Resumen de Quantum control at the boundary

Ángel Aitor Balmaseda Martín

• The main goal of this dissertation is to present and prove the viability of a non-standard method for controlling the state of a quantum system, whose dynamic is governed by the Schrödinger equation, by modifying its boundary conditions instead of relying on the action of external fields to drive the state of the system.

  Control of the states of a quantum system is becoming more and more significative because of the outstanding experimental achievements that are taking place as the race to reach effective quantum information technologies picks up.

  The standard approach to quantum control bases on the use of an external field to manipulate the system. From a technological point of view, some difficulties appear when controlling a quantum system in this way, related with the complications of manipulating a system made of few particles while maintaining the quantum correlations. As a consequence the quantum systems need to be kept at very low temperatures and the interactions with them have to be performed very fast, which is inconvenient for applications.

  The Quantum Control at the Boundary scheme takes an approach which is radically different to the standard one. Instead of seeking the control of the quantum system by directly interacting with it through an external field, the control is achieved by manipulating the boundary conditions of the system. The spectrum of a quantum system, for instance an electron moving in a box, depends on the boundary conditions imposed on it (usually Dirichlet or Neumann boundary conditions). Hence, a modification of such boundary conditions modifies the state of the system allowing for its manipulation and, eventually, its control. This kind of interaction is weaker, which makes one to expect that it may help maintaining the quantum correlations.

  The Quantum Control at the Boundary paradigm has been used to show how to generate entangled states in composite systems by modifications of the boundary conditions (Ibort, Marmo, and Pérez-Pardo, 2014). The relation of the paradigm and topology change has been explored in (Pérez-Pardo, Barbero-Liñán, and Ibort, 2015) and recently used to describe the physical properties of systems with moving walls (Garnero, 2018). But, in spite of its intrinsic interest, some basic issues such as the controllability of simple systems within this scheme had never been addressed.

  In order to show the viability of the Quantum Control at the Boundary method, a family of boundary control systems on Quantum Circuits (a generalization of quantum grahs) is introduced. Before being able to address the problem of controllability, the problem of existence of solutions for the Schrödinger equation with time-dependent boundary conditions is addressed.

  The approximate controllability of the systems under study is proven basing on a controllability result by T. Chambrion et al. (2009) and a stability result which constitutes another original contribution of this dissertation.