Hidden nonmacrorealism: Reviving the Leggett-Garg inequality with stochastic operations

The Leggett-Garg inequality (LGI) distinguishes nonmacrorealistic channels from macrorealistic ones by constraining the experimental outcomes of the underlying system. In this work, we propose a class of channels which, initially, cannot violate the LGI (in the form of the temporal Bell inequality) but can violate it after the application of stochastic pre- and post-operations (SPPOs). As a proof-of-principle experiment, we demonstrate the stochastic pre- and post-operations in an amplitude-damping channel with photonic qubits. We denote the above phenomenon as hidden nonmacrorealistic channels. We also discuss the relationship between these hidden nonmacrorealistic channels [in terms of the temporal Clauser-Horne-Shimony-Holt (CHSH) inequality] and the strongly nonlocality-breaking channel, which breaks the hidden spatial CHSH nonlocality for arbitrary states. In general, if the channel satisfies hidden nonmacrorealism, it is not a strongly CHSH nonlocality-breaking channel.

Figure 1

Schematic diagram. Without the process in the dashed-line boxes, the schematic diagram represents the standard nonmacrorealism test. Hidden nonmacrorealism can be observed by introducing the stochastic operations in the dashed-boundary boxes.

Figure 2

Experimental setup. BP: Band-pass filter; SPD: single-photon detection modules. BW: Brewster window plate. The time digitizer is triggered by the horizontally polarized photons. The coincidence counts are recorded once the measurements $M_{b|y}$ lead to clicks at Stop 1 or Stop 2.

Figure 3

Observation of the hidden CHSH nonmacrorealism in an amplitude-damping channel with visibility $v$. The grey shaded area at the bottom represent the macrorealistic regime. Without applying the SPPOs (blue dots with solid curves being the theoretical prediction), the temporal CHSH inequality is violated $B_{\mathrm{T}\text{−}\mathrm{CHSH}}>2$ up to $v=0.48$. With the SPPOs (red triangles with dashed curves), the violations are observed up to $v=0.64$. The green dot-dashed curve represents the theoretical prediction with loss $D\approx 1$ (where $D=0.99$). The error bars take into account the finite accuracy of the waveplate's angle ($\pm 1^{\circ }$), imperfect Brewster windows ($D\pm 2\%$), misaligned incident polarization ($\pm 1^{\circ }$), and nonideal visibility ($96–98\%$) of the Sagnac interferometer.