Operational Derivation of Born's Rule from Causal Consistency in Generalized Probabilistic Theories

Abstract: We present an operational derivation of Born's rule within finite-dimensional generalized probabilistic theories (GPTs), without assuming Hilbert-space structure. From a single causal requirement, namely causal consistency, together with sharp measurements, reversible symmetries, and no-signaling, we show that any admissible state-to-probability map must be affine under mixing; otherwise, its curvature enables superluminal signaling via steering. Using standard reconstruction results, affinity forces the probability assignment to coincide with the quadratic transition function of complex quantum theory. Our three-stage argument (operational assignment, causal-consistency constraints, and structural reconstruction) recovers complex quantum theory and identifies Born's rule as a causal fixed point among admissible probabilistic laws. We discuss limitations of the derivation and outline steering-based experiments that could bound deviations from affinity.

• The paper's main contribution is the derivation of Born's rule as the unique linear probability assignment enforced by causal consistency in GPTs.
• It employs operational axioms such as no-signaling, purification, and sharp measurements to connect quantum structure with probabilistic theory.
• The findings imply that any deviation from linearity would yield observable signaling, highlighting the robustness of quantum probability.

Operational Derivation of Born’s Rule from Causal Consistency in Generalized Probabilistic Theories

Introduction

The paper "Operational Derivation of Born's Rule from Causal Consistency in Generalized Probabilistic Theories" (2512.12636) addresses the longstanding foundational question regarding the origin of the quadratic Born rule $P(i) = |\langle \phi_i | \psi \rangle|^2$ in quantum theory. Traditional derivations—such as Gleason’s theorem, decision-theoretic approaches, and envariance—presuppose Hilbert-space geometry or introduce interpretational overhead. This work employs the framework of finite-dimensional Generalized Probabilistic Theories (GPTs) and identifies the origin of Born's rule with causal-consistency constraints, specifically the no-signaling principle, in conjunction with operational mixing (affinity), reversible symmetries, and sharp measurements. The analysis strictly decouples the operational probability assignment from quantum structural reconstruction, revealing that the quadratic Born rule emerges as a causal fixed point among admissible probabilistic laws in GPTs possessing purification and entanglement.

GPT Framework and Operational Foundations

GPTs generalize both classical and quantum probabilistic models by formulating states as points in a convex set and effects as affine functionals, emphasizing operational accessibility. The state space $\Omega$ resides in a finite-dimensional ordered vector space $V$ with a positivity cone $V^+$, and measurements are convex decompositions over this set. Composite systems are required to be locally tomographic—joint statistics fully specify the joint state.

The paper adopts six core operational axioms standard in contemporary GPT reconstructions:

1. No Superluminal Signaling (NSS): Local outcome distributions are invariant under remote measurement choices.
2. Local Tomography (LT): Composite states are characterized entirely by local joint statistics.
3. Purification: Every mixed state admits a purification, unique up to reversible transformations.
4. Continuous Reversibility: The pure state manifold is connected via reversible transformations forming a continuous group.
5. Spectrality/Sharpness: Any state decomposes into perfectly distinguishable pure states; sharply distinguishing measurements exist.
6. Strong Symmetry: All maximal distinguishable sets of pure states are interrelated by reversible operations.

These axioms ensure the presence of entanglement, reversible physical dynamics, and a classical-like structure for measurements, forming the basis for information processing in GPTs.

Operational Transition Probability

Central to the derivation is the operational transition probability $\tau(\psi, \phi)$, defined as the maximal acceptance probability of a state $\psi$ in a test designed to accept another pure state $\phi$ with certainty:

$$\tau(\psi, \phi) = \sup\{ e(\psi) : e(\phi) = 1, \ e \text{ is an effect} \}$$

This generalizes $|\langle \phi | \psi \rangle|^2$ in quantum theory but is defined strictly in terms of operational measurement and convex structure, independent of any Hilbert-space machinery.

Key properties, under the axioms, include:

• $\tau(\psi, \phi) \in [0, 1]$, with equality to $1$ if and only if $\psi = \phi$.
• For sharp two-outcome measurements separating $\phi$ and its orthogonal $\phi^\perp$, sum rules $\tau(\psi, \phi) + \tau(\psi, \phi^\perp) = 1$ hold.

Causal Consistency and Uniqueness of the Probability Law

The assignment of probabilities to measurement outcomes, given by $P(\phi | \psi) = \Phi(\tau(\psi, \phi))$ for some admissible function $\Phi$, is permitted to be arbitrary, subject only to normalization and monotonicity. The uniqueness result follows from enforcing:

• Normalization on sharp tests: For any $\psi$, the sum over outcome probabilities for measurements distinguishing $\phi, \phi^\perp$ is $1$, i.e., $\Phi(p) + \Phi(1 - p) = 1$.
• Operational affinity: Probability assignments must be affine under convex mixing (coarse-graining) of states.
• No-signaling: Local marginals cannot be affected by remote actions.

The combination of affinity and normalization forces $\Phi$ to be linear, i.e., $\Phi(p) = p$. If $\Phi$ deviates from affinity (i.e., is nonlinear), steering-based protocols exploiting ensemble decompositions and sharp effects generate operationally detectable superluminal signaling, thereby violating NSS. Thus, only the identity mapping is causally consistent, establishing $P(\phi | \psi) = \tau(\psi, \phi)$.

Structural Reconstruction: From GPTs to Quantum Theory

Having fixed the probability rule, established reconstruction theorems guarantee that any finite-dimensional GPT satisfying local tomography, purification, continuous reversibility, spectrality, and strong symmetry is operationally equivalent to complex quantum theory. Self-duality of the state cone, spectral decomposability, and homogeneity ensure that the transition probability $\tau$, now representing true quantum overlaps, corresponds precisely to $|\langle \phi | \psi \rangle|^2$.

Consequently, Born’s rule is not an artifact of Hilbert-space structure, but an inevitability due to the causal and operational demands encapsulated by the framework.

Implications and Future Directions

Theoretical Implications

This derivation subsumes previous approaches by reducing reliance on Hilbert spaces or non-operational premises. The quadratic nature of quantum probabilities emerges as a direct requirement of causal consistency and operational affinity in the presence of entanglement and purification. The GPT approach indicates that post-quantum probability laws generically threaten relativistic causality unless constrained as above. The separation between operational assignments and quantum structure clarifies foundational roles: causal principles fix the rule, reconstruction programs identify the underlying algebraic and geometric structure.

Practical Implications

Experimentally, any non-affine deviation from Born’s rule predicts observable signaling in appropriately designed steering experiments. The approach not only identifies potential avenues for falsification but also quantifies the permissible “robustness” of quantum probability laws under hypothetical post-quantum additions.

Extensions

The finite-dimensionality condition is a technical limitation; the extension to infinite-dimensional systems and quantum field theory would require new analytical tools, possibly involving topological GPTs or generalizations of the operational axioms. The necessity and sufficiency of each axiom in enforcing causal consistency can be further scrutinized. Future work may also address the minimal operational foundations required for the uniqueness of the probability law and quantify empirical bounds for deviation from affinity.

Conclusion

The paper provides a rigorous operational derivation of Born's rule in finite-dimensional GPTs, establishing that causal consistency, operational affinity, and sharp measurement normalization uniquely mandate the quadratic probability law. Standard reconstruction theorems subsequently yield complex quantum theory. Any departure from affinity under mixing inevitably triggers operationally detectable signaling via steering, stabilizing the Born rule as a causal fixed point. This consolidates the universality of quantum probabilities as a fundamental consequence of information-theoretic and relativistic principles rather than the peculiarities of Hilbert-space structure.