This book introduces universal bounds to quantum unitary dynamics, with applications ranging from condensed matter models to quantum metrology and computation. Motivated by the observation that the dynamics of many-body systems can be better unraveled in the Heisenberg picture, we focus on the unitary evolution of quantum observables, a process known as operator growth and quantified by the Krylov complexity. By means of a generalized uncertainty relation, we constrain the complexity growth through a universal speed limit named the dispersion bound, investigating also its relation with quantum chaos. Furthermore, the book extends the framework of quantum speed limits (QSLs) to operator flows, identifying new fundamental timescales of physical processes. Crucially, the dynamics of operator complexity attains the QSL whenever the dispersion bound is saturated. Our results provide computable constraints on the linear response of many-body systems out of equilibrium and the quantum Fisher information governing the precision of quantum measurements.
Chapter 1.Introduction.
Chapter 2.Operator growth in Krylov space.
Chapter 3.Dispersion bound on Krylov complexity.
Chapter 4.A brief history of quantum speed limits in isolated systems.
Chapter 5.QSLs on operator flows.
Chapter
6. QSLs on correlation functions.
Chapter 7.A geometric operator quantum speed limit.
Chapter 8.Conclusions.
Nicoletta Carabba undertook her PhD studies at the University of Luxembourg in 2021-2024 under the supervision of Prof. Adolfo del Campo. She previously received bachelor's and master's degrees, both cum laude, at the University of Pisa. The subject of her master thesis was the study of U(1) axial condensates in the high temperature, chirally-restored phase of QCD, with the purpose of shedding light on a long-standing question: the fate of the U(1) axial symmetry above the chiral transition. The work, carried out under the supervision of Prof. Enrico Meggiolaro, resulted in a publication in Physical Review D. Motivated by a deep fascination with the quantum world, during her PhD, she investigated the nature of time in quantum mechanics aiming at assessing the fundamental timescales and the amount of complexity of the dynamics. Her research focused on the field of quantum speed limits (QSL) and operator growth. Nicoletta also performed numerical studies during her research visits at the Donostia International Physics Center and the Ecole Normale Supérieure in Paris. Her PhD thesis is based on three publications. The first, published in Communications Physics, establishes a universal constraint to the growth of operator complexity and was recognized with the best theoretical poster award at the Bristol Quantum Information Technology Workshop of 2022. Two other papers, published in Quantum, generalize the notion of QSL to the evolution of operators.
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