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NOTE-1226: Di Biagio & Rovelli — Stable facts, relative facts

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PaperLIT-1666 — Stable Facts, Relative Facts

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Contribution

It gives relational quantum mechanics (RQM) a definition of the two kinds of fact it had been using informally. A relative fact is defined by an interaction (§1.1, p. 3). A stable fact is defined by a probability rule: facts are stable for a system when that system may sum over them classically, eq. (3). It then derives, in a few lines (eqs. 5–8, p. 4), that decoherence makes facts stable for a third system to order ε. The paper is short and mostly conceptual. Its technical content is that derivation. The rest applies the distinction to the measurement problem, to two no-go theorems for absolute facts (Frauchiger–Renner, and Bong et al.’s Local Friendliness inequalities), and to quantum logic.

Key insight

The classical sum rule P(b) = Σᵢ P(b|aᵢ)P(aᵢ) is the mark of a “stable” fact, and in quantum theory it holds without condition only when b and the aᵢ are facts relative to the same system (eq. 2). Used across systems (eq. 3) it fails, and “The failure of (3) is easily understood in terms of the standard language of quantum theory: it is the presence of interference effects” (p. 3). Decoherence restores (3) for a system that cannot reach the environment. It does so only approximately, and only for that system: “Stability is only approximate and relational. Approximate, because no fact is exactly stable for any finite 𝜖. Relational, because it depends on how the putative ‘observer’ system couples to the system and the environment” (p. 6).

Assumptions

  • Quantum theory is universal. “There are no exactly classical objects, strictly speaking” (p. 2). Collapse models and other limits on quantum theory are set aside: “there is no empirical evidence of its failure” (p. 2).
  • A fact is fixed by dependence in an interaction (p. 3): “If an interaction affects a variable L_F of F in a manner that depends on the value of a certain variable L_S of S, then the value of L_S is a fact relative to F.” “This is true by definition irrespectively of whether F is a classical system.” Nothing about F being an observer, macroscopic or classical is assumed (p. 4).
  • Eq. (2) holds always. For facts relative to one system F, P(b^(F)) = Σᵢ P(b^(F)|aᵢ^(F))P(aᵢ^(F)), “it is always the case” (p. 3). This is stated, not derived; it is RQM’s rule.
  • Systems are given. The definitions start from “two systems S and F” and their variables (p. 3). What a system is, or how the world divides into them, is not discussed.
  • The two conditions for decoherence (p. 4): “(a) 𝜖 is vanishing or very small and (b) a system W does not interact with E”. Both are needed: “assumption (b) above is just as crucial as assumption (a) in deriving (8)” (p. 5).
  • The quantum state is not ontic. “The state is not a ‘thing’, nor a condition of a system. Rather, it is what a physicist uses to calculate probabilities for relative facts” (p. 7).

Key results

§1: stable facts defined (pp. 2–3)

  • Eq. (1), p. 2: P(b) = Σᵢ P(b|aᵢ)P(aᵢ), for N mutually exclusive facts aᵢ. “We take Eq. (1) as a characterisation of stable facts.” The illustration: after L_z = ℏ/2, the probability of L_θ = ℏ/2 is cos²(θ/2) (p. 2).
  • Relative facts (§1.1, p. 3). The context: “The interaction with F is the context in which that variable takes a specific value; we call the system F, in this role, a ‘context’.” Footnote 1 takes “context” from Grangier and Auffèves [17], “However we do not require the context to be classical”, and avoids “observable” because it carries “the ideas of observation and a complex observer” (p. 3).
  • Eqs. (2) and (3), p. 3. With facts labelled by their context, (2) holds always; for W ≠ F, “it is in general not the case” that P(b^(W)) = Σᵢ P(b^(W)|aᵢ^(F))P(aᵢ^(F)) (3). “If (3) holds, we say that the facts aᵢ^(F) are stable for W.” “Stable facts are a strict subset of the relative facts” (p. 3).
  • Eq. (4), p. 3. For an isolated S–F, the state c₁|a₁⟩⊗|F_a1⟩ + c₂|a₂⟩⊗|F_a2⟩ gives W probabilities with interference terms, “because what sums is amplitudes, not probabilities”, so the value of L_S is not a stable fact for W.

§1.2: decoherence makes facts stable for a third system (pp. 4–5)

  • “the key phenomenon that makes facts stable is decoherence [36, 39, 40]: the suppression of interference that happens when some information becomes inaccessible” (p. 4).
  • Eqs. (5)–(8), p. 4. For |ψ⟩ = Σᵢ cᵢ|F_ai⟩⊗|ψᵢ⟩ (5) with normalized environment states |ψᵢ⟩, let 𝜖 = max_{i≠j} |⟨ψᵢ|ψⱼ⟩|² (6). Under (a) and (b), W’s probabilities for facts from an interaction with F can be computed from ρ = tr_E|ψ⟩⟨ψ| = Σᵢ|cᵢ|²|F_ai⟩⟨F_ai| + O(𝜖) (7). With P(F_ai^(E)) = |cᵢ|², P(b^(W)) = Σᵢ P(b^(W)|F_ai^(E))P(F_ai^(E)) + O(𝜖) (8). “Hence the value of the variable L_F is a fact relative to E that is stable for W to the extent to which one ignores effects of order 𝜖. In the limit 𝜖 → 0, the variable L_F of the system F is exactly stable for W.” (The bound O(𝜖) is stated, not computed. Since 𝜖 is defined as the squared overlap, the off-diagonal elements of ρ are bounded by |cᵢcⱼ|√𝜖, not by 𝜖; the paper does not discuss this. A reader’s point, not the paper’s.)
  • Which variables decohere are “those variables that commute with the interaction Hamiltonian”, and the decoherence time “is typically extremely short for macroscopic variables of macroscopic objects” (p. 5). Both are cited as “well understood” [41], not shown.
  • “First, decoherence is not an absolute phenomenon, but a relational one: it depends on how the third system W interacts with the combined system F − E” (p. 5). “Another system W′ that interacts differently with F − E might be able to detect interference effects.”
  • “Second, decoherence implies that an event regarding two systems F and E is stable for at third system W. Hence, a fact stable for W is not necessarily a fact relative to W” (p. 5). This is what licenses W to say the state of F “has collapsed” before W has interacted with F.
  • “decoherence does not imply that there is a perfectly classical world of absolute facts, although it does explain why (and when) we can reason in terms of stable, hence approximatively classical, facts” (p. 5).
  • Footnote 2, p. 5: stability without decoherence. If one amplitude has |c₁| close to 1 and W does not interact with E, W may use ρ′ = |F_a1⟩⟨F_a1| + O(η), η = 1 − |c₁|. “Thus, when a fact relative to a system has very high probability, then it is stable for other systems.” Contrasted with the EPR criterion of reality: it holds only while W cannot interact with E, “and there needs to be a fact relative to E in the first place”.

§1.3: a measurement in three stages (pp. 5–6)

  • If L_S and L_F are entangled and L_F is stable for W, “it follows immediately from the definitions that also L_S is stable for W” (p. 5).
  • The laboratory measurement, told twice (pp. 5–6). In standard terms: the system–apparatus interaction entangles L_S with a pointer L_F; L_F correlates with many microscopic variables E inaccessible to W; W reads the pointer. In relative facts: “1. A relative fact is established between S and F. 2. A relative fact is established between F and E. Since W does not interact with E, this stabilises the previous fact for W. 3. A relative fact is established between F and W.”
  • At stage 2, “neither the value of L_S nor that of L_F is a fact for W at this stage. Stability simply allows W to “de-label” facts relative to F” (p. 6). At stage 3, L_F becomes a fact for W, but “the value of L_S is still not a fact relative to W, it is merely a stable fact for W” (p. 6).
  • “It is the way that W, F and E couple to each other that makes F a measuring apparatus for W” (p. 6).

§2.1: why the ontology is relative facts, not stable ones (pp. 7–8)

  • Stable facts are the textbook link to the world, and “facts stable with respect to us are ubiquitous because of the ubiquity of decoherence” (p. 7). Two difficulties in taking them as the ontology: “First, stability is relational … Therefore one does not avoid relationalism by restricting to stable facts. Second, more seriously, stability is only approximate in general. At no point the interference terms perfectly vanish” (p. 7).
  • “How stable does a fact need to be before it is real? And with respect to what systems does it have to be stable, in order to be real? Any answer to these questions is bound to be as unsatisfactory as the textbook interpretation that requires a classical world” (p. 7).
  • “RQM notices that every interaction is in a sense a measurement, in that it results in the value of a variable to become a fact … And there is no ‘special context’: any system can be a context for any other system” (p. 7).
  • No universal wavefunction: it “would be a tool to calculate probabilities of facts relative to something that does not exist: a system that is not part of the universe” (p. 8).
  • “the ontology of RQM is realist … Realist in the sense that it is not about agents, beliefs, observers, or experiences: it is about real facts of the world and relative probabilities of their occurrence. The ontology is relational, in the sense that it is based on facts established at interactions and are labelled by physical contexts” (p. 8).

§2.2: the no-go theorems (pp. 8–9)

  • Frauchiger and Renner [12] (§2.2.1). Their assumption (C), “If W, applying quantum theory, concludes that F knows that L_S = a, then W can conclude that L_S = a”, becomes in relative facts the use of facts relative to F as if relative to W, which “holds only if every fact relative to F is stable for W” (p. 8). The superobservers have “full quantum control” of the friends’ labs, so “by definition, what is stable for the friends is not stable for W. Hence the contradiction follows from inappropriately mixing contexts” (p. 8). That no contradiction follows if the friends’ labs also decohere for W is credited to [38] and Relaño [26, 27], not shown. And: “The analysis of how an agent should reason about an experiment that will be performed on him has not been done within RQM yet” (pp. 8–9), with QBism’s analysis [9, 10] commended.
  • Bong et al. [5] (§2.2.2). Absoluteness of observed events, no superdeterminism and Bell locality imply the Local Friendliness inequalities, which quantum theory violates. “Absoluteness of observed events means that if W deems that F is an observer, then W can use (3)—even if W has full quantum control on F” (p. 9), which RQM denies. On the photon-friend experiments: dismissing them because “photons are not observers” “opens the problem of deciding which systems give rise to facts” (p. 9). “The violation of the LF inequalities is no way in tension with the relational interpretation. The opposite is true: the result is taken as evidence that the facts quantum theory deals with are facts relative to systems” (p. 9).

§2.3: conclusions (pp. 10–11)

  • The measurement problem: projection and linear evolution “refer to facts relative to different systems” (p. 10). “In a slogan: ‘Wigner’s facts are not necessarily his Friend’s facts’.”
  • Consistency between observers: after S and F interact they are entangled relative to W, “Therefore Wigner will always agree with his Friend about the value of L_S once he too interacts with them. In this sense, relative facts correspond to real events, they have universal empirical consequences” (p. 10), citing [28] (Rovelli 1996). An argument from the correlation in the entangled state, not a general proof.
  • Macroreality: “Such a macroreality only emerges approximately, relative to systems for which decoherence is sufficiently strong” (p. 10).
  • Decoherence: “there is consensus that decoherence alone is not a solution of the measurement problem … Decoherence needs an ontology. Relative facts provide such a general ontology that is well defined with or without decoherence. Decoherence clarifies why a large class of relative facts are stable for us” (p. 10).
  • Quantum logic, eqs. (9)–(10), p. 10: the violation of (1) has been read as P(b and (a₁ or a₂)) ≠ P((b and a₁) or (b and a₂)), against distributivity; RQM reads it as “forgetting that facts are relative: labelled by a context, as Bohr has repeatedly pointed out”.
  • Superposition: “Saying “Friend is in a quantum superposition” does not mean anything more than saying that Wigner would be mistaken in using (3)”; “Schrödinger’s cat has no reason to feel ‘superposed’” (p. 11).

Claims

idclaimstrengthsupport
C1Stable facts are a strict subset of relative facts, and the failure of eq. (3) across contexts is interferencestrong (definitional)eqs. (1)–(4), pp. 2–3; the interference example is standard
C2If F’s branches are entangled with an environment E with overlap ε and W does not interact with E, facts relative to E about L_F are stable for W up to O(ε)strongthe reduced-density-matrix step, eqs. (5)–(8), p. 4; the order of the error is stated, not bounded
C3Stability is relational: a system W′ coupled differently to F–E can see interferencestrongfollows from condition (b) in the derivation (p. 5); not illustrated with a model
C4A fact can be stable for W without being a fact relative to Wstrong (definitional)p. 5; §1.3’s stage 2 (p. 6)
C5Which variables decohere is fixed by the interaction Hamiltonian, and decoherence is fast for macroscopic variablesstrong, citedp. 5, from Zurek [41]; not shown here
C6Stable facts cannot serve as the ontology, because “how stable” and “stable for whom” have no non-arbitrary answer; relative facts canmoderate (interpretive)argument, §2.1, p. 7
C7The Frauchiger–Renner contradiction comes from assumption (C), which is eq. (3) applied where facts are not stable for Wmoderatethe translation of (C), p. 8; the no-contradiction result under decoherence is cited [26, 27, 38], not shown
C8The Local Friendliness violation is evidence that facts are relative to systemsweakit shows that one of three assumptions fails (p. 9); RQM drops absoluteness, but so do other views (cf. NOTE-174, C8)
C9Wigner will always agree with his friend about L_S once he interacts with themweakone sentence from the entangled state’s correlation, citing [28] (p. 10); no general argument
C10The projection and linear-evolution postulates are compatible because they apply to facts relative to different systemsmoderate§2.3, p. 10; restates RQM’s rule (THEORY-621)
C11A fact relative to E with probability near 1 is stable for other systems even without decoherencestrongfootnote 2, p. 5, with error O(η), η = 1 −

Concepts

  • Relative fact — the value of a variable L_S, relative to a system F whose variable L_F an interaction makes depend on it (p. 3). Written a
  • Context — the system F in that role (p. 3), from Grangier and Auffèves but not required to be classical (footnote 1).
  • Stable for W — facts aᵢ^(F) for which W may use eq. (3), the classical sum rule over them (p. 3).
  • Stable fact — a relative fact that is stable for the systems in question; “those whose relativity can effectively be ignored” (abstract). Always stable for some system; never absolutely (pp. 6–7).
  • 𝜖 — the largest squared overlap of the environment states correlated with F’s alternatives (eq. 6); the measure of how far a fact is from stable for a W that cannot reach E.
  • De-label — what stability lets W do with facts relative to F (p. 6).
  • Macroreality — absolute facts, in the Bell literature’s sense; here emergent “only approximately, relative to systems for which decoherence is sufficiently strong” (p. 10).

Connections

  • Rovelli 2021 (LIT-177, read in NOTE-174) cites this paper (its [18]) for stable facts, and NOTE-174’s C3 marked the decoherence result “cited, not shown”. It is shown here, at the level of eqs. (5)–(8). NOTE-174’s open question, “Can ‘stable facts’ be defined without an observer-relative choice of which degrees of freedom count as the environment?”, is answered in the negative by the paper itself: condition (b) names the system for which the environment is inaccessible, and the authors make that the point (p. 5).
  • Rovelli 1996 (LIT-1635, read in NOTE-1227). The 1996 paper’s “decoherence depends on which observation P will make” (§II.B, p. 5 there) is this paper’s “decoherence is not an absolute phenomenon, but a relational one” (p. 5). What is new here is the vocabulary of facts and contexts, the probability-rule definition of stability, and the explicit ontology: “based on facts established at interactions” (p. 8). The 1996 paper treated Bohr’s privileged classical observer as “consistent” with RQM; here an ontology of stable facts alone is “as unsatisfactory as the textbook interpretation that requires a classical world” (p. 7).
  • Grangier and Auffèves (LIT-077, read in NOTE-044) is ref. [17], the source of “context”, with the difference that the context here need not be classical (footnote 1).
  • QBism (LIT-1182, read in NOTE-955): the paper points to QBism’s analysis of an agent reasoning about an experiment on himself, “since also QBism holds that assumption (C) fails” (p. 9).
  • Zurek is cited for decoherence [39–41]; quantum Darwinism and redundancy are not mentioned. The record’s registered Zurek 2009 (LIT-1640) is where redundancy would come from.
  • Di Biagio and Rovelli 2025 (LIT-1676, unread) is the registered sequel.

Bearing on the record

  • QUESTION-285: the “stable record” rung is misattributed, and the paper supplies a different, sharper rung. The question’s table gives this paper as the source for “stable record: the outcome stays accessible far longer than the interaction took”. Nothing in the paper is about duration or accessibility in time. Stability here is a probability rule holding for a third system (eq. 3), and it comes from inaccessibility, not accessibility: “the suppression of interference that happens when some information becomes inaccessible” (p. 4). Persistence and redundancy are not discussed. What the paper does give the question is a three-step measurement in its own terms (pp. 5–6): a relative fact (S–F), its stabilization for W by a further fact (F–E) that W cannot reach, and a fact for W (F–W), which W then uses to update its state for S. These come apart: at stage 2 L_S is stable for W but not a fact for W, and even after stage 3 it is “merely a stable fact for W” (p. 6). And “It is the way that W, F and E couple to each other that makes F a measuring apparatus for W” (p. 6): apparatus is a role fixed by a three-way coupling, relative to W, near TERM-089’s operational sense. On the question’s first rung the paper is RQM’s answer as the question already states it: “every interaction is in a sense a measurement” (p. 7). The exchange was right that the paper gives observation without a conscious observer: the ontology “is not about agents, beliefs, observers, or experiences” (p. 8), and footnote 1 avoids “observable” for its connotation of “a complex observer” (p. 3).
  • The exchange’s gloss (A19 §3, A23 §VII), checked. “Facts relative to interacting systems” against “stable facts whose relativity can effectively be ignored”, connected to decoherence: right, nearly verbatim (abstract; §1.2). “A relationship established through interaction becomes stable enough to support a more nearly observer-independent effective description”: wrong in two respects. (i) It is not the S–F relation that becomes stable by being established. The fact relative to F becomes stable for a third system because F interacts further with an environment that third system cannot reach (p. 4; stage 2, p. 6). (ii) The result is not observer-independent, more or less. It is label-independent for a given W, and the paper insists on the difference: “decoherence is not an absolute phenomenon, but a relational one” (p. 5); “one does not avoid relationalism by restricting to stable facts” (p. 7); even macroreality “only emerges approximately, relative to systems for which decoherence is sufficiently strong” (p. 10). The defensible form is: for a system that cannot reach the environment, the relativity of a decohered fact can be ignored up to O(ε).
  • QUESTION-280: a worked case of approximate absoluteness, whose authors decline to promote it to ontology. The paper is an instance of the question’s “approximation for ontological claims” for facts, not for individuals. A relational feature (facts indexed to contexts) has a limit (ε → 0, with W cut off from E) in which it behaves as the absolute, label-free facts of classical physics, with a stated error O(ε) (p. 4). So it has the quantitative half of what the question asks for: a grade (ε) and a limit. But the authors take the question’s second horn for the ontology: stable facts are not the primary reality, because “How stable does a fact need to be before it is real? And with respect to what systems does it have to be stable?” have no satisfactory answer (p. 7), and “At no point the interference terms perfectly vanish”. And they add a feature the question does not name: the approximation is itself relative to a system (to W, which must not reach E), so even the approximately absolute is approximately absolute for something. Separable enough, in CLAIM-157’s terms, is here separable enough for a stated W, and the paper says that does not make it independent. It argues this; it does not prove that no account of approximate ontology could succeed.
  • THEORY-621: the stable-fact step is now shown, and the rule and consistency gaps stand. THEORY-621 records that the decoherence result is “cited, not shown” in Rovelli 2021. Here it is derived, at the level of a reduced density matrix (eqs. 5–8, p. 4). The promote_when asks for two things this paper does not give: eq. (2), the same-system rule, is asserted (“it is always the case”, p. 3), not derived from independent principles; and inter-observer consistency is one sentence, “Wigner will always agree with his Friend about the value of L_S once he too interacts with them” (p. 10), from the entangled state. The paper also concedes an open case: how an agent should reason about an experiment performed on himself “has not been done within RQM yet” (pp. 8–9). Its reading of the no-go theorems is the same as Rovelli 2021’s and has the same limit (C8 above). Filed as THEORY-1075 and THEORY-1081.
  • CLAIM-478: a constraint made by inaccessibility, not by slowness. The paper’s effective rule, the classical sum (3), appears for W because information in E is out of W’s reach, not because some variable is slow relative to another. Time appears once: the decoherence time, “the time needed for 𝜖 to become so small that interference effects become undetectable by given observational methods”, short for macroscopic variables (p. 5). One can read that as a separation (decoherence fast, W’s later interaction slower), but the paper does not, and its condition (b) is about coupling, not rates. So it does not support the claim’s mechanism. It is a different route to an effective constraint, with a graded error (ε), that the claim’s “Not one mechanism” list could note. The variables that become stable are selected by the interaction Hamiltonian (p. 5), which is a case of organization (who couples to what) fixing which constraint appears.
  • THEORY-1071 and NOTE-1227: the systems still precede the facts, and the ontology of facts is here by 2020. NOTE-1227 found no ontology of events in Rovelli 1996, and THEORY-1071’s note traced Faye and Jaksland’s “sparse ontology of (relational) quantum events” to Rovelli 2018. This paper states an ontology of facts outright: “The ontology is relational, in the sense that it is based on facts established at interactions and are labelled by physical contexts” (p. 8). The word “sparse” does not occur. And the definition of a relative fact presupposes the systems and their variables (“Let us consider two systems S and F. If an interaction affects a variable L_F of F …”, p. 3), so Faye and Jaksland’s “the systems precede the interactions” holds of this text as well, again by assumption and without argument. The paper does not take up what a system is.
  • THEORY-1080: its finding stands for 2021. Consistency between observers is still “the third system predicts correlated outcomes” (p. 10), asserted for the general case.
  • QUESTION-021. Nothing new beyond NOTE-1227’s point: stable facts are the nearest thing to an invariant, and they are invariant only for the systems that cannot reach the environment.
  • No instruction for machine-learning practice.

Limitations

  • The derivation is a sketch. Eqs. (5)–(8) state the error as O(𝜖) without a bound; 𝜖 is a squared overlap, so the residual off-diagonal elements of ρ are bounded by |cᵢcⱼ|√𝜖, which the paper does not discuss. Condition (b), “W does not interact with E”, is all-or-nothing; partial access is not treated.
  • The decoherence facts are cited: the pointer variables commuting with the interaction Hamiltonian and the short decoherence times (p. 5) come from Zurek [41].
  • The core rule is assumed. Eq. (2) holding always for same-context facts is the interpretation’s postulate (p. 3).
  • Consistency between observers is one sentence (p. 10). Not in this text, and not checked in this reading: it is standardly reported that later RQM work (Adlam and Rovelli, 2022) added a postulate of “cross-perspective links” to secure such agreement. The record holds no LIT for it.
  • The no-go analyses rely on cited work: Relaño [26, 27] and Żukowski and Markiewicz [38] for the claim that no contradiction follows when the friends’ labs decohere for W (p. 8).
  • Typos in the published text: “phase pace” (footnote 1, p. 3); “stable for at third system” (p. 5); the heading “The Link Between the Theory the World” (§2.1, p. 7); the conclusions are numbered §2.3 within §2. The acknowledgements thank “Eric Wiseman” for a point on [5], whose author list has H. M. Wiseman; which person is meant cannot be settled from the text.
  • The v1 preprint was not read; what changed between v1 and the journal text is unknown.

Open questions

  • Is there a quantitative account of partial access, where W interacts weakly with E, giving how far (3) fails as a function of both 𝜖 and W’s coupling?
  • If stability is relative to W, is the decomposition into S, F, E and W also relative, or given? The paper takes the systems as given.
  • How should an agent in RQM reason about an experiment performed on himself (pp. 8–9)? The paper names it as not yet done.
  • Can the agreement between Wigner and his friend (p. 10) be derived from the definitions, or does it need an added postulate?

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