Contextuality from the Projector Overlap Matrix
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arXiv
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2026
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| author | Günhan, Ali Can Aksoy, Semahi Serhat Gedik, Zafer |
| author_facet | Günhan, Ali Can Aksoy, Semahi Serhat Gedik, Zafer |
| contents | We place several known indicators of Kochen--Specker contextuality -- the KCBS correlator $χ$, the contextual fraction $\CF$, the Shannon-entropic $n$-cycle inequality of Chaves and Fritz, and the operational commutator witness $D$ of Paper~I -- into a single projector-geometric framework organized around the overlap matrix $\Tcal_{ij} = d^{-1}\tr[(\hat P_i \hat Q_j)^2]$, where $\hat P_i$ and $\hat Q_j$ are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\Tcal$ is carried by two independent contractions: the mutual information energy $E = \sum_{ij}\Tcal_{ij}$ of Paper~I (equivalently, its logarithmic form $S_2 = -\log_2 E$), and the Maassen--Uffink extremal overlap $c_\MU = \max_{i,j}|\langle a_i,b_i | c_j,b_j\rangle|$. We prove that $S_2$ is non-increasing under coarse-graining, that $S_2(\Gcal) > 0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\Gcal) = \sum_αS_2(\Gcal_α)$ is exact for the KCBS pentagon. We further show that in the spin-$1$ realization of the KCBS pentagon, a shared $m_s=0$ eigenstate in each context forces $c_\MU = 1$, rendering every Maassen--Uffink-type bound trivial -- a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_\MU$ are silent on KCBS contextuality, while $S_2 \approx 2.7266$~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\Gcal) > 0$ by an amount set by the projector geometry alone. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23898 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Contextuality from the Projector Overlap Matrix Günhan, Ali Can Aksoy, Semahi Serhat Gedik, Zafer Quantum Physics We place several known indicators of Kochen--Specker contextuality -- the KCBS correlator $χ$, the contextual fraction $\CF$, the Shannon-entropic $n$-cycle inequality of Chaves and Fritz, and the operational commutator witness $D$ of Paper~I -- into a single projector-geometric framework organized around the overlap matrix $\Tcal_{ij} = d^{-1}\tr[(\hat P_i \hat Q_j)^2]$, where $\hat P_i$ and $\hat Q_j$ are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\Tcal$ is carried by two independent contractions: the mutual information energy $E = \sum_{ij}\Tcal_{ij}$ of Paper~I (equivalently, its logarithmic form $S_2 = -\log_2 E$), and the Maassen--Uffink extremal overlap $c_\MU = \max_{i,j}|\langle a_i,b_i | c_j,b_j\rangle|$. We prove that $S_2$ is non-increasing under coarse-graining, that $S_2(\Gcal) > 0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\Gcal) = \sum_αS_2(\Gcal_α)$ is exact for the KCBS pentagon. We further show that in the spin-$1$ realization of the KCBS pentagon, a shared $m_s=0$ eigenstate in each context forces $c_\MU = 1$, rendering every Maassen--Uffink-type bound trivial -- a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_\MU$ are silent on KCBS contextuality, while $S_2 \approx 2.7266$~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\Gcal) > 0$ by an amount set by the projector geometry alone. |
| title | Contextuality from the Projector Overlap Matrix |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2604.23898 |