Testing Contextuality on Quantum Ensembles with One Clean Qubit

We present a protocol to evaluate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems, and use it to experimentally demonstrate—under an assumption of fair sampling—the violation of an inequality that is satisfied by any noncontextual hidden-variables theory. The experiment is performed on an ensemble of molecular nuclear spins in the solid state, using established nuclear magnetic resonance techniques for quantum-information processing.

Figure 1

A quantum network to encode the correlation between the outcomes of measurements $\{S_k{\}}_{k=1\dots m}$ on a $d$-dimensional system, in the phase of a probe qubit state. Repeating this procedure for the same preparation $\rho$ and averaging the outcome of the measurement on the probe qubit gives the ensemble average $⟨S_1{S}_2\cdots S_m{⟩}_{\rho }$. Alternatively, for an ensemble of quantum systems initially prepared according to $\rho$, on which operations are applied in parallel to the individual systems, an ensemble measurement readily produces $⟨S_1{S}_2\cdots S_m{⟩}_{\rho }$.

Figure 2

The quantum networks for the six experiments to estimate $\beta$ as given in (2). The ensemble is initially prepared according to ${\rho }_{ϵ}\otimes \frac{{\mathbb{1}}_2}{2}\otimes \frac{{\mathbb{1}}_2}{2}$, where ${\rho }_{ϵ}=(1-ϵ)\frac{{\mathbb{1}}_2}{2}+ϵ|+⟩⟨+|$, and $\frac{{\mathbb{1}}_2}{2}$ is the single-qubit maximally mixed state.

Figure 3

Malonic acid (${\mathrm{C}}_3{\mathrm{H}}_4{\mathrm{O}}_4$) molecule and Hamiltonian parameters (all values in kHz). Elements along the diagonal represent chemical shifts, ${\omega }_i$, with respect to the transmitter frequency (with the Hamiltonian $\sum _i\pi {\omega }_i{Z}_i$). Above the diagonal are dipolar coupling constants [$\sum _{i<j}\pi D_{i,j}(2Z_i{Z}_j-X_i{X}_j-Y_i{Y}_j)$], and below the diagonal are $J$ coupling constants [$\sum _{i<j}\frac{\pi }{2}{J}_{i,j}(Z_i{Z}_j+X_i{X}_j+Y_i{Y}_j)$]. An accurate natural Hamiltonian is necessary for high fidelity control and is obtained from precise spectral fitting of (also shown) a proton-decoupled ${}^{13}\mathrm{C}$ spectrum following polarization transfer from the abundant protons. The central peak in each quintuplet is due to natural abundance ${}^{13}\mathrm{C}$ nuclei present in the crystal at $\sim 1\%$. (For more details see 19, 20 and references therein.)

Figure 4

Summary of experimental results. Shown are (in dashed blue) the spectrum produced by the initial preparation procedure, ${\rho }_{ϵ}\otimes \frac{{\mathbb{1}}_2}{2}\otimes \frac{{\mathbb{1}}_2}{2}$, establishing a reference for $ϵ$, and (in solid red) the sum of the six spectra corresponding to the six terms in $\beta$ of inequality (2) with the appropriate signs, scaled by $\frac{1}{6}$ for ease of comparison with the reference.

Figure 5

Numerical simulation results of a simple model of decoherence (inset) showing the expected variation of $\beta$ as a function of the ratio of the pulse length, $t$, to an effective dephasing time, $T_2$. The dashed lines indicate bounds on the expected performance of the current experiment; for pulse length of 1.5 ms, and effective decoherence times of 2 ms ($\sim T_2^{*}$) and 30 ms ($\sim$ intrinsic coherence times [19]), the value of $\beta$ is expected to be 1.1 and 5.3, respectively.