Degrees, Levels, and Profiles of Contextuality

1. Problem Statement

Current approaches to quantifying contextuality in systems of random variables typically rely on a single scalar value representing the "degree of contextuality." While this distinguishes contextual systems from non-contextual ones, it fails to capture the structural complexity of how contextuality emerges as the system is analyzed at different scales.

Specifically, existing methods often stop analysis at the "first level of contextuality" (the lowest order $n$ where the system becomes contextual). The authors argue that this is insufficient because:

1. It ignores how contextuality evolves as higher-order joint distributions (marginals) are included.
2. It treats systems with identical final degrees of contextuality as indistinguishable, even if their internal structures differ.
3. There is a need to compare different measures of contextuality (e.g., distance-based vs. quasi-probability-based) not just by their final values, but by how they respond to increasing levels of system complexity.

2. Methodology

A. Level-wise Representations

The authors introduce a framework where a system $R$ is represented at different levels $n$ ($1 \le n \le N$, where $N$ is the maximum number of variables in a context).

• Level $n$ Representation ($R[n]$): The system is characterized by all joint distributions of $k$ variables where $k \le n$.
• Construction: To create $R[n]$ from a row of variables, the authors generate all possible subsets of size $n$ (and lower, cumulatively) and create distributional copies.
• Properties:
  • Level 1 is always non-contextual (degree = 0).
  • For $n > 1$, the representation includes $n$-tuples.
  • If $n$ exceeds the number of variables in a specific context, the representation stabilizes (coincides with the full system).

B. Contextuality Profiles

Instead of a single number, a system is characterized by a contextuality profile: a vector (or curve) of contextuality degrees $d_n$ across levels $n=1$ to $N$.
$$\text{Profile} = (d_1, d_2, \dots, d_N)$$
where $d_n$ is the degree of contextuality calculated for the level-$n$ representation using a specific measure.

C. Concatenated Systems

To analyze how contextuality grows between levels, the authors propose a concatenation method:

1. System A: A system with a known profile where contextuality exists at level $n$ (degree $d_n$).
2. System B: A system that is non-contextual at all levels up to $n$, but becomes contextual only at level $n+1$ (contributing an increment $\Delta_{n+1}$).
3. Concatenation ($A * B$): System $A$ and $B$ are combined such that their questions and contexts are disjoint.
   • At level $n$, the combined system's contextuality is determined solely by $A$ (since $B$ is non-contextual).
   • At level $n+1$, the combined system includes the $(n+1)$-tuples from $B$.
4. Analysis: By comparing the resulting degree $d_{n+1}$ of the combined system against the sum $d_n + \Delta_{n+1}$, the authors determine if the measure is additive, superadditive, or subadditive.

D. Measures Evaluated

The study applies this framework to three established measures of contextuality (extended to disturbed systems):

1. CNT2 (Distance Measure): Based on the $L_1$-distance between the system's probability vector and the non-contextuality polytope.
2. CNT3 (Quasi-Probability Measure): Based on the minimal sum of absolute values of a solution to the coupling matrix equation (allowing negative probabilities).
3. CNTF (Contextual Fraction): Based on the maximal weight of a non-contextual component in a decomposition of the system.

3. Key Contributions

1. Definition of Contextuality Profiles: The paper formalizes the concept of a profile, shifting the analysis from a static scalar to a dynamic function $n \mapsto d_n$.
2. Concatenation Methodology: A novel experimental design using disjoint concatenated systems to isolate the contribution of specific variable orders (e.g., pairs vs. triples) to the total contextuality.
3. Characterization of Measure Behavior: The authors rigorously demonstrate that the three major measures behave fundamentally differently regarding how contextuality accumulates:
   • CNT2 is Additive: The degree of contextuality increases linearly with the addition of new independent contextual components.
   • CNT3 and CNTF are Subadditive (Rule of Maximum): The degree of contextuality for the combined system is determined by the maximum of the individual components, not their sum.
4. Independence of Measures: The paper proves that these three measures are not functionally dependent on one another, even within the same system at different levels. A system can change its profile in one measure while remaining constant in another.

4. Results

• Additivity of CNT2: For the distance measure ($L_1$), the contextuality of a concatenated system $A * B$ at level $n+1$ is exactly the sum of the contextuality of $A$ at level $n$ and the new contribution from $B$.
  $$d_{n+1}(A*B) = d_n(A) + \Delta_{n+1}(B)$$
  This is attributed to the orthogonal nature of the subspaces (pairwise vs. triple probabilities) in the $L_1$ metric.

• Subadditivity of CNT3 and CNTF: For the quasi-probability and contextual fraction measures, the contextuality follows a "Rule of Maximum":
  $$d_{n+1}(A*B) = \max(d_n(A), \Delta_{n+1}(B))$$
  Consequently, if a system already has a high degree of contextuality at level $n$, adding new contextual components at level $n+1$ often results in a plateau (no increase in the measured degree), rather than an increase. This explains why these measures often show "flat" profiles compared to the distance measure.

• Hypercyclic Systems Analysis: The authors applied these methods to hypercyclic systems (highly structured cyclic systems). They confirmed that:

  • Undisturbed hypercyclic systems are non-contextual at all levels except the final one.
  • The profiles generated by the three measures are distinct; no single measure can predict the profile of another.

5. Significance

• Granular Understanding: The concept of contextuality profiles provides a much richer description of quantum and classical systems than a single scalar. It reveals at which structural level (pairs, triples, etc.) contextuality arises.
• Measure Selection: The results highlight that the choice of measure is critical. If a researcher is interested in the cumulative effect of contextuality, the $L_1$-distance (CNT2) is appropriate. If they are interested in the "bottleneck" or the strongest source of contextuality, CNT3 or CNTF may be more suitable.
• Resource Theory Implications: The authors suggest that contextuality profiles could offer new insights into resource theories (e.g., in quantum computing), where different levels of contextuality might correspond to different computational resources.
• Disturbance Handling: The framework successfully extends to "disturbed" systems (where marginal distributions change across contexts), a common scenario in empirical data that previous level-wise analyses struggled to handle uniformly.

In conclusion, the paper establishes that contextuality is not a monolithic property but a multi-layered phenomenon. The "profile" approach allows researchers to map the topology of contextuality, revealing that different mathematical measures capture fundamentally different aspects of this phenomenon.