Quantitative comparison of quantum pseudo-telepathy games and Bell inequalities

TL;DR

This work quantitatively compares quantum-contextuality demonstrations via two quantum pseudo-telepathy games (MPMG and DG) with violations of Bell inequalities (CHSH and CG) for two-qubit states. By analyzing two state families, modified Werner and Bell-diagonal states, it computes the regions in state space that yield a quantum advantage in the games or violate the inequalities, using volume measures and eigenvalue-based criteria. The results show a consistent inclusion chain R_MPMG ⊂ R_DG ⊂ R_CG ⊂ R_CHSH, with Bell inequalities generally probing larger entangled-state regions than the games; DG covers more states than MPMG, but both remain smaller than Bell-violation regions. The study highlights that, despite the intuitive appeal of pseudo-telepathy, Bell inequalities are more effective at detecting entanglement across the considered families, and the typicality of quantum-game advantages shrinks as the state parametrization expands.

Abstract

Quantum pseudo-telepathy games, such as the Mermin-Peres magic square and the doily game, theoretically allow players to win with unit probability when using entangled quantum strategies. We quantitatively characterize the quantum advantage in these games and compare it with violations of two Bell inequalities: the Clauser-Horne-Shimony-Holt and the Collins-Gisin inequalities. The analysis is restricted to two families of two-qubit states: modified Werner states and Bell-diagonal states. For each case, we identify and quantify the regions of quantum state space that exhibit either a quantum advantage or a Bell inequality violation, relative to the set of all entangled states. Within these families, the doily game captures a larger fraction of entangled states than the Mermin-Peres magic square game, though both are significantly more limited than the regions associated with Bell inequality violations. Although both approaches are fundamentally linked to quantum contextuality, our analysis of the examined two-qubit state families indicates that Bell inequalities are more effective at revealing entanglement, even if pseudo-telepathy games offer a more intuitive and conceptually appealing perspective.

Figures (5)

• Figure 1: The structure of the doily game (DG). Possible questions correspond to two-qubit Pauli observables connected by solid lines and curves. Observables lying on the same line or curve commute. Black lines and curves represent positive parity questions, while red lines indicate negative parity questions.
• Figure 2: Schematic illustrations of (a) $\mathcal{S} \cup \mathcal{R_{\text{MPMG}}}$ and (b) $\mathcal{R_{\text{MPMG}}}$ are shown. The tetrahedron represents the convex set of all Bell-diagonal states, with its four vertices corresponding to the maximally entangled Bell states, see Eq. \ref{['eq:Belldiagonal']}. When $a = 1$, the average probability of winning the MPMG is equal to one.
• Figure 3: Schematic illustrations of (a) $\mathcal{S} \cup \mathcal{R_{\text{DG}}}$ and (b) $\mathcal{R_{\text{DG}}}$ are shown. For the description of the tetrahedron, see Fig. \ref{['fig:regions_PM']}. When $a = 1$, the average probability of winning the DG is equal to one.
• Figure 4: Schematic illustrations of the tetrahedron excluding the Steinmetz solid, representing the region $\mathcal{R_{\text{CHSH}}}$ associated with the CHSH inequality. The dark red region at the vertex $a=1$, $\mathcal{R_{\text{DG}}}$, indicates the quantum states that provide a quantum advantage in the DG. For comparison, see Fig. \ref{['fig:regions_D']}. (The small gray spikes in the corners are just artifacts of the rendering algorithm.)
• Figure 5: Schematic illustrations of the tetrahedron excluding the Steinmetz solid, showing the region $\mathcal{R_{\text{CHSH}}}$ associated with the CHSH inequality, along with the convex body corresponding to the CG inequality. The green region represents the set difference $\mathcal{R_{\text{CHSH}}} \setminus \mathcal{R_{\text{CG}}}$. For comparison, see Fig. \ref{['fig:Ronly_CHSH_2']}. The black lines mark the boundaries of the six regions defined by Eq. \ref{['eq:CGineq']}.