Experimentally Witnessing the Quantumness of Correlations

The quantification of quantum correlations (other than entanglement) usually entails labored numerical optimization procedures also demanding quantum state tomographic methods. Thus it is interesting to have a laboratory friendly witness for the nature of correlations. In this Letter we report a direct experimental implementation of such a witness in a room temperature nuclear magnetic resonance system. In our experiment the nature of correlations is revealed by performing only few local magnetization measurements. We also compared the witness results with those for the symmetric quantum discord and we obtained a fairly good agreement.

Figure 1

(a) Schematic representation of the operation sequence used to witness the nonclassical nature of correlations. (b) Equivalent pulse sequence employed in our experiment. The thicker filled bars represent $\pi /2$ pulses, the thinner bars indicate $\pi /4$ pulses, and the grey bars indicate $\pi /6$ pulses with the phases as shown (negative pulse phases are described by a bar over the symbol). The pulses represented as unfilled dashed bars are modified to achieve the different rotations necessary for the witness protocol. The dashed gradient pulse is applied to obtain the classically correlated Bell-diagonal state. The time periods $\frac{1}{2J}$, $\frac{3}{2J}$, and $\frac{1}{4J}$ represent free evolutions under the $J$ coupling [18].

Figure 2

(a) The ${}^1\mathrm{H}$ spectra (normalized by the thermal equilibrium state spectrum) obtained after the witness circuit execution (with rotations $R_{n_1}$, $R_{n_2}$, and $R_{n_3}$), (b) witness expectation value, (c) quantum discord (light blue columns) and classical correlation (dark blue columns) measured in three different initial states, ${\rho }_{\mathrm{QC}}$ quantum correlated, ${\rho }_{CC}$ classically correlated, and ${\rho }_T$ thermal equilibrium state. The dashed line represents the experimental error bound for determination of classically correlated (zero discord) states. The witness was measured directly performing the circuit depicted in Fig. 1a, while the classical correlation and the symmetric quantum discord was computed after full QST and numerical extremization procedures. The correlations are displayed in units of (${\epsilon }^2/\ln 2$)bit [18].

Figure 3

Real (left) and imaginary (right) parts of the deviation matrix elements reconstructed by QST for the two initial prepared Bell-diagonal states: (a) ${\rho }_{\mathrm{QC}}$ quantum correlated (equivalent to ${\mathcal{C}}_1=2\epsilon$, ${\mathcal{C}}_2=2\epsilon$, and ${\mathcal{C}}_3=-2\epsilon$); (b) ${\rho }_{CC}$ classically correlated (equivalent to ${\mathcal{C}}_1=0$, ${\mathcal{C}}_2=0$, and ${\mathcal{C}}_3=-4\epsilon$); and also for (c) ${\rho }_T$ the thermal equilibrium state. The deviation matrix elements are displayed in the usual computational basis, where $|0⟩$ and $|1⟩$ represent the eigenstates of ${\sigma }_z$ for each qubit. The accuracy of prepared initial states can be estimated by the normalized trace distance from the ideal ones, $\delta ({\rho }_{\mathrm{ideal}},{\rho }_{\mathrm{prep}.})/\epsilon =\mathrm{tr}|\Delta {\rho }_{\mathrm{ideal}}-\Delta {\rho }_{\mathrm{prep}.}|/2\approx 0.1$ (for both ${\rho }_{\mathrm{QC}}$ and ${\rho }_{CC}$).

Figure 4

Witness and correlations decoherence dynamics. The panel displays the measured witness and computed correlations for ${\rho }_{\mathrm{QC}}$ [Fig. 3a] relaxed during a time interval, $t_n=n\delta t$ ($\delta t=55.7\mathrm{ms}$, $n=0,2,\dots ,11$), before performing the witness measurement protocol. The red tick bars represent the witness expectation value (scale on the right), the dark grey section represents the amount of classical correlation, the light grey section represents the symmetric quantum discord, the entire grey bars (light and dark sections summed) display the quantum mutual information (scale on the left). The classicality bound is represented by the blue dotted line. The inset image shows the real part of the deviation matrix elements reconstructed by QST for an intermediate classically correlated state. The effective transversal relaxation times are $T_2^{*}=0.31\mathrm{s}$ and $T_2^{*}=0.12\mathrm{s}$, for ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ nuclei, respectively. The correlations are displayed in units of (${\epsilon }^2/\ln 2$)bit.