# Application of geometric methods to quantum control

## Project manager

Zbigniew Puchała (IITiS PAN)

## Participants

- Piotr Gawron (IITiS PAN)
- Jarosław Adam Miszczak (IITiS PAN)

## Project meetings

## Publications

- [1] Z. Puchała, J.A. Miszczak, "Probability measure generated by the superfidelity",
*J. Phys. A: Math. Theor.*, Vol. 44 (2011): 405301. arXiv:1107.2792.*We study the probability measure on the space of density matrices induced by the metric defined by using superfidelity. We give the formula for the probability density of eigenvalues. We also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure.* - [2] C.F. Dunkl, P. Gawron, J.A. Holbrook, J.A. Miszczak, Z. Puchała, K. Życzkowski, "Numerical shadow and geometry of quantum states",
*J. Phys. A: Math. Theor.*, Vol. 44 (2011): 335301. arXiv:1104.2760.*The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.* - [3] B. Gardas, Z. Puchała, "Stationary states of two-level open quantum systems",
*J. Phys. A: Math. Theor.*, Vol. 44 (2011): 215306. arXiv:1006.3328.*A problem of finding stationary states of open quantum systems is addressed. We focus our attention on a generic type of open system: a qubit coupled to its environment. We apply the theory of block operator matrices and find stationary states of two-level open quantum systems under certain conditions applied on both the qubit and the surrounding.*

## Project information

- Project funded by Ministry of Science and Higher Education
- Number: IP 2010 0334 70
- Dates: 23.12.2010 - 31.12.2011