Abstract
Among various definitions of quantum correlations, quantum discord has attracted considerable attention. To find an analytical expression for quantum discord is an intractable task. Exact results are known only for very special states, namely two-qubit X-shaped states. We present in this paper a geometric viewpoint, from which two-qubit quantum discord can be described clearly. The known results on X state discord are restated in the directly perceivable geometric language. As a consequence, the dynamics of classical correlations and quantum discord for an X state in the presence of decoherence is endowed with geometric interpretation. More importantly, we extend the geometric method to the case of more general states, for which numerical as well as analytical results on quantum discord have not yet been obtained. Based on the support of numerical computations, some conjectures are proposed to help us establish the geometric picture. We find that the geometric picture for these states has an intimate relationship with that for X states. Thereby, in some cases, analytical expressions for classical correlations and quantum discord can be obtained.
Export citation and abstract BibTeX RIS
GENERAL SCIENTIFIC SUMMARY Introduction and background. The correlations between the parties of bipartite or multipartite quantum systems have attracted considerable attention. Various measures, which are measurement-oriented and different from entanglement measures, have been proposed to quantify quantum correlations. Among them, quantum discord (QD) has attracted extensive studies both in theory and experiments. An astonishing phenomenon, occurring in some computational models, is that the QD rather than the entanglement is responsible for the quantum speed-up. However, finding an analytical expression for QD is an intractable task. Exact results are known only for very special states. We propose a geometric picture to study the QD of two-qubit states. A concrete geometric object, named the 'quantum steering ellipsoid' (QSE), helps us interpret the dynamics of QD as well as simplify the calculations.
Main results. In the geometric picture, the optimal postmeasurement ensemble, which gives rise to the QD, is illustrated clearly in QSE: either horizontal or vertical decomposition. In some cases, the dynamics of QD (e.g. sudden transition between quantum and classical correlation) can even be 'seen' in QSE. Moreover, by numerical approaches and theoretical analysis, we give the exact results for the QD of another class of states. Finally, we make some conjectures on the QD of more general states.
Wider implications. This geometric picture for QD deserves further study. There may be a certain relationship between the concept of QSE and the quantum channel. Additionally, the conjectures given in the paper will be helpful in seeking analytical results for the QD of general states.