Exploration of an augmented set of Leggett-Garg inequalities using a noninvasive continuous-in-time velocity measurement

Macroscopic realism (MR) is the view that a system may possess definite properties at any time independent of past or future measurements and may be tested experimentally using the Leggett-Garg inequalities (LGIs). In this work we advance the study of LGIs in two ways using experiments carried out on a nuclear magnetic resonance spectrometer. First, we addresses the fact that the LGIs are only necessary conditions for MR but not sufficient ones. We implement a recently proposed test of necessary and sufficient conditions for MR which consists of a combination of the original four three-time LGIs augmented with a set of 12 two-time LGIs. We explore different regimes in which the two- and three-time LGIs may each be satisfied or violated. Second, we implement a recent proposal for a measurement protocol which determines the temporal correlation functions in an approximately noninvasive manner. It employs a measurement of the velocity of a dichotomic variable $Q$, continuous in time, from which a possible sign change of $Q$ may be determined in a single measurement of an ancilla coupled to the velocity. This protocol involves a significantly different set of assumptions to the traditional ideal negative measurement protocol, and a comparison with the latter is carried out.

Figure 1

Two of the LG3s as functions of $\omega t/\pi$ for measurements made at equidistant time intervals. The red and green regions indicate where the LG3s are and are not, respectively, violated.

Figure 2

A simulation of the effect of the decoherence on the correlation functions as a function of $\omega t$.

Figure 3

Equations (8) and (9) are plotted with the dampening effect of the decoherence. As $\omega t$ increase the LG3s gradually have larger regions in which they exist above the LGI bound. The decoherence is exaggerated for clarity.

Figure 4

The results of a search over the values of $v_y,v_z$ for which all LG2s are, or are not, satisfied for $\omega t=\pi /2$. The blue region (dark gray labeled A) depicts the initial states in which the LG2s are all satisfied, the red region (gray labeled B) the states in which the LG2s are not satisfied, and the gray (light gray labeled C) depicts initial states which do not exist (i.e., outside the bound of $v_y^2+v_z^2\le 1)$.

Figure 5

One of the 12 LG2s and the LG3s [Eqs. (8) and (9)] are plotted as functions of $\omega t$ with the effect of the decoherence. The figure shows that there exists a region of $\omega t$ in which at least one of the LG2s is violated (exists below the bound) while the LG3s are satisfied (exist above the bound). The same decoherence parameter is used as in Fig. 3.

Figure 6

The quantum circuit used for implementing the INM protocol. The evolutions $U\left(t_i\right)=e^{-iH{t}_i}$ and $U(t_j-t_i)=e^{-iH(t_j-t_i)}$ behave exactly as defined in Sec. 3a.

Figure 7

The probability of $Q$ undergoing multiple sign changes as a function of $\omega t$.

Figure 8

The probability of error from the back action of the ancilla on the system as a function of $\lambda$ and $\omega t$.

Figure 9

The values of $\lambda$ and $\omega t$ for which the value of $p\left(1\right)$ from Eq. (41) is greater than 0.01 are highlighted in blue.

Figure 10

The effect of the different values of $\lambda$ on the LG3s.

Figure 11

The pulse sequence used for producing the pseudo-pure state. $X\left(n\right)$ and $Y\left(n\right)$ depict rotations of $n$ radians around the $X$ and $Y$ axis $(X\left(n\right)=e^{-iX\frac{n}{2}}$ and $Y\left(n\right)=e^{-iY\frac{n}{2}})$. $ZZ\left(n\right)$ depicts the free evolution of the system that will provide an $n$ radian rotation of $ZZ\left(ZZ\left(n\right)=e^{-iZZ\frac{n}{2}}\right)$, and $\mathcal{G}$ represents the application of a gradient.

Figure 12

A pulse sequence for implementing the cnot gate in NMR. The full sequence will be referred to as $U_c$.

Figure 13

A pulse sequence for implementing the time delay while undoing the effect of the $j$ coupling. $D\left(\frac{a\tau }{2}\right)$ depict waiting for time $\frac{a\tau }{2}$. The full sequence will be referred to as ${\mathcal{D}}_{a\tau }$.

Figure 14

The pulse sequence used to implement the system-detector evolution. The parameters ${\alpha }_1=4.9751,{\beta }_1=1.8335$, and ${\gamma }_1=0.1035$ are used for implementing $U_{v1}$, and the parameters ${\alpha }_2=5.2433,{\beta }_2=2.1018$, and ${\gamma }_2=0.1998$ are used for implementing $U_{v2}$.

Figure 15

The first set of experiments. Experiments 1–3 are used to measure the values of $\langle Q_i\rangle$, and experiments 4–9 are used to measure the values of $C_{ij}$ using the INM protocol.

Figure 16

The second set of experiments. Experiments 10–12 are used to measure $\langle Q_i\rangle$, experiments 13–18 are used to measure the values of $C_{ij}$ using the INM protocol, and experiments 19–21 are used to measure the values of $C_{ij}$ using the CTVM protocol.

Figure 17

The values of the LG2s and LG3s constructed from the first set of experiments with their corresponding error bars. The labels in green highlight which inequalities were satisfied, and the labels in red highlight which were violated.

Figure 18

The values of the LG2s and LG3s constructed from the second set of experiments with their corresponding error bars. The labels in green highlight which inequalities were satisfied, and the labels in red highlight which were violated.

Figure 19

The two quantum circuits used to implement the INM procedure.