Entanglement and Quantum Cryptography Joonwoo Bae Barcelona, January, 2007 Universitat de Barcelona Departament d’Estructura i Constituents de la Mat`eria Tesis presentada para optar al grado de doctor, January 2007. Programa de doctorado: F´ısicaAvanzada. Bienio: 2003-2005. Thesis advisor: Dr. Antonio Ac´ınDal Maschio( Assistant Professor in ICFO and ICREA Junior) Dear friend, I pray that you may enjoy good health And that all may go well with you, Even as your soul is getting along well. Acknowledgement I am grateful to all the people who helped me throughout my education. My advisor, Antonio Ac´ın,has patiently guided me with his creative insights. In particular, many thanks for his kind care and ceaseless encouragement. He allowed me to have the journey in quantum information when I was just a beginner, and I am grateful for his clear, ideaful, and well-motivated expla- nations. I thank Ignacio Cirac who allowed me to join Barcelona quantum information. Also, thanks for the opportunity to visit MPQ. I thank Jos´e Ignacio Latorre for his kind support as my university tutor. I would also like to thank collaborators, Maciej Lewenstein, Anna San- pera, Emili Bagan, Ramon Mu˜noz-Tapia, Maria Baig, Lluis Masanes, Ujjwal Sen, Aditi Sen De, and Sofyan Iblisdir. I have enjoyed attending discussions; these helped me improve immensely. Many thanks go to Nicolas Gisin, So- fyan Iblisdir, Renato Renner, Andreas Buchleitner, Anna Sanpera, Maciej Lewenstein, and Ramon Mu˜noz-Tapia, for being on the thesis committee. I am very grateful for the people who assisted me while I was traveling as well. Many thanks go to Andreas Buchleitner and Florian Mintert duing my visit to the MPI-PKS; especially for their discussions and hospitality. Leong Chuan Kwek kindly invited me to Singapore and gave me opportunities to interact with people there. Many thanks to Leong Chuan Kwek, and the members of the group for discussions and hospitality: Dagomir Kaszlikowski, Berthold-Georg Englert, Artur Ekert, Tzu-Chieh Wei, Thomas Durt, Wee Kang Chua, Darwin Gosal. In 2006, many thanks to Sangchul Oh because he recommended me to pass a conference in my home country. Also, thanks to Jaewan Kim for inviting me, and also to Hyunchul Nha and Jaewon Lee for their hospitality and discussions. Many thanks go to Yoonho Kim for inviting me to his group and for all his hospitality. Muchas gracias a mi grupo: Miguel Navascu´es,Artur Garcia, Mafalda Almeida, Daniel Cavalcanti, y tambi´enAlessandro Ferraro y Stefano Piro- nio. I am very grateful to have the opportunity to interact with them, and appreciate all their kind help. I would like to thank some people for the discussions: Toby Cubitt, Nor- bert Lutkenhaus, Barbara Kraus, and Joseph Renes. I would like to thank Giacomo Mauro D’Ariano for selecting my proposed open problem in the arrangement at MPI-PKS, and many thanks go to Reinhard Werner for sug- 7 8 Please, neither Chinese nor Japanese, this is Korean(Core`a,Coreano, Cor´een,Koreanisch,...), meaning acknowledgement. Thanks! gesting it be included in the open problem list on his website. Thanks to Marco Piani, Rafal Demkowicz-Dobrzanski, Leandro Aolita, Giulio Chiri- bella, and Oriol Romero-Isart for the discussions as well. I would like to thank the former and the present Icfonians: Lluis Torner, who founded the institute; the secretaries, who kindly take care of my things; and my Ph.D friends. In particular I would like to express a deep gratitude to Yuan Liang Lim, friend in faith, for sharing his soul, and also for kindly explaining his work. I would like to thank Younghun Kwon for his kind advice and support each time I returned to Korea. I would like to thank people in my church: pastors (the Jang and the Yun families), the Ko family, the Jeon family, and the Cho family, for their loving care and help during my stay in Barcelona. Thanks to Charis and Ashley for correcting my English. Last but not least, my deep gratitude goes to Seeun Kim and my family for their heartful support, encouragement, and love; nothing can even com- pare to it. And to God, who loves me and so kindly guides me; he is always my strength and my foundation. I can do everything through him, who gives me strength. (NIV) Abstract Quantum cryptography is one of the most important quantum information applications. The present thesis covers several topics on quantum cryptog- raphy, such as the security analysis of quantum channels for key distribution protocols and the study of quantum cloning. First, we introduce a general formalism to characterize the cryptographic properties of quantum channels in the realistic scenario where the two hon- est parties employ prepare and measure protocols and the known two-way communication reconciliation techniques. We derive a necessary and suffi- cient condition to distill a secret key using this type of schemes for arbitrary bipartite quantum systems of finite dimension. The obtained results suggest that there may exist weakly entangling channels useless for key distribution using prepare and measure schemes. Next, we consider Gaussian states and Gaussian operations for crypto- graphic tasks and derive a new security condition. As it happens for quan- tum systems of finite dimension, our results suggest that there may also exist weakly entangled Gaussian states useless for key distribution, using Gaussian operations. Finally, we study the connection between cloning and state estimation. It was a long-standing problem to show whether state estimation becomes equivalent to quantum cloning in the asymptotic limit of an infinite number of clones. The equivalence is proven here using two known results in quan- tum information theory, the monogamy of quantum states and the properties of entanglement-breaking channels. 11 Thesis Publications The thesis is based on the following publications: ² J. Bae and A. Ac´ın, Key distillation from quantum channels using two- way communication protocols, Physical Review A 75 012334 (2007) ² J. Bae and A. Ac´ın, Asymptotic quantum cloning is state estimation, Physical Review Letters 97 030402 (2006) ² A. Ac´ın,J. Bae, E. Bagan, M. Baig, Ll. Masanes and R. Mu˜noz-Tapia, Secrecy properties of quantum channels, Physical Review A 73 012327 (2006) ² M. Navascu´es,J. Bae, J. I. Cirac, M. Lewestein, A. Sanpera and A. Ac´ın, Key distillation from Gaussian states by Gaussian operations, Physical Review Letters 94 010502 (2005) 13 Resumen La teor´ıacu´antica describe la f´ısicade sistemas microsc´opicos.Describiendo la f´ısicade la peque˜naescala, explota un formalismo diferente al utilizado por la f´ısicacl´asica,ya que las cantidades observables en teor´ıacu´antica no conmutan. Esto conduce a que la realidad f´ısicaes no determinista, y deriva en resultados antiintuitivos. Una de las caracter´ısticaspeculiares de la teor´ıa cu´antica es la correlaci´onpresente en sistemas cu´anticos, el entanglement. Desde hace una d´ecada,la teor´ıade la informaci´oncu´antica ha empezado a reconocer la importancia de considerar el punto de vista inform´atico- te´oricoen la teor´ıa cu´antica. El mejor ejemplo que ilustra la teor´ıa de la informaci´oncu´antica puede ser la teleportaci´oncu´antica, un protocolo para transmitir un estado cu´antico a una distancia arbitraria, y el algoritmo de factorizaci´oncu´antico, que soluciona el problema de la factorizaci´on 1 en tiempo polin´omico.De hecho, resulta que el entanglement permite a la teor´ıade la informaci´oncu´antica superar al correspondiente cl´asico.Muchas investigaciones importantes en teor´ıade la informaci´oncu´antica se realizan te´oricay experimentalmente, y afectan hoy en d´ıaotras ´areasde investi- gaci´onen f´ısica. Uno de los principales usos, asi como logros, de la teor´ıade la infor- maci´oncu´antica es la criptograf´ıacu´antica, i.e. la distribuci´oncu´antica de claves, que distribuye una clave secreta mediante estados cu´anticos y proce- sado cl´asico,ambas tecnolog´ıasactualmente factibles. Recientemente, ´esta tecnolog´ıaha abierto un nuevo mercado en la criptograf´ıa. La actual tesis sigue la l´ıneadel desarrollo reciente en teor´ıade la informaci´oncu´antica, estudiando el papel del entanglement en escenarios criptogr´aficoscu´anticos: nuevo an´alisisde seguridad de los canales cu´anticos para los escenarios crip- togr´aficos,posibilidad de aplicar estados gausianos en los escenarios de dis- tribucion de claves, y la prueba de equivalencia entre la clonaci´on asint´otica cu´antica y la estimaci´on del estado. 1Cl´asicamente el problema se considera muy duro, en el sentido que puede no ser generalmente soluble en tiempo polin´omico. 15 16 0.1 Introducci´on Esta tesis cubre diversos aspectos de la criptograf´ıacu´antica, principalmente nuevas condiciones de seguridad y el estudio de la clonacion cu´antica. La tesis demuestra condiciones de seguridad que indican si una clave secreta se puede destilar de un canal cu´antico dado, que puede ser un canal de un qubit, un canal de un qudit, o un canal gausiano. Esta condici´onprueba la seguridad de protocolos ya conocidos tales como el protocolo BB84 [8]. La estimaci´ony la clonaci´onde estados cu´anticos aparece naturalmente en el contexto de un escenario criptogr´aficocon presencia de un esp´ıa, y se ha conjeturado la equivalencia entre la clonaci´onasint´oticacu´antica y la estimaci´on. La tesis prueba esta conjetura realizada hace muchos a˜nos, demostrando que son de hecho equivalentes. 0.2 Motivaci´on 0.2.1 Key distillation de canales cu´anticos Los m´etodos criptogr´aficosexistentes que utilizan recursos cl´asicosbasan su seguridad en asunciones t´ecnicasrelativas al espia, llamado a menudo Eve, capacidades tales como potencia de c´alculofinita o memoria limitada [70].