Measuring Bipartite Quantum Correlations of an Unknown State

We report the experimental measurement of bipartite quantum correlations of an unknown two-qubit state. Using a liquid state Nuclear Magnetic Resonance setup and employing geometric discord, we evaluate the quantum correlations of a state without resorting to prior knowledge of its density matrix. The method is applicable to any $2\otimes d$ system and provides, in terms of number of measurements required, an advantage over full state tomography scaling with the dimension $d$ of the unmeasured subsystem. The negativity of quantumness is measured as well for reference. We also observe the phenomenon of sudden transition of quantum correlations when local phase and amplitude damping channels are applied to the state.

Figure 1

Block diagrams for the deviation density matrix related to the Bell diagonal initial states. For ${\rho }_1$ in (a) we have $|c_1^{(1)}|=|c_2^{(1)}|=|c_3^{(1)}|=0.2$, while ${\rho }_2$ in (b) reads $|c_1^{(2)}|=0.5$, $|c_2^{(2)}|=0.06$, and $|c_3^{(2)}|=0.24$.

Figure 2

Time evolution of $Q$ and $D_G$ for the state ${\rho }_1$ in (a) and (b), and ${\rho }_2$ in (c) and (d). The green dots represent the direct measurement and the black open squares corresponds to tomography results. The red lines depict the theoretical predictions. The relaxation times retrieved from experimental data are: $T_1=3.57\mathrm{s}$ and $T_2=1.2\mathrm{s}$ for the hydrogen, and $T_1=10\mathrm{s}$ and $T_2=0.19\mathrm{s}$ for the carbon. Quantum correlations are displayed in units of ${\epsilon }^2$.

Figure 3

Time evolution of $|c_1|$ (upward triangle), $|c_2|$ (downward triangle), and $|c_3|$ (hexagon) for state ${\rho }_1$ in (a), and ${\rho }_2$ in (c). The open symbols represent the tomography results and the filled symbols represent the direct measurement. Note that for each $|c_i|(i=1,2,3)$ both forms of measurement are superimposed. In (b), for the initial state ${\rho }_1$, and (d), for the initial state ${\rho }_2$, the green dots represent the direct measurement and the black squares represent the tomography results. The red lines are the theoretical predictions for $Q_N^A$. Quantum correlations are displayed in units of $\epsilon$.