Source
Di Biagio and Rovelli (2025), LIT-1676, arXiv 2510.11349v2, §§II–V (pp. 1–4) and §IX (p. 7); read in NOTE-1214. Pages are the PDF’s.
What was actually shown
The measures. I_A = I_A^max − H_A (eq. 1). The relative information I_{A|B} = Σ_b p(b) I_{A|b} is the expected information about A after B’s value is learnt (eq. 6). It satisfies I_{A|B} = I_A + I_{A:B} (eq. 7), which is the Shannon identity H(A|B) = H_A − I(A:B) in this notation.
The definitions. “the value of a variable A is a fact when the probability distribution p(a) is entirely concentrated on one value” (p. 2). A is a fact relative to B = b when I_{A|b} = I_A^max (eq. 9), and relative to B when I_{A|B} = I_A^max (eq. 10). So A can be a fact relative to B without being a fact relative to us: Robert’s location is a fact relative to Mary, or to his smartphone, when we know only that she knows (pp. 2–3). “B has information about A” means I_{A:B} ≠ 0, “in this general and nonmental sense” (p. 3).
Why the relatum is a commutative subalgebra. In quantum theory the joint distribution p_ρ(a, b) = tr Π_a Π_b ρ (eq. 12) exists only for commuting variables, “so only these can simultaneously be a fact according to our definitions”. A commuting collection is a “classical subsystem”, and “the collection of facts relative to A is the perspective associated to A” (p. 3). A qubit has non-commuting variables, so “we cannot associate a perspective to a qubit” (p. 4). The authors call this “a novelty with respect to the RQM literature”, whose earlier attempt “to define facts relative to entire quantum systems” led “either to ambiguities … or to complementary observables being facts simultaneously” (p. 4). Among the works cited for that attempt is their own Stable Facts paper.
Why the relativity iterates. “Every fact is a relative fact” (p. 3), because information is stored in physical variables and accessed by other systems. The joint distribution behind any relative fact is computed from a state relative to some classical subsystem: “Every quantum state is a quantum state of a system relative to some classical … system” (p. 4). “When we say that a variable B has information about another variable A, we are saying that this is the case relative to us” (p. 7). The authors accept the iteration and refuse the regress: it arises “only … if one wishes to exit relationality” (p. 7).
What could have come out otherwise: a definition of relative fact for a whole quantum system that avoids the ambiguity. The paper argues that none exists on these definitions, because complementary variables have no joint distribution.
What this does not say
- That the definitions dissolve RQM’s problems. That is the authors’ claim (“several previous problems … are dissolved”, abstract), and it is argued in a paragraph each (pp. 4–5).
- That observers play no role in what counts as a fact. Facts are graded in practice. Something is a fact “when the information is high enough for the level of accessible resolution or confidence needed” (p. 3), and the paper does not say what sets that level.
- That the perspective is fixed by the physics. A system is “a set of physical variables, represented … as distinguished operators” (p. 4). Which operators are distinguished, and so which subalgebra holds a perspective, is assumed. Decoherence is cited for the classical case (§VIII), not derived.
- That a perspective is an observer. Any commuting set has one, such as the spin-z of an electron, “which is clearly not an observer” (p. 4).
- That relative facts happen only at interactions. On these definitions a relative fact is a property of the joint distribution at any time, and the paper says it “allows talk of relative facts also in-between interactions” (p. 7). THEORY-621, from the 2021 chapter, has facts only at interactions.