Probing the limits of correlations in an indivisible quantum system

We employ a trapped ion to study quantum contextual correlations in a single qutrit using the 5-observable Klyachko, Can, Binicioğlu, and Shumovsky inequality, which is arguably the most fundamental noncontextuality inequality for testing quantum mechanics (QM). We quantify the effect of systematics in our experiment by purposely scanning the degree of signaling between measurements, which allows us to place realistic bounds on the nonclassicality of the observed correlations. Our results violate the classical bound for this experiment by up to 25 standard deviations, while being in agreement with the QM limit. In order to test the prediction of QM that the contextual fraction increases with the number of observables, we gradually increase the complexity of our measurements from 5 up to 121 observables. We find stronger-than-classical correlations in all prepared scenarios up to 101 observables, beyond which experimental imperfections blur the quantum-classical divide.

Figure 1

Left: “Qutrit sphere” spanned by arbitrary superpositions of the type $\alpha \left|0\right.〉+\beta \left|1\right.〉+\gamma \left|2\right.〉$, where $\alpha ,\beta ,\gamma \in \mathbb{R}$ and such that ${\alpha }^2+{\beta }^2+{\gamma }^2=1$. The directions along which projective measurements are performed are indicated by states $\left|{\psi }_i\right.〉$. When $\theta ={\theta }_5$, states connected by dashed lines are orthogonal. Experimentally relevant rotations are shown in dark red (see text). Right: Predicted values of $S_5$ [Eq. (2)] and $S_5^{\text{(ext)}}$ [Eq. (4)] as a function of $\theta$. The hashed region above $-3$ shows the space where $S_5^{\text{(ext)}}$ is consistent with NC models. The inset shows how a finite number of experiments (here $5\times {10}^4$ per data point) leads to a necessary deviation from theory around ${\theta }_5$ (dashed line). The analytical formulas for the plotted curves are given in [6].

Figure 2

Sequential measurement of observables $M_i$ and $M_j$. Red lines illustrate the parts of the sequence where the ion is hot and/or outside the computational basis. In the pump stage, the ion is Doppler cooled and pumped into the $S_{1/2}$ manifold, which includes state $\left|0\right.〉$. In the detect stage, the state is projected either onto the $\{\left|1\right.〉,\left|2\right.〉\}$ manifold (without affecting its motional state), or onto the $S_{1/2}$ manifold (heating it back to the Doppler limit). In the cool SP stage, the $S_{1/2}$ states are ground-state cooled and pumped into $\left|0\right.〉$, while states $\left|1\right.〉$ and $\left|2\right.〉$ remain unaffected. Each unitary $U_i$ is decomposed into a sequence of $(2i+1)$ coherent rotations on $\left|0\right.〉\leftrightarrow \left|1\right.〉$ and $\left|0\right.〉\leftrightarrow \left|2\right.〉$ transitions. Every sequence produces outcomes $A_i^{\left(1\right)}$ and $A_j^{\left(2\right)}$, from which we calculate the correlator $A_i^{\left(1\right)}{A}_j^{\left(2\right)}$. For every setting, the measurement is repeated 10 000 times.

Figure 3

Results of the KCBS measurement as a function of the opening angle $\theta$. Each data point results from $10000$ measurements on each of the five correlators $\langle A_i{A}_j\rangle$, either in normal $\left(j=i+1\right)$ or reverse order $\left(j=i-1\right)$. Blue and dashed red lines represent theoretical expectations for $S_5$ and $S_5^{\text{(ext)}}$, respectively (see Fig. 1, right). Note that all measurements of $S_5^{\text{(ext)}}$ violate the NC bound of ${\overline{S}}_5^{\text{NC}}=-3$. Error bars here and in the remainder of the Rapid Communication show the standard error in the mean, with sample standard deviation obtained directly from the data [6].

Figure 4

Results of measurements of $N$-cycle witnesses. Solid and dashed lines show QM expectations for relevant quantities (red dashed line includes shot noise), and hashed regions below 0 correspond to results explainable by classical models. The top plot illustrates the fractional gap between quantum and classical witnesses, which decreases for large $N$. Our data shows contextuality up to $N=101(N=61)$ for $S_N\left(S_N^{\text{(ext)}}\right)$. The bottom plot shows the calculated contextual fraction. Ideally the system becomes fully contextual at $N\to \infty$, but experimental imperfections lead to $\text{CF}<0$ for large enough $N$. We measure ${\text{CF}}_{31}=0.800\left(4\right)$. Error bars are generally smaller than the point size.