Local witness for bipartite quantum discord

Recently, we proposed a method for the local detection of quantum correlations on the basis of local measurements and state tomography at different instances in time [Phys. Rev. Lett. 107, 180402 (2011)]. The method allows for the detection of quantum discord in bipartite systems when access is restricted to only one of the subsystems. Here, we elaborate the details of this method and provide applications to specific physical models. In particular, we discuss the performance of the scheme for generic complex systems by investigating thermal equilibrium states corresponding to randomly generated Hamiltonians. Moreover, we formulate an ergodicity-type hypothesis which links the time average to the analytically obtained average over the group of unitary operators equipped with the Haar measure.

Figure 1

The plot shows the dependence of the unitary average value $\mu$ and the variance $s^2$ on the parameter $z$ for ${\rho }_z$. The relative error is constant at $s/\mu \approx 0.58$.

Figure 2

Comparison of the unitary average with the actual time evolution for the Gibbs states of six randomly picked two-qubit Hamiltonians ($d_A=d_B=2$) at fixed temperature $\beta =1$. The pictures show the value of the Hilbert-Schmidt distance after applying the local detection method to the Gibbs state.

Figure 3

Comparison of the unitary average with the actual time evolution for the Gibbs states of four randomly picked Hamiltonians for a qubit coupled to environments with different dimensions at fixed temperature $\beta =1$.

Figure 4

Dependence of the discord of a randomly picked fixed thermal state on the temperature for $d_B=8$. For higher temperatures (lower $\beta$), discord diminishes. The picture on the lower right shows the average value and one standard deviation as a function of the inverse temperature $\beta$.