Resource Theory of Causal Influence

• RTCaus is a framework that quantifies causal influence as a physical resource by integrating information-theoretic, functional, and process-theoretic methods in both classical and quantum settings.
• It employs rigorous measures such as relative entropy and signalling robustness to assess causal strength, ensuring operational consistency and enabling resource conversion and distillation.
• RTCaus defines free operations and monotonicity conditions, providing clear guidelines for manipulating causal structures while addressing limitations of traditional causal inference approaches.

The resource theory of causal influence (RTCaus) provides a rigorous, quantitative, and operational framework for treating causal strength as a resource, both in classical and quantum domains. It unifies several approaches—including entropy-based, functional, and process-theoretic methods—into a consistent theory that clarifies which features of a system’s causal architecture can be viewed, manipulated, or converted as valuable physical resources.

1. Formal Definition and Motivating Frameworks

RTCaus formalizes the quantification of causal strength using information-theoretic or functional measures, and supports a resource-theoretic structure on allowable operations and monotones. The fundamental setting is a composite system with well-specified causal relations (a directed acyclic graph (DAG) in the classical case or a quantum process matrix), for which one seeks to quantify and manipulate “how much” causal power flows along a given arrow or through a channel.

In the classical case, as elaborated in “Quantifying causal influences” (Janzing et al., 2012), the resource is a triple $(G, P, S)$: a DAG $G$ with variables $(X_1, \dots, X_n)$, a joint distribution $P$ realizing the Markov factorization, and a set of edges $S$. To measure the causal strength of $S$, one considers the impact of cutting those edges and feeding in statistically independent inputs. The quantitative measure is the relative entropy

$C_S(P) := D(P \| P_S)$

where $P_S$ is the distribution after cutting edges in $S$. This definition satisfies a set of axiomatic postulates—CMC-consistency, mutual information recovery for single arrows, locality, a lower bound with conditional mutual information, and heredity—that together single out $C_S$ as the unique measure meeting these natural criteria.

In the quantum and process-theoretic generalization, as developed in “Resource theory of causal connection” (Milz et al., 2021), the basic objects are quantum processes or process matrices (generalized channels). Here, the resource is the capacity of one party to send information to another—formalized as the possibility of signalling—and quantified by monotones like the signalling robustness $R_s$.

A complementary functional approach is presented in “The resource theory of causal influence and knowledge of causal influence” (Ansanelli et al., 12 Dec 2025), which focuses on the compositional and convertibility structure for deterministic functions and distributions over causal mechanisms, with particular clarity in the two-variable, no-hidden-confounders regime.

2. Quantitative Measures of Causal Strength

For classical systems (DAGs with joint distributions), the primary monotone is the relative entropy between the true joint law and the distribution with specified causal edges cut: $C_{i \to j}(P) = D(P \| P^{\mathrm{do}(i \to j)})$ This coincides with mutual information in the two-node case (Postulate P1), is locally computable (P2), lower bounds conditional mutual information (P3), and is strictly monotonic under edge deletion and mixing (heredity, P4) (Janzing et al., 2012).

For deterministic functional dependence, as in (Ansanelli et al., 12 Dec 2025), the monotone is $\log_2|\mathrm{Im}(f)|$, corresponding to the “causal range” of $f: X \to Y$. This quantity decreases under pre- and post-processing by deterministic functions, and characterizes the convertibility relation: $f$ can be converted into $g$ by free operations iff $|\mathrm{Im}(f)| \geq |\mathrm{Im}(g)|$.

When the causal mechanism is uncertain (a distribution over functions), a complete set of monotones is required. In the binary case, these are the weight on causally connected functions $\beta$, the polarization $|\alpha|$ between identity and flip, and a quantity $M_{|\gamma|,\beta}$ capturing residual polarization in the reset sector (Ansanelli et al., 12 Dec 2025).

In the quantum resource theory, the signalling robustness for a process $W$ is defined as: $$R_s(W) = \min \left\{ \lambda \geq 0: \frac{W+\lambda T}{1+\lambda} \in \mathsf{Free},\; T \in \mathsf{Proc} \right\}$$ where the free set consists of non-signalling (parallel) processes (Milz et al., 2021). This measure is convex and monotonic under non-signalling adapters.

A separate approach distinguishes “signalling” and “causal influence” for quantum gates, providing two distinct measures $\Sigma(U)$ and $C(U)$ for a bipartite unitary $U$, both defined using the diamond norm and characterized operationally; SWAP achieves maximal causal influence, while CNOT exhibits a finite gap between causal influence and signalling for single use, which vanishes in the asymptotic many-use limit (Barsse et al., 20 May 2025).

3. Resource Theoretic Structure and Free Operations

The resource theory is organized by specifying which transformations (“free operations”) can be performed at no cost. In the classical setting, these are edge deletions and probabilistic mixing of DAGs/laws (Janzing et al., 2012), or, for functions, pre- and post-processing by (possibly random) deterministic functions that cannot increase the output image size (Ansanelli et al., 12 Dec 2025).

For process-theoretic and quantum settings, three nested families of free adapters are defined:

• Admissible Free-Preserving ($\Theta_{AFP}$): All supermaps that map $\mathsf{Free} \to \mathsf{Free}$.
• Non-Signalling ($\Theta_{NS}$): Adapters respecting no-signalling from Alice to Bob and vice versa, with linear constraints on their Choi matrices.
• Local Operations with Shared Entanglement ($\Theta_{LOSE}$): Compositions of local CPTP maps and pre-shared entanglement.

Monotonicity of resource monotones under these adapters is ensured, and the causal order is preserved under non-signalling adapters. For function-based RTCaus, resource convertibility and equivalence classes are characterized by the monotones, with the non-total pre-order necessitating a complete tuple in the binary knowledge case (Ansanelli et al., 12 Dec 2025).

4. Resource Conversion, Distillation, and Maximal Resources

Resource conversion scenarios—such as parallelization and distillation—are central to RTCaus:

• Parallelization: Combining several independent causal channels to boost an edge’s effective strength; the relative entropy monotone is superadditive for parallel edges.
• Distillation: Concentrating many weak instances of causal influence into fewer, more deterministic uses, with the asymptotic rate corresponding to the channel capacity:

$$\lim_{n\to\infty} \frac{1}{n} C_{\{\text{n copies}\}}(P^{\otimes n}) = \max_{Q(X_i)} I_{Q \otimes P}(X_i; X_j)$$

• Maximally resourceful processes: In the ordered quantum setting, “chain of identities” combs (e.g., the process sending full information deterministically along the arrow) saturate dimension-dependent bounds for signalling robustness (Milz et al., 2021).

A key observation is that in scenarios with indefinite causal order, no universal maximal resource exists. Causal non-separability generates only a partial order among resources, which cannot be totally ordered by any single monotone (Milz et al., 2021).

5. Comparison with Alternative Notions and Failure Modes

RTCaus addresses limitations in earlier quantifications of causal influence:

• Average Causal Effect (ACE) and ANOVA fail in the presence of more complex dependencies, such as XOR relations or strong confounding, and generally violate the postulates required for a robust resource theory.
• Mutual Information and Conditional Mutual Information are not sufficient in the presence of confounding or copy-chain topologies, as they can be nonzero even when no direct causal link exists.
• Transfer Entropy and Directed Information do not capture causal influence in copy-chains or fork structures where deterministic dependencies can mask or eliminate apparent signalling (Janzing et al., 2012).
• Distinction of Signalling and Causal Influence: In the quantum setting, there is a provable gap between the ability to signal (as in channel discrimination) and causal influence (which includes the effect of local operations on distant correlations), as shown for CNOT (Barsse et al., 20 May 2025). This gap closes only in the asymptotic regime.

RTCaus, by contrast, ensures all structural postulates are satisfied and confers operational monotonicity to its monotones.

6. Special Cases: Functional and Knowledge-Theoretic RTCaus

In the two-variable, no-confounders regime, the resource is a deterministic mapping $f: X \to Y$, or (in the knowledge-theoretic setting) a distribution over such functions (Ansanelli et al., 12 Dec 2025). Monotonicity is determined by image size and—when knowledge is incomplete—by a minimal complete set of monotones ($\beta, |\alpha|, M_{|\gamma|,\beta}$) in the binary case, capturing all essential trade-offs between causal influence, bias, and memory in the resource.

The convertibility of resources is reduced to image cardinality in the known function scenario, and to linear programming over convex polytopes for distributions in the knowledge setting.

7. Extensions, Open Directions, and Significance

RTCaus facilitates the analysis of causal strength in systems ranging from classical stochastic processes, through discrete functional relations, to general quantum processes. It supports operational tasks such as resource conversion, distillation, and discrimination, and provides a set of sharp no-go theorems for total ordering among resources, uniqueness of maximal elements, and sufficiency of monotones (Janzing et al., 2012, Milz et al., 2021, Ansanelli et al., 12 Dec 2025, Barsse et al., 20 May 2025).

Notable extensions include:

• Asymptotic resource conversion rates and their analogy to channel coding.
• The delineation between causally separable and non-separable processes as separate resource theories, each with its own free operations and robustness monotones (Milz et al., 2021).
• A precise distinction between signalling and more general causal influence in bipartite quantum channels, with characterization of the operational gap (Barsse et al., 20 May 2025).

In all cases, RTCaus provides a mathematically and operationally robust framework for assessing, quantifying, and transforming causal structure as a physical resource.