Quantum discord for general X and CS states: A piecewise-analytic-numerical formula

Quantum discord is a function of density-matrix elements (and through them, e.~g., of temperature, applied fields, time, and so forth). The domain of such a function in the case of two-qubit system with X or centrosymmetric (CS) density matrix can consist at most of three subdomains: two ones, where the quantum discord is expressed in closed analytical forms (Q_0 and Q_{\pi/2}), and an intermediate subdomain in which for determining the quantum discord Q_\theta it is required to solve numerically a one-dimensional minimization problem to find the optimal measurement angle \theta\in(0,\pi/2). Exact equations for determining the boundaries between these subdomains are obtained and solved for a number of models. The Q_\theta subdomains are discovered in the anisotropic spin dimers in external field. On the other hand, coinciding boundaries and therefore sudden transitions between optimal measurement angles \theta=\pi/2 and \theta=0 are observed in dynamics of spin currying particles in closed nanopore and also in phase flip channels. In latter cases the solutions are entirely analytical.