Analysis and Design of Complex-Valued Linear Systems

This paper studies a class of complex-valued linear systems whose state evolution dependents on both the state vector and its conjugate. The complex-valued linear system comes from linear dynamical quantum control theory and is also encountered when a normal linear system is controlled by feedback containing both the state vector and its conjugate that can provide more design freedom. By introducing the concept of bimatrix and its properties, the considered system is transformed into an equivalent real-representation system and a non-equivalent complex-lifting system, which are normal linear systems. Some analysis and design problems including solutions, controllability, observability, stability, eigenvalue assignment, stabilization, linear quadratic regulation (LQR), and state observer design are then investigated. Criterion, conditions, and algorithms are provided in terms of the coefficient bimatrices of the original system. The developed approaches are also utilized to investigate the so-called antilinear system which is a special case of the considered complex-valued linear system. The existing results on this system have been improved and some new results are established.Comment: 19 page

Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $y_0 \in H$ we associate the minimal "energy" needed to transfer $y_0$ to $0$ in a time $T \ge T_0$ ("energy" of a control being the square of its $L^2$ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $0$ for $T\to+\infty$. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.Comment: In this version we have also added a section on examples and applications of our main results. This version is similar to the one which will be published on "SIAM Journal on Control and Optimization" (SIAM

The Damped String Problem Revisited

We revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it, employing methods familiar from supersymmetric quantum mechanics. We also derive completeness and Riesz basis results (with parentheses) for the associated root functions under less smoothness assumptions on the coefficients than usual, using operator theoretic methods (rather than detailed eigenvalue and root function asymptotics) only.Comment: 39 page

Embedding quantum simulators

79 p.Quantum simulations consist in the reproduction of the dynamics of a quantum systemon a controllable platform, with the goal of capturing an interesting feature of the considered model. It is broadly believed that the advent of quantum simulators will represent a technological revolution, as they promise to solve several problems whichare considered intractable in a classical computer. Although there are strong theoretical bases confirming this claim, several aspects of quantum simulators have still to bestudied, in order to faith fully prove their feasibility. More over, the general question on which features of the considered models are simulatable is an attractive research topic,whose study would help to define the limits of a quantum simulator.In this Thesis, we develop several algorithms, which are able to catch relevant properties of the simulated quantum model. The proposed protocols follow a new concept named embedding quantum simulator, in which the simulated Schrodinger equation is mapped onto an enlarged Hilbert space in a nontrivial way. Via this embedding, weare able to retrieve, by measuring few observables, quantities that generally require full tomography in order to be evaluated. Moreover, we pay a special attention to the experimental feasibility, defining mappings which are space efficient, and do not require the implementation of challenging Hamiltonians. The presented algorithms are general,and they may be implemented in several quantum platforms, e.g. photonics, trappedions, circuit QED, among others.First, we propose a protocol which simulate the dynamics of an embedded Hamiltonian,allowing for the efficient extraction of a class of entanglement monotones. This is done using an embedding that is able to implement unphysical operations, as is the case of complex conjugation. The analysis is accompanied with a study of feasibility in a trapped-ion setup, which can be generalised to other platforms following similar computational models. Second, we propose an algorithm to measure n-time correlation functions of spinorial, fermionic, and bosonic operators, by considerably improving previous versions of the same result. We apply this protocol to the computation of magnetic susceptibilities, as well as to the simulation of Markovian and non-Markovian dissipative processes in a novel way, without the necessity of engineering any bath. All the proposed protocols are designed with a single ancillary qubit, minimising the needed experimental resources.We believe that embedding quantum simulators have a potential to become a powerful tool in the quantum simulation theory, since they pave the way for improving the flexibility of a quantum simulator in diferent experimental contexts

Quantitative Unique Continuation and Applications

PhD Thesi

Velocity averaging for diffusive transport equations with discontinuous flux

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. As a corollary, we show the existence of a solution for the Cauchy problem for nonlinear degenerate parabolic equation with discontinuous flux. In order to achieve the results, we introduce a new variant of micro-local defect functionals which are able to "recognise" changes of the type of the equation