Theory-Independent Context Incompatibility: Quantification and Experimental Demonstration

TL;DR

This work proposes the notion of theory-independent context compatibility, a concept that is trivially satisfied by classical statistical theory but is found in conflict with quantum mechanics, and experimentally demonstrates that quantum systems can exhibit pronounced degrees of violation.

Abstract

The concept of compatibility originally emerged as a synonym for the commutativity of observables and later evolved into the notion of measurement compatibility. In any case, however, it has remained predominantly algebraic in nature, tied to the formalism of quantum mechanics. Recently, still within the quantum domain, the concept of context incompatibility has been proposed as a resource for detecting eavesdropping in quantum communication channels. Here, we propose a significant generalization of this concept by introducing the notion of theory-independent context compatibility, a concept that is trivially satisfied by classical statistical theory but is found in conflict with quantum mechanics. Moreover, we propose a figure of merit capable of quantifying the degree of violation of theory-independent context incompatibility, and we experimentally demonstrate, using a quantum optics platform, that quantum systems can exhibit pronounced degrees of violation. Besides yielding a concept that extends to generic probabilistic theories and retrieving the notion of measurement incompatibility in the quantum domain, our results offer a promising perspective on evaluating the role of incompatibility in the manifestation of non-local correlations.

Figures (4)

• Figure 1: Schematic representation of context compatibility. A context $\mathbbm{C}=\{\mathscr{E},\mathscr{X},\mathscr{Y}\}$ consists of a preparation $\mathscr{E}$ and generalized measurements $\mathscr{X}$ and $\mathscr{Y}$ with outcomes $x_i$ and $y_j$. Each measurement induces a nonselective map on the preparation, and the context is compatible when the resulting distributions of all $x_i$ and $y_j$ match those obtained directly from $\mathscr{E}$.
• Figure 2: Experimental scheme used to obtain the photon-counting probabilities. HWP$_p$ is the element that implements the unitary modifying the parameter $p$, HWP$_1$ modifies $\theta$, and QWP$_1$ changes $\phi$. HWP$_A$ implements $\theta_A$ and, together with PBS$_1$, perform the first measurement $A$. HWP$_B$ implements $\theta_B$ and, together with PBS$_2$, performs the second nonselective measurement $B$. A final simplification involved the removal of the second HWP$_B$, which would come after PBS$_2$, as the final photon detection is performed in the polarization basis ${H, V}$, without the need for the final rotation of $\theta_B$. The elements D1 and D2 are avalanche photodetectors, and CC is the photon coincidence electronic circuit.
• Figure 3: Theoretical prediction of $\mathcal{I}_{\mathbbm{C}}$ (red line) and experimental data (blue dots) for the context (a) $\{\rho_x,\sigma_x,\sigma_z\}$, (b) $\{\rho_y,\sigma_x,\sigma_z\}$, and (c) $\{\rho_z,\sigma_x,\sigma_z\}$ as a function of $p$. The $y$-axis error bars are included but not visible (order $10^{-2}$). The calculations were performed using base-$e$ logarithms.
• Figure 4: Theoretical prediction of the context incompatibility $\mathcal{I}_{\mathbbm{C}}$ (red line) and experimental results (blue dots) for the context $\mathbbm{C} = \{\rho_z, A, \sigma_z\}$ with fixed state interpolated at $p=0$ (top), $p=1$ (middle), and $p=2$ (bottom), as a function of the Half-Wave Plate angle $\theta_A$ that determines the observable $A$. Error bars are shown in each plot, but are not visible (on the order of $10^{-2}$). The calculations were performed using base-$e$ logarithms.

Theorems & Definitions (1)

• Definition