres les he conegut a punt d’acabar-lo, però totes i cada una d’elles m’han influenciat tant acadèmicament com personalment. La primera d’elles és en José Ignacio, a qui considero més que un director, un mentor. Gràcies per les teves lliçons. Crec que tant jo com qualsevol dels teus estudiants recordarem sempre les llargues i inspiradores converses davant la pissarra de guix del teu despatx. I mil gràcies per donar-me l’oportunitat d’embarcar-me en aquesta aventura científica i de treballar tot el possible perquè no l’abandonés. En segon lloc, vull agrair al que va ser també el meu director durant uns mesos. Gràcies Juan per convidar-me a treballar amb tu a Oxford. Van ser uns mesos clau on vaig prendre la decisió de continuar amb aquest doctorat com sigui. Estic segura que sense aquella experiència avui no seria on sóc. Aprofito per agrair també a totes aquelles persones que vaig conèixer durant la meva estància al Rudolph Pierls Institute: Marco, Nathan, Stefano, Emanuele, Valerio i, per descomptat, Luca, amb qui he treballat, sofert i gaudit amb un dels projectes d’aquesta tesi. Grazie mille. També vull agrair als meus coautors, Karol, Dardo, Germán i Albert, per les seves aportacions i interessants discussions. Durant aquests quatre anys, he tingut companys al meu voltant amb els que he compartit alegries i frustracions, i que s’han convertit en veritables amics. Gràcies Dani per compartir la teva experiència i feina amb mi. Les llargues tardes al despatx parlant de política i física i les lliçons d’escacs al Petronilla fan que et perdoni per no poder deixar sense vigilància els meus Kinder. Gràcies també a Javier, Adrià, Clara, Marc, Isma i Elis per compartir aquests anys de doctorat i els anteriors de la carrera. Les converses i cafés a mig matí, mitja tarda i/o al mig dia han fet suportables els dies més pesats. Gracias Ivan por los mismos motivos y por nuestras discusiones sobre física de partículas: aprendí mucho contigo. També voldria agrair a totes aquelles persones que han format part de la meva vida fora de la Universitat, en especial les que sempre han estat disponibles per fer unes birres o uns vermuts per desconnectar i gaudir de la vida. Gràcies Sergi, Irene, Fran, Marc, Josep, ii Toni, Gemma, Joan, Marc, Alba i molts més. Aquesta tesi ha sigut el reflex de significants canvis al voltant de la investigació en informació quàntica a Barcelona. El motiu pel qual he dedicat els últims dos anys al camp de la computació quàntica ha sigut la creació d’un grup d’investigació fruit de la col·laboració entre la Universitat de Barcelona i el Barcelona Supercomputing Center (BSC). Voldria agrair al BSC per donar-me suport econòmic en aquests darrers anys i per apostar per aquest camp. En especial vull donar les gràcies a tots els que formen part del grup Quantic o hi col·laboren: Pol, Artur, Luca, Sofyan, Chris, Carlos, David, Adrian, Diego, Sergi, Elies i Josep. També voldria agrair a l’empresa IBM per atorgar-me el premi “Teach Me QISKit” i donar-me així visibilitat en aquest camp. Gracias Carlos por plantar la semilla de todo esto. Fuiste una fuente de inspiración para mi y muchos otros. Gracias por ser el primero en mostrarme la belleza de la física. Gràcies també als meus pares, Jordi i Gemma, i al meu germà, Gerard, per animar-me sempre a estudiar allò que més m’agradés, per animar-me en els moments més baixos i per compartir amb mi les alegries que han anat sorgint. Sense el seu suport res del que he fet hagués estat possible. Gràcies també als meus avis, Roser, Llorenç, Cleo i Luis, per tota una vida de sacrificis que han fet possible que jo ara hagi arribat on hagi volgut. Sé que algun d’ells hagués triat aquest mateix camí si hagués pogut, però les circumstàncies de l’època no ho van fer possible. Per aquest motiu els hi dedico especialment aquesta tesi, que no és més que el fruit del seu esforç. Por último, a la persona que mejor conoce los entresijos no escritos de esta tesis. Gracias Miguel, por tu apoyo, por tu optimismo, por tus cuidados, por soportar los malos momentos, por celebrar los buenos, por tu infinita paciencia, por escucharme, por tu dedicación y, en definitiva, por quererme. Esta tesis también va dedicada a ti. List of publications • D. Alsina, A. Cervera, D. Goyeneche, J. I. Latorre and K. Życzkowski, Operational Approach to Bell Inequalities: Application to qutrits, Physical Review A 94, 032102 (2016). • A. Cervera-Lierta, J. I. Latorre, J. Rojo and L. Rottoli, Maximal Entanglement in High Energy Physics, SciPost Physics 3, 036 (2017). • A. Cervera-Lierta, A. Gasull, J. I. Latorre and G. Sierra, Multipartite entanglement in spin chains and the Hyperdeterminant, Journal of Physical A: Mathematical and Theoretical 51, 505301 (2018). • A. Cervera-Lierta, Exact Ising model simulation on a quantum computer, Quantum 2, 114 (2018). Resum L’entrellaçament és una de les principals característiques de la mecànica quàntica. És probablement un dels fenòmens quàntics més debatuts i estudiants degut, en part, a la seva naturalesa antiintuïtiva i, més recentment, a les seves aplicacions en el camp de la informació quàntica. És precisament una propietat que va més enllà del que la física clàssica pot explicar. La motivació d’aquesta tesi és estudiar l’entrellaçament en general i sota quines circumstàncies és màxim en particular. Primerament, analitzem el paper de l’entrellaçament en la construcció de la frontera entre la física clàssica i la física quàntica. Els experiments de Bell ens permeten calcular una sèrie de correladors que ens ajudaran a distingir si les partícules implicades en l’experiment obeeixen el que es coneix com a realisme local. La violació de les desigualtats de Bell demostra que no hi ha una teoria clàssica de variables ocultes que expliqui els resultats de l’experiment, és a dir, la física quàntica subjacent no pot ser explicada des de la física clàssica. La caracterització de desigualtats de Bell per a qualsevol nombre de partícules i dimensions locals és un problema obert en informació quàntica. En aquesta tesi, estudiem i deduïm noves desigualtats de Bell en termes d’operadors, focalitzant-nos especialment en aquelles que involucren qutrits. Les desigualtats per qubits són violades màximament pels estats altament entrellaçats coneguts com a GHZ. Les desigualtats de qutrits, o altres dimensions més grans, són violades màximament per estats que són una deformació dels GHZ. Aquest resultat mostra l’estreta relació, però no equivalència, entre no-localitat i màxim entrellaçament. Seguidament, estudiem l’entrellaçament multipartit i la seva aplicació en la detecció de transicions de fase quàntiques. En particular, estudiem l’entrellaçament entre quatre partícules de dimensió dos, és a dir, entre quatre qubits. Com a figura de mèrit, utilitzem l’hiperdeterminant i els dos invariants polinòmics que el formen, anomenats S i T . Analitzem uns quants estats quàntics rellevants per acabar concloent que aquesta figura de mèrit capta un tipus concret d’entrellaçament multipartit. Quan calculem l’hiperdeterminant en cadenes d’espins 1/2, obtenim un pic pronunciat al voltant de la transició de fase en el vi cas del model d’Ising. En el cas del model XXZ, l’hiperdeterminant és sempre zero. En aquest cas, utilitzem els invariants S i T . El resultat és que el valor dels invariants canvia bruscament en els punts on hi ha transició de fase. Finalment, estudiem la funció d’ona de Haldane-Shastry i obtenim resultats similars als del model XXZ. En una segona part de la tesi, ens centrem en el camp de la computació quàntica. Gràcies als avenços que s’han dut a terme en els darrers anys en relació al control dels àtoms, fotons i processos quàntics en general, la computació quàntica ha esdevingut una realitat. Actualment, diverses empreses estan en procès de construcció dels seus propis ordinadors quàntics. Per aquest motiu, es fa necessària la recerca de mètodes per testejar i comparar aquests primers prototips d’ordinadors. Per una banda, proposem i testegem un mètode que consisteix en la simulació exacta del model d’Ising.
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