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 c o n d as a function of measurement angle θ ∈ [ 0 , π / 2 ] . Numerical calculations show that the function S c o n d ( θ ) for X states can have at most one local extremum in the open interval from zero to π / 2 (unimodality property). If the extremum is a minimum, the quantum discord displays region with variable (state-dependent) optimal measurement angle θ ∗ . Such θ -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 , π / 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.