Any theory that admits a Wigner's Friend type multi-agent paradox is logically contextual

TL;DR

This work develops a theory-independent framework to analyze Wigner’s Friend-type multi-agent paradoxes and demonstrates that any theory admitting such paradoxes necessarily supports logically contextual empirical models. By explicitly defining compatibility of agents, generalized Heisenberg-cut notions, and a language for atomic outcomes and inferences, the authors show that paradoxes correspond to logical contextuality, linking foundational paradoxes to contextual resources. They instantiate the framework in quantum theory via a KCBS-based five-cycle (5-cycle) contextuality scenario, illustrating a concrete paradox that does not require Bell non-locality and highlighting post-selection as a necessary feature in quantum n-cycle paradoxes. The paper also derives structural results: paradoxes map to liar cycles with cyclic reference graphs, extremal vertices in n-cycle polytopes enable post-selection-free paradoxes in general theories, and quantum theory prohibits such post-selection-free n-cycle paradoxes. Collectively, these findings illuminate how multi-agent reasoning tests reveal deep, theory-dependent contextual structures and offer a pathway to classify and understand non-classical resources across physical theories.

Abstract

Wigner's Friend scenarios push the boundaries of quantum theory by modeling agents, along with their memories storing measurement outcomes, as physical quantum systems. Extending these ideas beyond quantum theory, we ask: in which physical theories, and under what assumptions, can agents who are reasoning logically about each other's measurement outcomes encounter apparent paradoxes? To address this, we prove a link between Wigner's Friend type multi-agent paradoxes and contextuality in general theories: if agents who are modeled within a physical theory come to a contradiction when reasoning using that theory (under certain assumptions on how they reason and describe measurements), then the theory must admit contextual correlations of a logical form. This also yields a link between the distinct fundamental concepts of Heisenberg cuts and measurement contexts in general theories, and in particular, implies that the quantum Frauchiger-Renner paradox is a proof of logical contextuality. Moreover, we identify structural properties of such paradoxes in general theories and specific to quantum theory. For instance, we demonstrate that theories admitting behaviors corresponding to extremal vertices of n-cycle contextuality scenarios admit Wigner's Friend type paradoxes without post-selection, and that any quantum Wigner's Friend paradox based on the n-cycle scenario must necessarily involve post-selection. Further, we construct a multi-agent paradox based on a genuine contextuality scenario involving sequential measurements on a single system, showing that Bell non-local correlations between distinct subsystems are not necessary for Wigner's Friend paradoxes. Our work offers an approach to investigate the structure of physical theories and their information-theoretic resources by means of deconstructing the assumptions underlying multi-agent physical paradoxes.

Figures (11)

• Figure 1: A simple example of a multi-agent paradox. The setting includes two agents, Alice and Bob. Alice says that Bob is lying; Bob says that Alice is telling the truth. Together, their statements are inconsistent with each other, meaning that there exist no global truth value assignment to their statements.
• Figure 2: Reference relation graphs for paradoxical chains of statements, finite and infinite.
• Figure 3: Frauchiger-Renner setup Frauchiger2018 setting and contextuality. In Frauchiger-Renner setup, as well as in Hardy's paradox, there are four contexts $C_{UB}, C_{BA}, C_{AW}, C_{WU}$ in which the outcome assignments are made. However, there exists an outcome assignment (illustrated above) which does not belong to any compatible family of assignments, as it does not agree on the intersection between contexts $C_{AW}$ and $C_{WU}$, where the value assignment on the latter is the result of post-selection. For the specific details of the setup, please refer to Figure \ref{['fig:fr-entanglement']}.
• Figure 4: Two views on measurement in quantum theory. Suppose that Bob is measuring a quantum system $S$ (measurement $\mathcal{M}_i$). There are two ways this measurement can be modelled in quantum theory, depending on whether Bob is considered to be a quantum system (under the Heisenberg cut) or not. This choice can be seen as a choice of a setting variable$s_i$. If $s_i=0$ then the evolution is modelled as a unitary one; if $s_i=1$ then it is assumed that classical outcomes are observed (identified by the measurement projectors). This formalisation of measurement models/Heisenberg cuts in terms of settings was first introduced for quantum theory in Vilasini2022. Here we consider an extension of the concept to general physical theories, and relate it to contextuality of the theory under other assumptions.
• Figure 5: Multi-agent setup. Consists of: a set of agents, a set of their memories, a set of additional systems, and a set of measurements agents perform. The measurement of one agent can act on any subset of the memories of other agents and the additional systems, and for simplicity we assume each agent carries out only one measurement. Each memory $L_i$ is initialised to some state $\rho^0_{L_i}$ and the systems start out in the initial state $\rho_{S_1,...,S_m}$.

Theorems & Definitions (31)

• definition 2: Joint measurability or compatibility of measurements
• definition 3: Measurement scenario
• definition 4: Empirical model
• definition 5: Compatible family of outcome assignments Abramsky15
• definition 6: Logical contextuality
• definition 7: Strong contextuality
• definition 8: Multi-agent setup
• definition 9: Default predictions of a multi-agent setup
• definition 10: Set of statements of a setup
• definition 11: Compatibility of measurements in a multi-agent setup