Comparing various formulations of macrorealism

Abstract

The Leggett–Garg inequality (LGI) is used to test incompatibility between the classical world view of macrorealism and quantum mechanics. Except for the LGI, other formulations for testing macrorealism have also been proposed, such as the entropic LGI and the no-signaling-in-time (NSIT) condition. And these formulations of macrorealism are not both the necessary and sufficient conditions for macrorealism. In this paper, analogous to the LGI, an equality for the energy change of a quantum system is given for testing macrorealism. It is called as the energy LG (ELG). Then, we study quantum violations of the ELG, the LGI, the entropic LGI and the NSIT condition for a two-level system. It is shown that for the projective measurement, the ELG can be violated for a wider parameter regime than the NSIT condition, and the NSIT condition can be violated for a wider parameter regime than the LGI and the entropic LGI. For the coarsening measurement reference and coarsening final resolution, we find that the quantum violations of the ELG and the NSIT condition provide the same robustness, which are both the most robust, and the quantum violation of the entropic LGI is the most vulnerable.

Appendices

ELG

From Eqs. (1), (5) and (9), the ELG under the coarsening final measurement resolution can be obtained as

$$\begin{aligned} E_{LG, \delta }= & {} \frac{1}{4} \alpha \omega \sin ^2\theta \left[ 3+4 (\delta -1) \delta -4 \sqrt{-(\delta -1) \delta } -\left( 4 (\delta -1) \delta \right. \right. \nonumber \\{} & {} +\left. \left. 4 \sqrt{-(\delta -1) \delta }-1\right) \left( \cos 2 \theta +2 \cos ^2\theta (\cos 2 \omega \tau -2 \cos \omega \tau )\right) \right] .\nonumber \\ \end{aligned}$$

(A1)

It can be found that in the case of $\alpha =0.3, \tau =\frac{\pi }{4\omega }$ and $\theta =\frac{\pi }{5}$, the ELG can be always violated for any value of coarsening degree $\delta$, i.e., $0<\delta <0.5$. In other words, in the case of the coarsening final measurement resolution, there are no conditions that make the ELG not be violated, which is listed in Table 2.

LGI

Using Eqs. (1), (2), (4), (9) and (10), the LGI under the projective measurement, the coarsening measurement reference and the coarsening measurement in final resolution, can be, respectively, given as

$$\begin{aligned} K_\mathrm{{LG}}= & {} \cos ^2\theta -\sin ^2\theta (\cos 2 \omega \tau -2 \cos \omega \tau ), \end{aligned}$$

(A2)

$$\begin{aligned} K_{\mathrm{{LG}}, \mathrm{{\Delta }}}= & {} e^{-2 \mathrm{{\Delta }} ^2} \left[ e^{\mathrm{{\Delta }} ^2} \cos ^2\theta -\sin ^2\theta (\cos 2 \omega \tau -2 \cos \omega \tau )\right] , \end{aligned}$$

(A3)

$$\begin{aligned} K_{\mathrm{{LG}}, \delta }= & {} (1-2 \delta )^2 \left[ \cos ^2\theta -\sin ^2\theta (\cos 2\omega \tau -2 \cos \omega \tau )\right] . \end{aligned}$$

(A4)

It can be found from Eq. (A2) that under the projective measurement, when one of the following conditions is satisfied: (1)$\theta = 0$; (2) $\frac{\pi }{2\omega }\le \tau \le \frac{\pi }{\omega }$, the LGI can be satisfied, which is listed in Table 1. From Eq. (A3), we find the LGI with the coarsening measurement in reference, can be satisfied, when $0.3073<\mathrm{{\Delta }}<1$ (suppose $\alpha =0.3, \tau =\frac{\pi }{4\omega }$ and $\theta =\frac{\pi }{5}$). That is to say, when $0<\mathrm{{\Delta }}\le 0.3073$, the LGI can be violated. For the coarsening measurement in final resolution, we from Eq. (A4) find that when $0.0323<\delta <0.5$, the LGI can be satisfied (suppose $\alpha =0.3, \tau =\frac{\pi }{4\omega }$ and $\theta =\frac{\pi }{5}$). And the non-violation conditions of the LGI under coarsening measurement both in reference and in final resolution are listed in Table 2.

Entropic LGI

From Eqs. (9), (10) and (14), the entropic LGI under the projective measurement can be expressed as

$$\begin{aligned} H_{LG}= & {} \frac{1}{16}\left[ -8 \left( \log (\alpha \cos \theta +1)+\left( \log (1-\alpha \cos \theta )+\log \left( \frac{1}{4}\right) \right) \right. \right. \nonumber \\{} & {} +2 \left. \alpha \cos \theta \tanh ^{-1}(\alpha \cos \theta )\right) -2e^{-i\omega \tau }\left( -2\left( -1+e^{i\omega \tau }\right) ^2 \left( \log (\alpha \cos \theta -1)\right. \right. \nonumber \\{} & {} +\log \left. \left( \sin ^2\theta \right) +\log \left( \sin ^2\left( \frac{ \omega \tau }{2}\right) \right) +\log \left( -\frac{1}{2}\right) \right) (\alpha \cos \theta -1)\sin ^2\theta \nonumber \\{} & {} +\left( \log (\alpha \cos \theta +1)+\log \left( \sin ^2\theta \right) +\log \left( \sin ^2\left( \frac{\omega \tau }{2}\right) \right) +\log \left( \frac{1}{2}\right) \right) \nonumber \\{} & {} \times 2 \left( -1+e^{i\omega \tau }\right) ^2 (\alpha \cos \theta +1)\sin ^2\theta +2 e^{i \omega \tau } (\alpha \cos \theta -1)\nonumber \\{} & {} \times (\cos 2 \theta +2 \sin ^2\theta \cos \omega \tau +3) \left( \log (\alpha \cos \theta -1)\right. \nonumber \\{} & {} +\log \left. \left( \cos 2 \theta +2 \sin ^2\theta \cos \omega \tau +3\right) +\log \left( -\frac{1}{8}\right) \right) -2 e^{i \omega \tau } (\alpha \cos \theta +1) \nonumber \\{} & {} \times \left( \log (\alpha \cos \theta +1)+\log \left( \cos 2 \theta +2 \sin ^2\theta \cos \omega \tau +3\right) +\log \left( \frac{1}{8}\right) \right) \nonumber \\{} & {} \times \left. (\cos 2 \theta +2 \sin ^2\theta \cos \omega \tau +3)\right) \!+\!e^{-2 i \omega \tau }\left( -2 \left( -1+e^{2 i \omega \tau }\right) ^2 \!(\alpha \cos \theta \!-\!1)\right. \nonumber \\{} & {} \times \sin ^2\theta \left( \log (\alpha \cos \theta -1)+\log \left( \sin ^2\theta \right) +\log \left( \sin ^2 \omega \tau \right) +\log \left( -\frac{1}{2}\right) \right) \nonumber \\{} & {} +2 \left( -1+e^{2 i \omega \tau }\right) ^2 \sin ^2\theta (\alpha \cos \theta +1) \left( \log (\alpha \cos \theta +1)+\log \left( \sin ^2\theta \right) \right. \nonumber \\{} & {} +\log \left. \left( \sin ^2 \omega \tau \right) +\log \left( \frac{1}{2}\right) \right) +2 e^{2 i \omega \tau } (\alpha \cos \theta -1) \left( \log (\alpha \cos \theta -1)\right. \nonumber \\{} & {} +\log \left. \left( \cos 2 \theta +2 \sin ^2\theta \cos 2 \omega \tau +3\right) +\log \left( -\frac{1}{8}\right) \right) \nonumber \\{} & {} \left( \cos 2 \theta +2 \sin ^2\theta \cos 2 \omega \tau +3\right) -2 e^{2 i \omega \tau } (\alpha \cos \theta +1)\nonumber \\{} & {} \times (\cos 2 \theta +2 \sin ^2\theta \cos 2\omega \tau +3)\left( \log (\alpha \cos \theta +1)\right. \nonumber \\{} & {} +\log \left. \left. \left. \left( \cos 2 \theta +2 \sin ^2\theta \cos 2\omega \tau +3\right) +\log \left( \frac{1}{8}\right) \right) \right) \right] . \end{aligned}$$

(A5)

From above expression, the non-violation conditions of the entropic LGI (i.e., Eq. (14)) for the projective measurement can be found, and then, these non-violation conditions are listed in Table 1.

NSIT condition

From Eqs. (9), (10) and (15–17), the NSIT condition with the projective measurement can be written as

$$\begin{aligned}&P(b_1=+1)-\!\sum _{{a_0=\pm 1}} P(a_0,b_1=+1) \!=\!-\!\left( P(b_1=-1) -\! \sum _{{a_0=\pm 1}} P(a_0,b_1=-1)\right) \nonumber \\&\quad =P(b_2=+1)-\sum _{{a_1=\pm 1}} P(a_1,b_2=+1)\nonumber \\&\quad =-\left( P(b_2=-1) - \sum _{{a_1=\pm 1}} P(a_1,b_2=-1)\right) =-\alpha \sin ^2\left( \frac{ \omega \tau }{2}\right) \sin ^2\theta \cos \theta ,\nonumber \\ \end{aligned}$$

(A6)

$$\begin{aligned}&\qquad -\left( P(b_2=+1)-\sum _{{a_0=\pm 1}} P(a_0,b_2=+1)\right) =P(b_2=-1)\nonumber \\&\qquad - \sum _{{a_0=\pm 1}} P(a_0,b_2=-1) =\alpha \sin ^2 \omega \tau \sin ^2\theta \cos \theta , \end{aligned}$$

(A7)

$$\begin{aligned}&P(b_1=+1,c_2=+1) - \sum _{{a_0=\pm 1}} P(a_0,b_1=+1,c_2=+1)\nonumber \\&\quad =- \left( P(b_1=-1,c_2=-1) - \sum _{{a_0=\pm 1}} P(a_0,b_1=-1,c_2=-1))\right) \nonumber \\&\quad =-\frac{1}{4} \alpha \sin ^2\theta \left( 3 +2\cos \omega \tau \sin ^2\theta +\cos 2 \theta \right) \cos \theta \sin ^2\left( \frac{\omega \tau }{2}\right) , \end{aligned}$$

(A8)

$$\begin{aligned}&P(b_1=-1,c_2=+1) - \sum _{{a_0=\pm 1}} P(a_0,b_1=-1,c_2=+1)\nonumber \\&\quad =-\left( P(b_1=+1,c_2=-1) - \sum _{{a_0=\pm 1}} P(a_0,b_1=+1,c_2=-1))\right) \nonumber \\&\quad =\alpha \sin ^4\left( \frac{\tau \omega }{2}\right) \sin ^4\theta \cos \theta , \end{aligned}$$

(A9)

$$\begin{aligned}&P(a_0=+1,c_2=+1) - \sum _{{b_1=\pm 1}} P(a_0=+1, b_1, c_2=+1)\nonumber \\&\quad =-\left( P(a_0=+1,c_2=-1) - \sum _{{b_1=\pm 1}} P(a_0=+1, b_1, c_2=-1)\right) \nonumber \\&\quad =\frac{1}{4} \sin ^2\left( \frac{ \omega \tau }{2}\right) \left[ \cos \omega \tau (\cos 2 \theta +3) +2 \sin ^2\theta \right] (\alpha \cos \theta -1)\sin ^2\theta , \end{aligned}$$

(A10)

$$\begin{aligned}&P(a_0=-1,c_2=+1) - \sum _{{b_1=\pm 1}} P(a_0=-1, b_1, c_2=+1)\nonumber \\&\quad =-\left( P(a_0=-1,c_2=-1) - \sum _{{b_1=\pm 1}} P(a_0=-1, b_1, c_2=-1)\right) \nonumber \\&\quad =\frac{1}{4} \left[ 2 \sin ^2 \omega \tau -\left( 3+2 \cos \omega \tau \sin ^2\theta +\cos 2 \theta \right) \sin ^2\left( \frac{\omega \tau }{2}\right) \right] (\alpha \cos \theta +1)\sin ^2\theta . \end{aligned}$$

(A11)

From Eqs. (A6) to (A11), we find that the NSIT condition under the projective measurement can be satisfied (i.e., Eqs. (15–17) is satisfied), if one of the following conditions is satisfied: (1) $\theta =0$; (2) $\theta =\frac{\pi }{2}$ and $\tau =\frac{\pi }{\omega }$. The results obtained for the NSIT condition in this case are summarized in Table 1.