Resource characterisation of quantum entanglement and nonlocality in multipartite settings. Caracterizaci´ondel entrelazamiento y no localidad cu´anticos como recursos en sistemas multipartitos Patricia Contreras Tejada Directores: Carlos Palazuelos Cabez´on Julio ´I~nigode Vicente Maj´ua Facultad de Ciencias Matem´aticas Universidad Complutense de Madrid DECLARACIÓN DE AUTORÍA Y ORIGINALIDAD DE LA TESIS PRESENTADA PARA OBTENER EL TÍTULO DE DOCTOR D./Dña.________________________________________________________Patricia Contreras Tejada ________, estudiante en el Programa de Doctorado _________________________Investigación Matemática ____________, de la Facultad de _____________________________Ciencias Matemáticas de la Universidad Complutense de Madrid, como autor/a de la tesis presentada para la obtención del título de Doctor y titulada: Resource characterisation of quantum entanglement and nonlocali ty in multipartite settings. Caracterización del entrelazamiento y no localidad cuánticos como recursos en sistemas multipartitos y dirigida por: Carlos Palazuelos Cabezón y Julio Íñigo de Vicente Majúa DECLARO QUE: La tesis es una obra original que no infringe los derechos de propiedad intelectual ni los derechos de propiedad industrial u otros, de acuerdo con el ordenamiento jurídico vigente, en particular, la Ley de Propiedad Intelectual (R.D. legislativo 1/1996, de 12 de abril, por el que se aprueba el texto refundido de la Ley de Propiedad Intelectual, modificado por la Ley 2/2019, de 1 de marzo, regularizando, aclarando y armonizando las disposiciones legales vigentes sobre la materia), en particular, las disposiciones referidas al derecho de cita. Del mismo modo, asumo frente a la Universidad cualquier responsabilidad que pudiera derivarse de la autoría o falta de originalidad del contenido de la tesis presentada de conformidad con el ordenamiento jurídico vigente. En Madrid, a ____14 de _________________________abril de 20___21 Firmado digitalmente por CONTRERAS TEJADA PATRICIA - 51144227R Fecha: 2021.04.14 Fdo.: _______________________________19:27:00 +02'00' Esta DECLARACIÓN DE AUTORÍA Y ORIGINALIDAD debe ser insertada en la primera página de la tesis presentada para la obtención del título de Doctor. Acknowledgments I am indebted to lots of people for making this thesis possible through their support, inspiration and encouragement. Here's a (certainly incomplete) tribute: I probably wouldn't have become interested in maths or physics had it not been for Mrs Robinson and Mr Combley. Their dedication and their determination to inspire curiosity was something I admired class after class, and which has stayed with me until today. Soon after, Roman Schubert, John Hannay, Sandu Popescu, Jasper van Wezel, Tony Short and Noah Linden all contributed to making quantum physics and quantum information tangible, and gave me a glimpse of this fascinating field that made me want to explore it further. Sin mis directores, Carlos y Julio, esta tesis no se habr´ıahecho realidad. De los dos he aprendido a leer, desarrollar y escribir matem´aticascasi desde cero. Julio, gracias especialmente por escuchar y apoyarme cuando lo he necesitado. My collaborators for Chapter 5, Gianni, Alek, Adam and Piero, introduced me to a whole new field. Thank you all for engaging in so many divergent discussions, and converging into an actual paper! Several people have been of great help during the development of the results. In particular, thanks go to Aleksander M. Kubicki and Elie Wolfe for enlightening discussions towards Chapter 3, Andreas Winter for a helpful communication about Chapter 4, and Eduardo Zambrano, Elie Wolfe, Valerio Scarani, Alex Pozas-Kerstjens, Miguel Navascu´es,Serge Fehr, Peter van Emde Boas and Ronald de Wolf for their comments and help towards Chapter 5. Gianni deserves a special mention, for introducing some much needed serenity at key points in the process. Helen provided immense encouragement by praising achievements of mine such as opening a box or building a block tower with a very sincere \well done!". Mam´ay Pap´a,vuestro apoyo incondicional, estos a~nos y siempre, hace que todo sea un poquito bastante m´asf´acil.Pap´a,me has servido de gran inspiraci´onen el camino. Mam´a,gracias por todas esas conversaciones hasta las tantas. Antonio, gracias por tu v entusiasmo con todo. <Nunca sospech´eque hablar´ıamosde matem´aticasen tanto detalle! Thanks also go to the mathQI group, past and present, for many an interesting chat. Sobre todo, gracias al despacho 251, por tantas conversaciones con mayor o menor sentido y sobre todo muchas risas. La fuerza del equipo BYMAT ha sido tremendamente inspiradora. Agata´ y Laura, gracias por abrirme una ventana al mundo y por bailar el corro de la patata con tanto convencimiento. Contents 1 Introduction 11 1.1 Quantum formalism . 13 1.1.1 Entanglement in bipartite systems . 17 1.1.2 Entanglement in multipartite systems . 24 1.2 Probability distributions . 25 1.2.1 Multipartite nonlocality . 33 1.3 Our contribution . 36 1.3.1 A nontrivial resource theory of multipartite entanglement . 36 1.3.2 Pure pair-entangled network states . 38 1.3.3 Mixed pair-entangled network states . 40 1.3.4 A physical principle from observers' agreement . 43 2 A nontrivial resource theory of multipartite entanglement 47 2.1 Definitions and preliminaries . 48 2.2 Non-triviality of the theories . 49 2.3 Existence of a maximally entangled state . 51 2.3.1 FSP regime . 51 2.3.2 BSP regime . 62 2.4 Comparison between the regimes . 64 2.5 Looking beyond . 65 3 Pure pair-entangled network states 69 3.1 Definitions and preliminaries . 69 3.2 GMNL from bipartite entanglement . 71 3.3 GMNL from GME . 82 3.4 Looking beyond . 92 vii 4 Mixed pair-entangled network states 95 4.1 Entanglement in mixed-state networks . 95 4.2 Locality in mixed pair-entangled networks . 110 4.3 Superactivation of GMNL in networks . 119 5 A physical principle from observers' agreement 129 5.1 Classical agreement theorem . 129 5.2 Mapping agreement to nonsignalling boxes . 132 5.3 Nonsignalling agents can agree to disagree . 140 5.4 Quantum agents cannot agree to disagree . 146 5.5 Quantum agents cannot disagree singularly . 152 6 Conclusions 157 6.1 Multipartite entanglement and nonlocality . 157 6.2 A physical principle from observers' agreement . 161 Bibliography 163 List of Figures 3.1 Connected network of bipartite entanglement. For each i 2 [n]; party Ai k k has input xi and output ai on the particle at edge k: Particles connected by an edge are entangled. 72 3.2 Element i 2 [n − 1] of the star network of bipartite entanglement created from a GME state jΨi : Parties fBjgj2[n−1];j6=i have already measured jΨi and are left unentangled. Alice and party Bi share a pure bipartite entangled state. Alice has input xi and output ai while each party Bj; i i j 2 [n − 1]; has input yj and output bj: ................... 83 List of Tables 4.1 Bounds for biseparability and GME in a Lambda network (Λ) and a triangle network (4) where the shared states are isotropic states with visibility p in dimension 2. 101 5.1 Parametrisation of 2-input 2-output nonsignalling boxes with common certainty of disagreement. Here, r; s; t; u 2 [0; 1] are such that all the entries of the box are non-negative, r > 0, and s − u 6= r − t: ....... 141 5.2 Parametrisation of 2-input 2-output nonsignalling boxes with singular disagreement. Here, r; s; t; u; 2 [0; 1] are such that all the entries of the box are non-negative, s > 0, and s + t 6= 0 and u + t 6= 1. 154 1 Abstract Quantum technologies are enjoying an unprecedented popularity, and some applications are already in the market. This thesis studies two phenomena that are behind a lot of quantum technologies: entanglement and nonlocality. We focus on multipartite systems, and ask what configurations of those systems are more useful than others. `Usefulness' takes on different meanings depending on the context, but, roughly speaking, we aim for more entanglement or more nonlocality. Chapter 2 is motivated by an important issue with traditional resource theories of multipartite entanglement: they give rise to isolated states and inequivalent forms of entanglement. We propose two new resource theories that do not give rise to these problems: the resource theory of non-full-separability under full separability-preserving operations, and the resource theory of genuine multipartite entanglement (GME) under biseparability-preserving operations. Further, the latter theory gives rise to a unique maximally GME state. Chapters 3 and 4 focus on quantum networks, that is, configurations where pairs of parties share entangled states, and parties are bipartitely entangled to one or more of the others. First, we assume all shared states are pure. It is known that all connected networks of bipartite pure entangled states are GME (which is a necessary requirement for being nonlocal) so we ask what networks give rise to genuine multipartite nonlocality (GMNL). Surprisingly, they all do: any connected network of bipartite pure entangled states is GMNL. Next, we allow for the presence of noise, and study networks of mixed states taking isotropic states as a noise model. Not even GME is guaranteed in these networks, so our first task is to find out what networks, in terms of both noise and geometry, give rise to GME. We find that, unlike in the case of pure states, topology plays a crucial role: for any non-zero noise, tree networks and polygonal networks become biseparable if the number of parties is large enough. In sharp contrast, a completely connected network of isotropic states is GME for any number of parties as long as the noise is below a threshold. We further deduce that, while non-steerability of the shared 3 states can compromise GMNL or even render the networks fully local, taking many copies of bilocal networks can restore GMNL.