Quantum Bilinear Optimization

A bilinear formulation for vector sparsity optimization

Sparsity plays an important role in many fields of engineering. The cardinality penalty function, often used as a measure of sparsity, is neither continuous nor differentiable and therefore smooth optimization algorithms cannot be applied directly. In this paper we present a continuous yet non-differentiable sparsity function which constitutes a tight lower bound on the cardinality function. Th...

Periodic excitations of a bilinear quantum system

A well known method to transfer the population of a quantum system from an eigenspace of the free Hamiltonian to another is to use a periodic control law with angular frequency equal to the difference of the eigenvalues. This paper gives a theoretical proof of this experimental result. We introduce a notion of efficiency and demonstrate its interest for the design of controls on the example of ...

Singular Bilinear Integrals in Quantum Physics

Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular vari...

Bilinear Summation Formulas from Quantum Algebra Representations

The tensor product of a positive and a negative discrete series representation of the quantum algebra Uq ( su(1, 1) ) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of li...

On the controllability of bilinear quantum systems