Efficient Computation of Generalized Noncontextual Polytopes and Quantum violation of their Facet Inequalities

Abstract

Finding a set of empirical criteria fulfilled by any theory satisfying the generalized notion of noncontextuality is a challenging task of both operational and foundational importance. This work presents a methodology for constructing the noncontextual polytope while ensuring that the dimension of the polytope associated with the preparations remains constant regardless of the number of measurements and their outcome size. The facet inequalities of the noncontextual polytope can thus be obtained in a computationally efficient manner. We illustrate the efficacy of our methodology through several distinct contextuality scenarios. Our investigation uncovers several hitherto unexplored noncontextuality inequalities and demonstrates applications of quantum contextual correlations in certification of non-projective measurements, witnessing the dimension of quantum systems, and randomness certification.

Figures (3)

• Figure 1: The extended polytope $(\mathbb{P}_\text{P})$ encompassing the probabilities specified by \ref{['pbar']}, is defined by the collection of vertices as, $\mathbb{P}_\text{P}=\{v_1,v_2,v_3,v_5,v_6,v_7\}$. The polytope adhering solely to the normalization condition \ref{['n']}, is defined as, $\mathbb{P}_{\text{NP}}=\{v_1,v_2,v_7,v_8,v_9\}$. $\mathbb{P}_{\text{NCP}}=\{v_1,v_2,v_3,v_4,v_7\}$ is the precise non-contextual polytope, which is the intersection of $\mathbb{P}_\text{P}$ and $\mathbb{P}_\text{NP}$.
• Figure 2: The x-z plane of the Bloch sphere is considered to pinpoint the quantum states and measurements that yield the maximum violations of some of the noncontextuality inequalities, as determined by the see-saw optimization technique for two-dimensional quantum systems. The symbols $\diamond$ and $\star$ represent the indistinguishable mixed state and the indistinguishable measurement effects in the respective scenario. In figure (\ref{['sc9fig']}), the length of the Bloch vectors representing $M_{0|0}$ and $M_{1|1}$ is approximately $0.3689$ and the length of the Bloch vectors representing $M_{1|0},M_{0|1}$ is approximately $0.674$.
• Figure 3: Randomness ($H_{min}$) as a function of $\mathcal{I}_7 \in [1.5,1.7321]$ from Table \ref{['sc7table']}.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof