On the quantum discord of general X states

Quantum discord Q is a function of density matrix elements. The domain of such a function in the case of two-qubit system with X density matrix may consist of three subdomains at most: two ones where the quantum discord is expressed in closed analytical forms ($$Q_{\pi /2}$$Q?/2 and $$Q_0$$Q0) and an intermediate subdomain for which, to extract the quantum discord $$Q_{\theta }$$Q?, it is required to solve numerically a one-dimensional minimization problem to find the optimal measurement angle $$\theta \in (0,\pi /2)$$??(0,?/2). Hence, the quantum discord is given by a piecewise analytical---numerical formula $$Q=\min \{Q_{\pi /2},Q_{\theta },Q_0\}$$Q=min{Q?/2,Q?,Q0}. It is shown that the boundaries between the subdomains consist of bifurcation points. The $$Q_{\theta }$$Q? subdomains are discovered in the dynamical phase flip channel model, in the anisotropic spin systems at thermal equilibrium, and in the heteronuclear dimers in an external magnetic field. We found that the transitions between $$Q_{\theta }$$Q? subdomain and $$Q_{\pi /2}$$Q?/2 and $$Q_0$$Q0 ones occur suddenly, but continuously and smoothly, i.e., nonanalyticity is hidden and can be observed in higher order derivatives of discord function.