A symmetric geometric measure and the dynamics of quantum discord

Abstract

A symmetric measure of quantum correlation based on the Hilbert—Schmidt distance is presented in this paper. For two-qubit states, we considerably simplify the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering from dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There exists a certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.

Analytically Computable Symmetric Quantum Correlations

“…To demonstrate the validity of our Theorem 3, we consider a state ρ M generated in the same way as that shown in Equation (51) …”

“…[12][13][14][15][16][17][18][19][20][21][22][23] In addition, as a quantum correlation that extends beyond entanglement, the quantum discord (QD), [24,25] which quantifies the discrepancy between the DOI: 10.1002/andp.201800178 quantum versions of two classically equivalent pieces of mutual information, can only be analytically calculated for a general state in 2 ⊗ n dimensions. To overcome this shortcoming, symmetric versions of the QD and the MIN are defined, that is, the symmetric quantum discord (SQD) [50][51][52] and the symmetric measurement-induced nonlocality (SMIN). [40] Recently, some progress has been made in regard to analytical expressions for certain classes of states or effective numerical methods.…”

“…[42][43][44][45][46][47][48][49] However, neither the QD nor the MIN of a bipartite system is symmetric in the sense that different results will be obtained (especially for some zero-discord states) if the two subsystems are swapped, which implies that the QD and MIN cannot completely quantify all of the quantum correlations present in a state. [51] However, the geometric SQD seems not to be a good measure because forming a composite state by taking the product with another mixed state will reduce the SQD of the state of interest. In particular, with the development of the resource theory of quantum coherence, [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70] it was shown that the coherence of a system could be converted through incoherent operations into the SQD, instead of the asymmetric QD, of a composite system.…”

Geometric global quantum discord of two-qubit states

“…In recent years more relevant measures such as geometric quantum discord [15][16][17] (GQD) have been suggested. It takes use of different quantities and offers analytical solutions in some conditions generally [18][19][20][21]. However, in the original definitions both the quantum discord and the geometric quantum discord are not symmetric with respect to the subsystems.…”

Analytic symmetric information-theoretical discord for one class of two-qubit states

“…50 Physically, the symmetric discord quanti¯es the minimal lost information induced by the two-side projective measurements. However, just like the asymmetric quantum discord, the good understanding of the symmetric quantum discord is only restricted to some particular systems [51][52][53] owing to the lack of the exact and analytic expression for general states. In particular, the considered di±culties for the potential analytic results for the symmetric geometric discord are analyzed in Refs.…”