Bipartite Leggett-Garg and macroscopic Bell-inequality violations using cat states: Distinguishing weak and deterministic macroscopic realism

We consider tests of Leggett-Garg's macrorealism and of macroscopic local realism, where for spacelike separated measurements the assumption of macroscopic noninvasive measurability is justified by that of macroscopic Bell locality. We give a mapping between the Bell and Leggett-Garg experiments for microscopic qubits based on spin-$1/2$ eigenstates and gedanken experiments for macroscopic qubits based on two macroscopically distinct coherent states $|\alpha \rangle$ and $|-\alpha \rangle$ (as $\alpha \to \infty$). In this mapping, the unitary rotation $U_{\theta }$ of the Stern-Gerlach analyzer or polarizing beam splitter is realized by a local interaction $H=\mathrm{\Omega }{\widehat{n}}^4$ where $\widehat{n}$ is the number of quanta. By adjusting the time of interaction, one alters the measurement setting $\theta$. We thus predict violations of Leggett-Garg and Bell inequalities in a macroscopic regime where coarse-grained measurements $\widehat{M}$ need only discriminate between two macroscopically distinct coherent states. To interpret the violations, we distinguish between different definitions of macroscopic realism. Deterministic macroscopic local realism (dMR) assumes the system is in a state with a definite outcome ${\lambda }_{\theta }$ for the measurement $\widehat{M}$ prior to the unitary rotation $U_{\theta }$, and is negated by the violations. Weak macroscopic realism (wMR) assumes a definite outcome for systems prepared in a superposition ${\psi }_{\text{pointer}}$ of two macroscopically distinct eigenstates of $\widehat{M}$, after the unitary rotation $U_{\theta }$. We find that wMR can be viewed consistently with the violations and with the predictions of quantum mechanics for two tests of wMR. A model is presented in which wMR holds and the macroscopic violations emerge over the course of the unitary dynamics occurring at both sites. Implications for the realism of the micro-macro state generated during a quantum measurement are discussed. Finally, we point out an EPR-type paradox, which demonstrates inconsistency between wMR and the completeness of quantum mechanics at a microscopic level.

Figure 1

Deterministic macroscopic realism versus weak macroscopic realism: The two-mode cat state $|{\psi }_{\text{Bell}}\rangle$ is prepared at time $t=0$. Spin measurements ${\widehat{S}}_{\theta }^{\left(A\right)}$ and ${\widehat{S}}_{\varphi }^{\left(B\right)}$ are made at sites $A$ and $B$. The measurements have two stages. The choice of measurement setting at each site corresponds to unitary interactions $U_{\theta }^{\left(A\right)}\equiv U_A\left(\theta \right)$ and $U_{\varphi }^{\left(B\right)}\equiv U_B\left(\varphi \right)$ which occur over a time interval. After the interactions, at time $t_f$, the system is in a superposition $|{\psi }_{\text{pointer}}\rangle$ of the macroscopic states $|\pm \alpha \rangle$. Final pointer measurements ${\widehat{M}}^{\left(A\right)}$ and ${\widehat{M}}^{\left(B\right)}$ are made, distinguishing between the states $|\pm \alpha \rangle$ at each site, to give outcomes $\pm 1$. Deterministic macroscopic realism implies that the values of the final measurements are determined by hidden variables ${\lambda }_{M_{\theta }}^{\left(A\right)}, {\lambda }_{M_{\varphi }}^{\left(B\right)}$ which describe the system at time $t_1$ prior to the interactions $U$. Weak macroscopic realism is the less restrictive assumption that the final outcomes of the pointer measurements ${\widehat{M}}^{\left(A\right)}$ and ${\widehat{M}}^{\left(B\right)}$ are determined by hidden variables ${\lambda }_{M_{\theta }}^{\left(A\right)}, {\lambda }_{M_{\varphi }}^{\left(B\right)}$ at a time $t_f$, after the transformations $U$. The hidden variables then apply to the state $|{\psi }_{\text{pointer}}\rangle$ prepared in the pointer basis.

Figure 2

A Leggett-Garg test of macrorealism using bipartite systems. Following the notation of Fig. 1, two systems prepared in the entangled state $|{\psi }_{\text{Bell}}\rangle$ at time $t_1$ undergo dynamical evolution according to local unitary operators, $U_A\left(t_i\right)\equiv U_i^{\left(A\right)}$ and $U_B\left(t_j\right)\equiv U_j^{\left(B\right)}$. Measurements ${\widehat{M}}_i^{\left(A\right)}$ and ${\widehat{M}}_j^{\left(B\right)}$ can be made at the times $t_i$ and $t_j$ ($i,j=1,2$ or 3) at $A$ or $B$, and each system is found to be in one or other of two macroscopically distinct states labeled $-1$ and $+1$, the outcomes being anticorrelated. One can infer the outcome for $A$ at $t_i$ by measuring at $B$. A possible macrorealistic evolution is shown by the dashed lines. Weak macroscopic realism asserts that the system $A$ ($B$) at time $t_i$ ($t_j$) after the interaction $U_A\left(t_i\right) \left(U_B\left(t_j\right)\right)$ is in a state with a definite outcome ${\lambda }_i^{\left(A\right)}({\lambda }_j^{\left(B\right)}$) for ${\widehat{M}}_i^{\left(A\right)}$ (${\widehat{M}}_j^{\left(B\right)}$).

Figure 3

An initial coherent state $|\alpha \rangle$ evolving according to the Hamltonian $H_{\text{NL}}^{\left(A\right)}, k=4, \alpha =3$. Contour plots of the $Q$ function $Q(x_A,p_A)$ are shown for the states created at a time $t$. For the times shown, a superposition of the states $|-\alpha \rangle$ and $|\alpha \rangle$ is formed.

Figure 4

The detailed dynamics of the creation of the superposition states shown in Fig. 3. The system is not a superposition of states $|-\alpha \rangle$ and $|\alpha \rangle$ at all times $t$. Here $\alpha =4$.

Figure 5

Distributions giving macroscopic Bell violations: Contour plots of $P(X_A,X_B)$ for the system depicted in Fig. 1 with an initial entangled state $|{\psi }_{\text{Bell}}\rangle$ (6) and unitary transformations $U_A$ and $U_B$ (20), for measurement settings $t_a$ and $t_b$. Here, $\alpha =\beta =3$ and time $t$ is in units of ${\mathrm{\Omega }}^{-1}$. For each setting $\theta$ and $\varphi$ given by ($t_a, t_b$), the outcomes $S_A$ and $S_B$ for ${\widehat{M}}^{\left(A\right)}$ and ${\widehat{M}}^{\left(B\right)}$ are $+1$ or $-1$ if the measurement of ${\widehat{X}}_A$ or ${\widehat{X}}_B$ yields a positive or negative value.

Figure 6

The violation of the macroscopic Bell inequality (2) for the system depicted in Fig. 1, as described in Fig. 5. We show (top) $B$ vs $\beta$ where $\alpha =\beta .$ In the lower plot, $\beta$ is varied for fixed $\alpha$. A violation is obtained when $\left|B\right|>2$.

Figure 7

Top: Contour plots of the $Q$ function marginal $Q(x_A,x_B)$ for the states created at the times relevant for the violation of the macroscopic Bell inequality (2), as in Fig. 5. The system is prepared at $t_a=t_b=0$ in the entangled Bell cat state $|{\psi }_{\text{Bell}}\rangle$ [Eq. (6)]. Lower: The corresponding plots if the system is prepared in the nonentangled mixture ${\rho }_{\text{mix}}$ [Eq. (26)], which does not violate the Bell inequality. Here, $\alpha =\beta =3$ and time $t$ is in units of ${\mathrm{\Omega }}^{-1}$.

Figure 8

The two-state nature of the systems created at the specific times given in Figs. 5 and 7 is evident on examining the contour plots of the marginal $Q(x_A,p_A)$. The plots for the Bell cat state and the mixed state ${\rho }_{\text{mix}}$ are indistinguishable.

Figure 9

The entangled cat state $|{\psi }_{\text{Bell}}\rangle$ (6) evolves while being measured at the different sites. We show the contour plots of $Q(x_A,x_B)$ corresponding to the four joint measurements leading to violation of the Bell inequality (2), as in Figs. 5 and 7. The top sequence shows the dynamics for the measurements with settings $(t_a=0,t_b=\pi /4)$ and $(t_a=0,t_b=3\pi /4)$. The center and lower sequences correspond to settings $(t_a=\pi /2,t_b=3\pi /4)$ and $(t_a=\pi /2,t_b=\pi /4)$, respectively. Here, $\alpha =\beta =3$ and time $t$ is in units of ${\mathrm{\Omega }}^{-1}$.

Figure 10

The plots are as for Fig. 9, but here the system evolves from a nonentangled state ${\rho }_{\text{mix}}$. There is no violation of the Bell inequality. The plots with $t_a=0$ (top) involving a unitary rotation at only one site are indistinguishable from those of Fig. 9. The center and lower plots involving transformations at two sites show a macroscopic difference at time $t_a=\pi /2$.

Figure 11

The $Q$ function $Q(x_A,p_A)$ for times $t_1, t_2$, and $t_3$ ($\alpha =3$). The top sequence assumes no measurement is made at $t_2$. The lower sequence assumes a measurement is made at $t_2$, of the type that collapses the system into one or other of two coherent states. While the difference in the $Q$ function at time $t_2$ due to measurement is negligible, this leads to distinct dynamics, and a macroscopic difference at time $t_3$.

Figure 12

Distributions giving the violation of the bipartite Leggett-Garg-Bell inequality (34). Two systems $A$ and $B$ are prepared in the entangled cat state $|{\psi }_{\text{Bell}}\rangle$ (6) and then evolve according to the unitary transformations $U_A\left(t_a\right)$ and $U_B\left(t_b\right)$ (20) as in Fig. 2. We show the contour plots of the joint probability of $P(X_A,X_B)$ for the measurements ${\widehat{X}}_A$ and ${\widehat{X}}_B$ made at the times $t_a$ and $t_b$, with $\alpha =\beta =3$. The last three distributions give the moments $\langle S_1^{\left(B\right)}{S}_2^{\left(A\right)}\rangle , \langle S_1^{\left(B\right)}{S}_3^{\left(A\right)}\rangle$, and $\langle S_2^{\left(B\right)}{S}_3^{\left(A\right)}\rangle$. Here $t_1=0, t_2=\pi /4\mathrm{\Omega }$, and $t_3=\pi /2\mathrm{\Omega }$. The times $t_a$ and $t_b$ are given in units of ${\mathrm{\Omega }}^{-1}$.

Figure 13

The violation of the Leggett-Garg-Bell inequality (34) and (5) for the system depicted in Figs. 2 and 12. We plot $B_{\text{lg}}$ given by (35) vs $\beta$ for the state $|{\psi }_{\text{Bell}}\rangle$ (6), where $\alpha =\beta .$ Violation is obtained when $B_{\text{lg}}>1$. The verification of $\langle S_i^{\left(A\right)}{S}_j^{\left(A\right)}\rangle =-\langle S_i^{\left(B\right)}{S}_j^{\left(A\right)}\rangle$ for $i=1,2$ is given by examination of the conditional distribution $P(S_i^{\left(A\right)}=1|S_i^{\left(B\right)}=-1)$ as shown.

Figure 14

$Q$-function dynamics as the entangled cat state $|{\psi }_{\text{Bell}}\rangle$ evolves through the three measurement sequences of the bipartite Leggett-Garg test. Plots show the marginal distribution $Q(x_A,x_B)$ after evolving for times $t_a$ and $t_b$ at the sites $A$ and $B$. The subsystems evolve acording to the same shared clock, with $t_a=t_b$, up to the time of measurement at the site $B$. Here $t_1=0, t_2=\pi /4\mathrm{\Omega }$, and $t_3=\pi /2\mathrm{\Omega }$. The top, center, and lower rows give a sequence of snapshots corresponding to the measurement of $\langle S_1^{\left(B\right)}{S}_2^{\left(A\right)}\rangle , \langle S_1^{\left(B\right)}{S}_3^{\left(A\right)}\rangle$, and $\langle S_2^{\left(B\right)}{S}_3^{\left(A\right)}\rangle$, respectively. The results agree with those of Fig. 11. The times $t_a$ and $t_b$ are given in units of ${\mathrm{\Omega }}^{-1}$.

Figure 15

The timing of the final collapse stage of the measurement $S_1^{\left(B\right)}$ at time $t_1$ on $B$ is immaterial, for $\beta >1$: The plots show a sequence of snapshots of $Q(x_A,x_B)$ as the system described in Fig. 14 evolves for the evaluation of $\langle S_1^{\left(A\right)}{S}_2^{\left(A\right)}\rangle$ and $\langle S_1^{\left(A\right)}{S}_3^{\left(A\right)}\rangle$. (a) The top two sequences: Here $\alpha =\beta =0.5$. The top sequence shows the evolution if no collapse of the measurement at $B$ has taken place before the time $t_3=\pi /2\mathrm{\Omega }$. The lower sequence shows the evolution assuming the collapse has taken place at time $t_b=t_1=0$. (b) The lower two sequences: As for (a), but here $\alpha =\beta =1.5$. The states for the top and lower sequences although different are visually indistinguishable. The times $t_a$ and $t_b$ are given in units of ${\mathrm{\Omega }}^{-1}$.