In the following equation, we have used a pure initial state ${|\psi ⟩}_{{\mathtt{S}}_i}$ to make the equations more concise, but it is easy to verify that analogous equations and the same arguments hold for mixed initial states ${\rho }_{{\mathtt{S}}_i}$.

Collapse theories, for instance, would violate this assumption and prescribe different probabilities and rules for computing them.

One can formally write down the probability expressions obtained in steps 3 and 4 through the Born rule. However, in the interests of conciseness, we decline from doing so at this point, as we show that these conventional predictions since are recovered as a special case of our to-be-described and rigorously defined general formalism (Appendix pp10-s2).

FR refer to $C$ as an interagent “consistency” assumption, hence the name. Following the formulation of this in terms of epistemic modal logic [4, 28], we find it more appropriate to view it as an inheritance of agents’ knowledge. However, we stick to the original acronym for simplicity. $D$ refers to the distributive axiom of logic, and $S$ corresponds to an assumption about measurements yielding “single outcomes” in every experimental run.

Recall that, in our framework, the settings $x_i=1$ for all outcomes $a_i$ belong to the outcome sets ${\overrightarrow{a}}_j$ or ${\overrightarrow{a}}_l$ appearing in the given prediction $P({\overrightarrow{a}}_j={\overrightarrow{\mathcal{a}}}_j|{\overrightarrow{a}}_l={\overrightarrow{\mathcal{a}}}_l,\overrightarrow{x}=\overrightarrow{\xi })$; the remaining components of $\overrightarrow{x}$ can be varied.

This aligns with a previous work, which also argued that FR’s conclusions hold only under a specific way of applying quantum theory [31]. However, our formalisation of this point via the assumption $\mathbf{I}$ allows us to further achieve a full resolution of FR-type paradoxes meeting the desiderata outlined in Sec. 1a.

We need not consider the case where Ursula and Wigner’s measurements are modeled unitarily (setting 0), as this is the case only if there were further super-superagents who measured Ursula and Wigner’s labs.

See Ref. [34] for further details on a general, theory-independent formalisation of this class of Wigner’s friend (apparent) paradoxes, derivable only by combining statements about compatible measurements, in addition to imposing $\mathbf{I}$, which we have shown to be necessary.

Keep in mind that each agent in our formalism is associated with a single time step, and a physical agent acting at many time steps would be modeled as multiple agents acting here; thus, one agent has to act later in time than another in order to act on the latter’s memory.

These structures and the setting independences they entail follow only from the properties of the EWFS protocol (and not agents’ subjective choices of cuts) and can be included in the initial protocol specification given to the agents.

This holds even when no projection postulate is used to set postmeasurement states; see Sec. 5b.

Commonly called indefinite causal order [46, 47], though this terminology is often debated.

The memory ${\mathrm{M}}_i$ of an agent ${\mathrm{A}}_i$ is chosen to be a system of at least the same dimension as the system ${\mathtt{S}}_i$ that the agent ${\mathrm{A}}_i$ measures.

In simple terms, a DAG is a graph with additional structure: (i) The edges have a direction associated with them. (ii) By “following the edges in the indicated direction,” one can never get back to the starting point, i.e., no “directed loops.”

In Ref. [4], the assumption C is replaced by what they call the trust axiom, which asserts that an agent $A^i$ can inherit the knowledge of another agent $A^j$ as per Eq. (c3) only if $A^i$ trusts $A^j$. In the FR setup, it is precisely these pairs of agents who can be said to trust each other. Other pairs such as Alice and Ursula need not trust each other, as one agent Hadamard’s the memory of the other.

The cited works are only representatives of the extensive research generated by the FR paper. For further references on previous responses to FR, see Ref. [11].

Although additional steps can be employed to simplify the reasoning by dropping redundant settings and parameters, as is also the case in classical multiagent scenarios (cf. Sec. 7a).

This fact is independent of whether or not different agents agree on the setting choices to model their perspective of the experiment, as, for instance, even if Wigner models the friend’s lab as a unitarily evolving closed quantum system ($x_F=0$), both Wigner and the friend can still use our augmented circuit to compute predictions for the case where $x_F=1$ and will arrive at the same answer.

Physically, if we regard the circuit as embedded in space-time such that the absence of directed paths between ${\mathrm{A}}_i$ and ${\mathrm{A}}_k$ correspond to spacelike separation, then the above transformation can be seen as transforming to another reference frame where the time order relative to the coordinate time is reversed.

Since the scenarios are operationally equivalent with a one-one-one correspondence between the outcome sets, we can use the same labels without loss of generality.