RTKnowCaus: Theory of Causal Knowledge

• RTKnowCaus is a theoretical framework that quantifies knowledge about causal influence using constructs such as quantum assemblages and probabilistic functional dependencies.
• It delineates free operations and resource monotones that maintain epistemic constraints, ensuring that classical and nonclassical knowledge are rigorously distinguished.
• The framework has practical implications in quantum cryptography, AI causality analysis, and networked reasoning by formalizing how agents assess and convert causal knowledge.

The Resource Theory of Knowledge of Causal Influence (RTKnowCaus) is a collection of mathematical frameworks that formalize how knowledge about causal relationships—not direct causal power itself—can be systematically quantified, manipulated, and compared. Developed across quantum information, causal inference, and categorical approaches, RTKnowCaus analyzes the limits and value of the knowledge possessed by agents about potential causal influence under well-specified operational constraints. The theory generalizes conventional resource theories, such as those for entanglement or signaling, by treating knowledge states (rather than physical states) as primary resources and delineating free operations that reflect natural epistemic or operational limitations. Key instantiations include quantum assemblage-based frameworks, convex measure encodings for networked causal structures, and linear-program-based orderings for classical knowledge in deterministic functional settings (Zjawin et al., 2021, D'Acunto et al., 13 Mar 2025, Ansanelli et al., 12 Dec 2025).

1. Fundamental Resource Structures: Assemblages and Probability Distributions

RTKnowCaus formalizes objects of knowledge in varying domains:

• Quantum Frameworks: Here, the central resource is a set of assemblages $\{\sigma_{a|x}\}$, indexed by measurement settings $x$ and outcomes $a$, where each $\sigma_{a|x}$ is an unnormalized density operator representing an inference about a quantum system given observed outcomes and settings. Assemblages must satisfy positivity and normalization constraints (tr$(\sigma_{a|x})\geq 0$, $\sum_a \mathrm{tr}[\sigma_{a|x}] = 1$ for each $x$). Assemblages admit quantum realizations through measurements on entangled states, but unsteerable (classical) assemblages are those admitting local-hidden-state (LHS) decompositions (Zjawin et al., 2021).
• Classical/Networked Frameworks: Knowledge states are represented by probability distributions over deterministic functional dependencies (e.g., for $X\rightarrow Y$, a distribution $P_F$ over functions $f:X\to Y$) or more generally, by convex sets of observational and interventional measures attached to nodes and edges in a network, as in the functor category construction for structural causal models (SCMs) (D'Acunto et al., 13 Mar 2025, Ansanelli et al., 12 Dec 2025). These representations allow uncertainty and abstraction in the knowledge of causal mechanisms.

2. Free Resources and Free Operations

The notion of "free” in RTKnowCaus is operationally defined and context-dependent:

• Quantum Setting: Free assemblages are those explainable via classical common causes (LHS models). The free operations are Local Operations and Shared Randomness (LOSR), encompassing classical post- and pre-processing of settings and outcomes, and CPTP maps on the system correlated only via shared randomness. LOSR cannot generate nonclassical correlations and preserves classicality (Zjawin et al., 2021).
• Classical/Networked Setting: Free knowledge states are convex mixtures supported only on constant functions (zero causal influence). Free operations are convex combinations of common-cause combs—preprocessing and postprocessing maps that do not induce direct causal influence—plus convex mixing of measures. In networked scenarios, free operations correspond to interventional-consistent (IC) abstractions and push-forward affine maps arising from abstraction of SCMs (D'Acunto et al., 13 Mar 2025, Ansanelli et al., 12 Dec 2025).

This guarantees that free operations cannot generate knowledge of nontrivial causal influence where none was known.

3. Quantification and Monotones

Resource monotones in RTKnowCaus are functions on resources that do not increase under free operations, providing partial or total orderings and measures of nonclassical or causal knowledge.

• Quantum Assemblage Monotones: Steerable weight and robustness of steering are classic monotones, with new Bell-derived "yield" monotones $M_\eta$ defined via lifted tilted CHSH inequalities. Each $M_\eta$ identifies a specific type of nonclassical resource, and these monotones are often strictly monotonic and can exhibit infinite incomparability—familes of resources neither convertible into each other under LOSR nor ordered by any simple scalar measure (Zjawin et al., 2021).
• Classical Binary Monotones: For binary variable scenarios, a complete set of three monotones $(M_\beta, M_{|\alpha|}, M_{|\gamma|,\beta})$ collectively characterizes the partial preorder of all knowledge states, directly determining convertibility (i.e., $P_B \to P'_B$ iff all monotones weakly decrease). Each monotone corresponds to interpretable attributes: the weight on causal knowledge, the degree of functional asymmetry, and a composite that resolves remaining ambiguities in cases of mixed knowledge (Ansanelli et al., 12 Dec 2025).
• Networked/Categorical Settings: Any convex, nonincreasing function under IC abstraction push-forwards is a monotone, including maximal intervention-effect (KL divergence), mutual information between interventions and outcomes, and total variation from observations. These quantify the discriminatory power and informativeness of knowledge transferred through the network (D'Acunto et al., 13 Mar 2025).

4. Resource Convertibility, Decision Procedures, and Order Structure

RTKnowCaus rigorously characterizes which resource states (i.e., knowledge states) can be transformed into others under free operations:

• Quantum SDPs: Convertibility between assemblages is decidable via a single semidefinite program that tests for feasible Choi operators and deterministic wirings, capturing all LOSR transformations. The decision problem is numerically tractable and, in the typical two-setting qubit scenario, shows infinite incomparability: families of resources $\Sigma^{\theta,1}$ indexed by $\theta$ are totally unordered, certified by Bell-derived monotones (Zjawin et al., 2021).
• Classical Linear Programs: Convertibility between knowledge states over functions is decided by a linear program determining the existence of convex coefficients for reproducing the target distribution from the allowed free-image polytopes. In binary cases, the triple monotone system gives a direct algebraic test for convertibility (Ansanelli et al., 12 Dec 2025).
• Networked Causal Knowledge: Transference and abstraction of causal knowledge across network nodes/edges are functorial, with sheaf and cosheaf structures determining the set of possible induced knowledge states and their monotonicity under abstraction mappings. The resource object is the convex set $\mathsf{CK}^{\rho_1, \sigma_k}$ of distributions accessible at one node given interventions at another (D'Acunto et al., 13 Mar 2025).

5. Physical and Epistemic Interpretation

RTKnowCaus distinguishes knowledge-of-causal-influence from ontic causal influence:

• In EPR and quantum steering scenarios, the theory rejects any assumption of Alice’s measurement influencing Bob’s system; Alice’s action merely updates her conditional knowledge via correlated common causes. Thus, resource value reflects epistemic nonclassicality, not signaling or physical action at a distance (Zjawin et al., 2021).
• In classical and SCM-based constructions, resource objects encode what one agent or subsystem knows about causal effects elsewhere under imposed abstraction or interventional constraints. Free resources correspond to knowledge purely explicable via shared randomness or ignorance, with nonfree resources arising from more informative, often nonclassically structured, dependencies (D'Acunto et al., 13 Mar 2025, Ansanelli et al., 12 Dec 2025).

A plausible implication is that RTKnowCaus provides a rigorous foundation for epistemic causality in contexts where the transmission or existence of causal influence is either prohibited or inherently ambiguous, supporting applications in quantum cryptography, causality analysis in AI, and distributed network reasoning.

6. Examples and Operational Significance

RTKnowCaus is illustrated concretely in binary-variable and assemblage scenarios:

• Binary Knowledge State Example: For $P_B^4 = \frac{1}{3}[F] + \frac{2}{3}[R_0]$ and $$P_B^5 = \frac{1}{3}[F] + \frac{1}{3}[R_0] + \frac{1}{3}[R_1]$$, their monotone vectors $(\beta, \alpha, \gamma)$ determine that $P_B^5 \to P_B^4$ but not vice versa, revealing strict incomparability (Ansanelli et al., 12 Dec 2025).
• Quantum Assemblage Example: The family $\{\Sigma^{\theta,1}\}$ measured in z/x bases exhibits total unorderedness under LOSR, with unique monotones $M_{\eta}$ isolating resource types—demonstrating that resource convertibility is fundamentally obstructed by epistemic constraints (Zjawin et al., 2021).
• Networked Sheaf Example: Filtering interventions across a path in a causal network produces a set of induced knowledge states at remote nodes. Mixing interventions at a source node can render them indistinguishable at other nodes under sufficient abstraction, as revealed by vanishing monotones (e.g., mutual information) (D'Acunto et al., 13 Mar 2025).

7. Connections to Broader Research Areas and Methodologies

RTKnowCaus unites resource theory concepts from quantum information, causal inference, and categorical epistemology:

• It extends quantum resource theories (entanglement, nonlocality, steering) to strictly knowledge-based operational regimes (Zjawin et al., 2021).
• The categorical approach via functor categories and sheaf theory situates causal knowledge as a distributed, relational resource within networks, aligning with contemporary approaches to AI and collaborative reasoning (D'Acunto et al., 13 Mar 2025).
• The classical probabilistic treatment with explicit monotone and LP characterizations provides direct computational and interpretive utility in applied settings (Ansanelli et al., 12 Dec 2025).

Given this synthesis, RTKnowCaus serves as a foundational platform for the analysis of causal knowledge, resource quantification, and order structure across classical, quantum, and multi-agent domains.