Contextuality from the Projector Overlap Matrix

• URL: http://arxiv.org/abs/2604.23898v2
• Date: Wed, 29 Apr 2026 06:28:14 GMT
• Title: Contextuality from the Projector Overlap Matrix
• Abstract summary: We place several known indicators of Kochen--Specker contextuality into a single projector-geometric framework.<n>$Tcal_ij = d-1tr[(hat P_i hat Q_j)2]$, where $hat P_i$ and $hat Q_j$ are the joint-eigenspace projectors of the two compatible observable pairs.<n>We show that in the spin-$1$ realization of the KCBS pentagon, a shared $m_s=0$ eigenstate in each context forces $c_MU =
• Score: 0.6372261626436676
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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