Gacs Algorithmic Complexity on Hilbert Spaces and Some of its Applications

We extend the notion of Gacs quantum algorithmic entropy, originally formulated for finitely many qubits, to infinite dimensional quantum spin chains and investigate the relation of this extension with two quantum dynamical entropies that have been proposed in recent years. Further, we prove an extension of Brudno’s theorem in quantum spin chains with shift dynamics. Acknowledgements I would like to express my special appreciation and thanks to my advisors Professor Ahmad Shafiei Deh Abad and Professor Fabio Benatti, you have been a tremendous mentor for me. I would like to thank you for encouraging my research and for allowing me to grow as a research scientist. Your advice on both research as well as on my career have been priceless. I would also like to thank my committee members, professor Vahid Karimipour, Dr. Ali Rezakhani, Dr. Majid Alizade and Dr. Mohammad Bager Asadi for serving as my committee members even at hardship. I also want to thank you for letting my defense be an enjoyable moment, and for your brilliant comments and suggestions, thanks to you. A special thanks to my family. Words cannot express how grateful I am to my mother for all of the sacrifices that youve made on my behalf. Your prayer for me was what sustained me thus far. I would like also express appreciation to my beloved wife Elham who spent sleepless nights with and was always my support in the moments when there was no one to answer my queries. At the end, I would like the support this thesis by the STEP programme of the Abdus Salam ICTP of Trieste.