Geometric quantum discord of an arbitrary two-qudit state: Exact value and general upper bounds

The geometric quantum discord of a two-qudit state has been studied in many papers; however, its exact analytical value in the explicit form is known only for a general two-qubit state, a general qubit-qudit state, and some special families of two-qudit states. Based on the general Bloch vectors formalism [E. R. Loubenets et al., J. Phys. A: Math. Theor. 54, 195301 (2021)], we find the explicit exact analytical value of the geometric quantum discord for an arbitrary two-qudit state of any dimension via the parameters of its correlation matrix and the Bloch vectors of its reduced states. This general analytical result includes all the known exact results on the geometric quantum discord only as particular cases and proves rigorously that the lower bound on the geometric discord presented in [S. Rana et al., Phys. Rev. A 85, 024102 (2012)] constitutes its exact value for each two-qudit state. Moreover, our general result allows us to find for an arbitrary two-qudit state, pure or mixed, the upper and lower bounds on its geometric quantum discord, expressed via the Hilbert space characteristics of this state.