Experimental detection of nonclassical correlations in mixed-state quantum computation

We report on an experiment to detect nonclassical correlations in a highly mixed state. The correlations are characterized by the quantum discord and are observed using four qubits in a liquid-state nuclear magnetic resonance quantum information processor. The state analyzed is the output of a DQC1 computation, whose input is a single quantum bit accompanied by $n$ maximally mixed qubits. This model of computation outperforms the best known classical algorithms and, although it contains vanishing entanglement, it is known to have quantum correlations characterized by the quantum discord. This experiment detects nonvanishing quantum discord, ensuring the existence of nonclassical correlations as measured by the quantum discord.

Figure 1

DQC1 circuit where the top qubit has a bias of $\varepsilon$ toward the ground state. Measurements of $\langle {\sigma }_x\rangle$ and $\langle {\sigma }_y\rangle$ yield the real and imaginary parts of $\varepsilon \text{Tr}\left(U_n\right)/2^n$, respectively. In the initial state, the identity terms are recognized as properly normalized states such that $\text{tr}\left(\rho \right)=1$.

Figure 2

Molecule trans-crotonic acid and a table with the parameters of the Hamiltonian given in Hz. The shaded diagonal elements represent the chemical shifts ${\omega }_i$ with the Hamiltonian ${\Sigma }_i\pi {\omega }_i{\sigma }_z^i$. The remaining elements indicate the scalar coupling constants $J_{ij}$ with the Hamiltonian ${\Sigma }_{i<j}\pi J_{ij}{\sigma }_z^i{\sigma }_z^j$.

Figure 3

Distributions of the singular values (and their cumulative distributions) computed for the experimentally determined correlation matrices of the (a) initial and (b) final states of a DQC1 algorithm implemented in NMR. These distributions are created by sampling from a normal distribution of the errors on each matrix element and calculating the singular values of the sampled matrix. There are $10000$ samples in each plot and the histogram bin size is $0.005$. The cumulative of each distribution is included to guide the reader in estimating the integral of portions of the distributions. We deduce that the correlation matrix of the initial state has a rank of 1 and the correlation matrix of the final state has a rank of at least 3, which is to say that we did not detect quantum discord in the initial state, but did witness the presence of quantum discord in the final state of the DQC1 computation. Quantum discord is witnessed in ${\rho }_{\text{pps}}$, which implies that the physical state ${\rho }_{\text{NMR}}=(1-\alpha )I/2^N+\alpha {\rho }_{\text{pps}}$ also has nonzero discord, regardless of the polarization $\alpha$.