Extremal properties of conditional entropy and quantum discord for XXZ symmetric quantum states

For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy S_{cond} as a function of measurement angle \theta\in[0,\pi/2]. Numerical calculations show that the function S_{cond}(\theta) for X states can have at most one local extremum in the open interval from zero to \pi/2 (unimodality property). If the extremum is a minimum the quantum discord displays region with variable (state-dependent) optimal measurement angle \theta^*. Such \theta-regions (phases, fractions) are very tiny in the space of X state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval (0,\pi/2). It is remarkable that the maxima exist in surprisingly wide regions and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum. Moreover, the found maxima can exceed the conditional entropy values at the ends of interval [0,\pi/2] more than by 1%. This instils hope in the possibility to detect such maxima in experiment.