Contextuality from the Projector Overlap Matrix

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.

• The paper introduces a unifying projector-geometric framework based on the projector overlap matrix to systematically assess both state-dependent and configuration-level quantum contextuality.
• The paper demonstrates that scalar contractions, including the quadratic contraction S2 and the Maassen–Uffink overlap, reveal critical geometric obstructions and monotonicity properties in contextuality scenarios.
• The paper reveals that exact additive decompositions in KCBS and CHSH scenarios enable a clear distinction between classical and nonclassical configurations, informing resource-theoretic insights.

Contextuality and Scalar Projector Overlap: A Projector-Geometric Framework

Introduction

This paper synthesizes and extends multiple operational and analytical approaches to quantum contextuality by introducing a unifying projector-geometric framework centered on the projector overlap matrix $$T_{ij} = d^{-1}\operatorname{Tr}\big[(\hat P_i \hat Q_j)^2\big]$$. Here, $\hat P_i$ and $\hat Q_j$ denote joint-eigenspace projectors associated with compatible observable pairs within a measurement context on a Hilbert space of dimension $d \ge 3$. The framework systematizes state-dependent and configuration-level (state-independent) contextuality indicators—most notably, the KCBS correlator, the contextual fraction, various entropic inequalities, and a direct commutator-based witness—within two scalar contractions of $T$: a quadratic contraction (mutual information energy $E$, or equivalently $S_2 = -\log_2 E$) and the Maassen–Uffink extremal overlap $c_{MU}$. The approach establishes structural, operational, and informational relations between these scalar quantities and the activation or silence of quantum contextuality indicators, with explicit analyses for the KCBS pentagon and CHSH scenarios.

The Projector Overlap Matrix and Scalar Contractions

Let $\{G_\alpha\}$ denote a contextuality scenario, where each context comprises a set of pairwise compatible observables $\{\hat A, \hat B, \hat C\}$ supporting complete families of joint-eigenspace projectors $\hat P_i$0 and $\hat P_i$1. The projector overlap matrix $\hat P_i$2, defined above, encodes the configuration-level geometry of the measurement setting.

The operational content of $\hat P_i$3 is captured by two scalar contractions:

• Quadratic contraction ($\hat P_i$4 and $\hat P_i$5): $\hat P_i$6 (mutual information energy (Günhan et al., 11 Dec 2025)) detects nontrivial geometric obstruction to noncontextuality at the configuration level. $\hat P_i$7's logarithmic form yields additive composition over contexts and a compelling comparison with entropic uncertainty relations.
• Maassen–Uffink extremal overlap ($\hat P_i$8): $\hat P_i$9 governs the tightest lower bound for outcome entropic uncertainty relations.

A fundamental inequality is established: $\hat Q_j$0, with saturation contingent on the structure of the measurement configuration.

Key Theoretical Results

1. Monotonicity under Coarse-Graining: $\hat Q_j$1 is non-increasing under coarse-graining (contractions) of the projector families. This monotonicity, reminiscent of resource-theoretic monotonicity, is restricted to this projector-geometric framework and is not guaranteed under arbitrary noncontextual wirings.
2. Necessity Condition: $\hat Q_j$2 is necessary and sufficient for classical (noncontextual hidden variable) realizability of the scenario. Any configuration with $\hat Q_j$3 is a prerequisite for observing contextuality via any quantum state.
3. Exact Additive Decomposition: For the KCBS pentagon and CHSH 4-cycle, $\hat Q_j$4 is exact. In KCBS, this is a consequence of cyclic orthogonality of projectors; in CHSH, it arises due to the distinct bases induced by Alice/Bob subsystem alternation.

State-Dependent vs. Configuration-Level Contextuality Witnesses

The paper explicitly compares standard state-dependent witnesses and configuration-level (state-independent) indicators, tabulating their values and activation patterns in representative scenarios.

• KCBS Scenario (spin-1 realization): For the mixing family $\hat Q_j$5, the KCBS correlator $\hat Q_j$6 and contextual fraction $\hat Q_j$7 are active only above a threshold $\hat Q_j$8, while standard entropic contextuality inequalities ($\hat Q_j$9) and the direct commutator witness $d \ge 3$0 are structurally or symmetrically silent. Critically, $d \ge 3$1 bits throughout the entire scenario, independent of quantum state.
• Structural Mechanism for $d \ge 3$2: The KCBS pentagon admits a shared $d \ge 3$3 eigenstate in each context, enforcing $d \ge 3$4 and rendering all Maassen–Uffink-type entropic uncertainty relations trivial. This outcome, proven in the general spin-1 realization and numerically on higher odd-$d \ge 3$5-cycle extensions, explains the absence of entropic contextuality violations in these scenarios.
• CHSH Scenario: At Bell-optimal measurement angles, the CHSH correlator $d \ge 3$6 achieves maximal quantum (Tsirelson) violation, while the entropic witness $d \ge 3$7 remains well within the classical bound. At entropic-optimal angles (identified by Chaves), the situation reverses: the CHSH value is subcritical but the entropic witness is active. In both cases, $d \ge 3$8 is strictly positive and sensitive to measurement configuration, not state. The scenario also demonstrates both saturated ($d \ge 3$9) and unsaturated geometries, depending on precise angle choices.

Implications and Theoretical Significance

This framework establishes that genuine quantum contextuality, as obstructiveness to classical global sections or noncontextual assignments, admits a geometric signature independent of quantum state through $T$0. Configuration-level contextuality is thus strictly broader than any state-dependent witness: $T$1 is necessary for the activation of state-dependent indicators, but the converse is not true. There exist configurations where all standard state-dependent witnesses are silent, yet the underlying projector geometry is irremediably contextual.

This clarifies the information-theoretic and operational content of contextuality: state-dependent indicators are necessary for empirical violation, but they probe only a subspace of the configuration's nonclassicality. The geometric quantity $T$2 registers a fundamental obstruction invisible to any measurement scenario whose configuration $T$3 has $T$4.

For resource theory, the monotonicity under projector-family coarse-graining and the existence of analytic (as opposed to purely combinatorial) invariants open the door to context-independent monotones that could serve as resource parameters. However, the full status of $T$5 under general noncontextual wirings and its operational or computational role—especially in MBQC and nonclassical computation—remains open.

Future Directions

• Alternative Contractions: Further analysis of $T$6-norms, spectral invariants, and normalized variants of $T$7 could yield finer discrimination of contextuality or monotonicity properties.
• Extension to State-Independent Scenarios: Generalizing the framework to, e.g., Peres–Mermin square, Yu–Oh configurations, or grid-based KS proofs may elucidate the structure of additivity and geometric versus combinatorial obstructions.
• Refined Uncertainty Relations: Exploring entropic or geometric uncertainty relations where the lower bound involves $T$8 rather than $T$9 may uncover nontrivial contextuality witnesses valid even in shared-eigenstate scenarios.

Conclusion

By systematizing both operational and geometric approaches to contextuality through the projector overlap matrix and its scalar contractions, this work delineates the boundary between state-dependent and configuration-level contextuality diagnostics. $E$0 emerges as a robust analytic witness—faithfully detecting geometric contextuality irrespective of quantum state—and highlights structural mechanisms leading to the silence of entropic (Maassen–Uffink) bounds in key scenarios. The work provides new tools for dissecting the interplay between quantum configuration geometry and observable nonclassicality, with significant implications for contextuality-based quantum advantage in computation and information.

Reference: "Contextuality from the Projector Overlap Matrix" (2604.23898)