Maps on quantum states preserving Bregman and Jensen divergences

2017. 03. 22. 16:00
Virosztek Dániel (BME Matematika Intézet)

Given a differentiable convex function $c$ on a convex subset $S$ of a real normed space, a natural quantity measuring the dissimilarity of the elements of $S$ appears. The Bregman divergence of $x$ and $y$ is the gap between $c(x)$ and $T_{c,y}(x)$, where $T_{c,y}$ denotes the first-order Taylor polynomial of $c$ with basepoint $y$. This notion of divergence is rather general. The Kullback-Leibler divergence, the Umegaki relative entropy, Stein’s loss, the logistic loss, and the squared Euclidean distance are just particular examples of Bregman divergences. We consider a finite-level quantum system and determine the structure of the transformations on its state space preserving the Bregman divergence generated by an arbitrary strictly convex differentiable function. We also derive a similar result concerning Jensen
divergences, which are in an intimate connection with Bregman divergences. The most direct motivation of our work is the paper of Molnár [L. Molnár, J. Math. Phys. 49, 032114 (2008)] which describes the preservers of the Umegaki relative entropy on the state space.

The talk is based on the paper [D. Virosztek, Lett. Math. Phys. 106, 1217–1234 (2016)].