Enhanced non-macrorealism: Extreme violations of Leggett-Garg inequalities for a system evolving under superposition of unitaries

Abstract

Quantum theory contravenes classical macrorealism by allowing a system to be in a superposition of two or more physically distinct states, producing physical consequences radically different from that of classical physics. We show that a system, upon subjecting to transform under superposition of unitary operators, exhibits enhanced non-macrorealistic feature - as quantified by violation of the Leggett-Garg inequality (LGI) beyond the temporal Tsirelson bound. Moreover, this superposition of unitaries also provides robustness against decoherence by allowing the system to violate LGI and thereby retain its non-macrorealistic behavior for a strikingly longer duration. Using an NMR register, we experimentally demonstrate the superposition of unitaries with the help of an ancillary qubit and verify these theoretical predictions.

Figures (6)

• Figure 1: (a) The action of a superposition of two unitaries $U_0 = \exp(-\mathrm{i} \hat{\sigma}_\phi \omega \delta/2)$ and $U_1 = \exp(-\mathrm{i} \hat{\sigma}_x \omega \delta/2)$ ($\phi=\pi/2$ here) on the Bloch sphere is two fold : Shifting rotation axis (thick arrow) of the effective unitary $\mathcal{U}$ to lie between $\hat{x}$ and $\hat{\phi}$ and making the SOE under it non-linear in time - as the Bloch vector moving slowly near the poles relatively faster near equator. (b) A quantum circuit for realizing the superposed unitary with SP $\alpha$ by using an ancillary qubit (A). Here $2\alpha_y$ describes a rotation of the Bloch vector by angle $2\alpha$ about $\hat{y}$. (c) The interferometric circuit to determine the two-time correlator $C_{ij}$ corresponding to observable $\hat{Q}$ using an additional qubit M.
• Figure 2: (a) When the system evolves under $U_1$, $K^{\rm max}_3$ is evaluated and color plotted against $\xi$ and $\eta$, which are the longitude and co-latitude, respectively, of rotation axis of $U_0$ on the Bloch sphere. The plot shows $K_3$ is upper bounded by TTB=$1.5$. (b) When the system evolves under superposition of unitaries, $K^{\rm max}_3$ is evaluated and color plotted against $\alpha$ (the amount of superposition) and $\phi$ (the angle between the rotation axis of the unitaries which are being superposed). For each $\phi \in (0,\pi)$, $K^{\rm max}_3$ increases with increaseing $\alpha$. (c) The increase of $K^{\rm max}_3$ with increase in SP $\alpha$ is clearly shown for $\phi=90^0$ (dashed line), $\phi = 135^0$ (solid line) and $\phi = 165^0$ (dotted line) .
• Figure 3: (a) The molecular structure of 13C-DBFM with its qubits labelled. (b) Values of the scalar coupling constants in Hz ($J_{ij}$ in $i,j$'th off-diagonal element), resonance offsets (diagonal) along with relaxation time constants ($T_1$ and $T_{2}^{*}$ in s). (c) Quantum circuit for determining $C_{ij}$ of S by measuring the coherence of M in the end. Here $\hat{\rm{H}}$ signifies a Hadamard gate and $2\alpha_y$ signifies rotation in the Bloch sphere by angle $2\alpha$ about $\hat{y}$. (d-f) The quantum circuits used to realize (d) controlled-$\hat{\sigma}_z$ gate (e) $U_{T0}$ and (f) $U_{T1}$, where $\tilde{\phi} = \phi - \pi$ and $\tilde{\theta}=(\pi - t)/2$. Delays represent evolution under the respective scalar couplings and $\tau = 1/2J_{\rm{SM}}$. See Appendix \ref{['append:Circuit']} for details.
• Figure 4: Experimentally measured values of $K_3$ (dots with vertical error bars) are plotted along with the their theoretically predicted curves (solid thin curves) for four different values of $\alpha$ and $\phi$. Error bars represent vector sum of systematic (RF in-homogeneity) and random errors (thermal noise).
• Figure 5: (a) Defining $\tau_{\alpha}$ as the time upto which $K_3 \geq 1$, the gain in robustness $\tau_{\alpha}/\tau_{0}$ is computed by solving Eq. (\ref{['eq:bloch']}) and plotted with increasing SP $\alpha$. The plots are showing increase of $\tau_{\alpha}$ with increasing SP for $\phi$ equals $90^{0}$(dashed line), $115^{0}$ (solid line) and $140^{0}$ (dotted line), demonstrating the robustness against decoherence due to the superposition of unitaries. (b) The same is plotted, but using a more modest model of Eq. (\ref{['eq:lind_blad']}), where the additional noise to be encountered in physical realization of the superposition of unitaries are also considered. Results show that the enhanced robustness to persist even in the presence of additional noise. Both computations are done for decay constant $\gamma=1/(4\pi)~\rm{s}^{-1}$.