Violation of entropic Leggett-Garg inequality in nuclear spins

We report an experimental study of recently formulated entropic Leggett-Garg inequality (ELGI) by Usha Devi et al. [Phys. Rev. A 87, 052103 (2013)]. This inequality places a bound on the statistical measurement outcomes of dynamical observables describing a macrorealistic system. Such a bound is not necessarily obeyed by quantum systems, and therefore provides an important way to distinguish quantumness from classical behavior. Here we study ELGI using a two-qubit nuclear magnetic resonance system. To perform the noninvasive measurements required for the ELGI study, we prepare the system qubit in a maximally mixed state as well as use the “ideal negative result measurement” procedure with the help of an ancilla qubit. The experimental results show a clear violation of ELGI by over four standard deviations. These results agree with the predictions of quantum theory. The violation of ELGI is attributed to the fact that certain joint probabilities are not legitimate in the quantum scenario, in the sense they do not reproduce all the marginal probabilities. Using a three-qubit system, we also demonstrate that three-time joint probabilities do not reproduce certain two-time marginal probabilities.

Figure 1

Circuits for measuring (a) single-event probabilities, (b) two-time joint probabilities, and (c) three-time joint probabilities. The gates grouped by dashed lines represent measurements in the $\{{\Pi }_0^t,{\Pi }_1^t\}$ basis. The pointer at the end in each circuit represents the measurement of diagonal elements of the density matrix.

Figure 2

The molecular structures of (a) chloroform and (c) trifluoroiodoethylene and the (b), (d) corresponding tables of relative resonance frequencies (diagonal elements, in Hz) and the $J$-coupling constants (in Hz). The pulse sequence for initializing trifluoroiodoethylene is shown in (e). In (e) the open pulses are $\pi$ pulses and the delay $\tau =1/\left(4J_{23}\right)$. Pulsed field gradient (PFG) pulses are used to destroy the transverse magnetization.

Figure 3

The lines indicate the theoretical joint probabilities (a) $P{(q_1,q_2)}_{\mathrm{th}}$ and (b) $P{(q_1,q_3)}_{\mathrm{th}}$, and the symbols indicate the mean experimental probabilities (a) $P(q_1,q_2)$ and (b) $P(q_1,q_3)$ obtained by the INRM procedure.

Figure 4

Information deficit ${\mathcal{D}}_3$ versus $\theta$ obtained with (a), (b) cnot; (c), (d) anti-cnot; and (e), (f) INRM procedure. The boxed areas in the left plots [(a), (c), (e)] are magnified in the right plots [(b), (d), (f)], respectively. The mean experimental ${\mathcal{D}}_3$ (in bits) values are shown as symbols. The curves indicate theoretical ${\mathcal{D}}_3$ (in bits). The horizontal lines at ${\mathcal{D}}_3=0$ indicate the lower bounds of the macrorealism territories.

Figure 5

(a) Joint probabilities $P(q_1,q_2)$ and marginal probabilities $P^{\prime }(q_1,q_2)$, and (b) joint probabilities $P(q_1,q_3)$ and marginal probabilities $P^{\prime }(q_1,q_3)$. The lines correspond to theoretical values and the symbols are mean experimental values.