Absence of local maxima for optimal control of two-level quantum systems

The goal of optimal control for a quantum system whose evolution is governed by Schrodinger equation is to find controls which maximize target objective functional, such as quantum average of some observable. Often numerical methods are used to find optimal controls. This makes important the problem of analysis of the existence or absence of local maxima (traps) of the target functional, since their presence may hinder the numerical search from finding true global maxima. Significant progress in the analysis of trap was made in recent works by H. Rabitz, A.N. Pechen, D.J. Tannor, R. Wu, C. Brif, P. de Fouquieres, S.G. Schirmer and others[1,2,3]. However, systems without traps were not known. In this talk we will present the proof of the absence of traps for a wide range of target functionals for two-level quantum systems governed by Schrodinger equation. The talk is based on the joint work with A.N. Pechen[4,5].

References

1. A. Pechen, C. Brif, R. Wu, R. Chakrabarti, H. Rabitz, “General unifying features of controlled quantum phenomena”, Phys. Rev. A, 82 (2010), 030101(R)      
2. A. Pechen, D. J. Tannor, “Are there traps in quantum control landscapes”, Phys. Rev. Lett., 106 (2011), 120402        
3. P. de Fouquieres, S.G. Schirmer, “A closer look at quantum control landscapes and their implication for control optimization”, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 16:3 (2013), 1350021          
4. A. Pechen, N. Il'in, “Trap-free manipulation in the Landau-Zener system”, Phys. Rev. A, 86 (2012), 052117        
5. A.N. Pechen, N.B. Il'in, “Coherent control of a qubit is trap free”, Proc. Steklov Inst. Math., 2014 (to appear)