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NOTE-1214: Di Biagio & Rovelli — Relative information, relative facts

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PaperLIT-1676 — Relative Information, Relative Facts

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Contribution

The paper gives relational quantum mechanics (RQM) quantitative definitions of information, relative information, fact and relative fact. They come from Shannon’s theory and Everett’s measure, and nothing is added to the quantum formalism (§§II–IV, pp. 1–4). The new move is the relatum. Facts are relative to commutative subalgebras of observables, not to whole quantum systems, which the authors call “a novelty with respect to the RQM literature” (p. 4). Their own earlier papers, the companion LIT-1666 among them, are named as attempts “to define facts relative to entire quantum systems”. With these definitions the paper treats measurement as continuous growth of relative information (§VII, App. B), shows perspectives merging in a three-system example (§VI), and re-reads Wigner’s friend, EPR and the extended Wigner’s friend scenario (Apps. C–E).

Key insight

A fact is a probability distribution concentrated on one value, and a relative fact is a conditional distribution concentrated on one value. That is a statement about correlations in a joint distribution, and only commuting variables have one, so what holds a perspective is a set of commuting variables, whatever system they belong to. Facts are then as continuous and as graded as information is: “call something a fact when the information is high enough for the level of accessible resolution or confidence needed” (p. 3). The joint distribution is always computed from a state relative to some further system, “generally us” (p. 3), so the relativity iterates. The paper declines the regress: “exiting relationality is not physics” (§IX, p. 7).

Assumptions

  • Probabilities for variables with finitely many values. I_A^max = log N_A (eq. 1 and footnote 1, p. 2). Continuous variables enter only through characteristic functions χ_[x0,x1] or projector-valued measures (footnote 2, p. 3). POVMs are left to ref. [18].
  • The Born rule for commuting variables. p_ρ(a) = tr Π_a ρ (eq. 11) and p_ρ(a, b) = tr Π_a Π_b ρ for [A, B] = 0 (eq. 12), p. 3.
  • A system is a set of distinguished variables, “represented in the theory as distinguished operators on the Hilbert space” (§V, p. 4). A spin-½ is the algebra generated by X, Y, Z, and a classical bit is the subalgebra spanned by 1 and Z. Which operators are distinguished is assumed, not derived.
  • Every state is relative to a classical subsystem. “Every quantum state is a quantum state of a system relative to some classical (in the sense above) system” (p. 4). In every worked example the state is given “relative to us” (eq. 13) or to an outside observer O (App. D).
  • Unitary evolution throughout. No collapse is postulated (p. 4).
  • Ideal measurements in the worked examples. These use pure ready states and perfect correlation. App. B says this idealization is what makes a conditional relative fact appear at once (p. 9).

Key results

Information measures (§II, pp. 1–2; App. A, p. 8)

  • I_A := I_A^max − H_A, with H_A = −Σ p(a) log p(a) (eqs. 1–2). It is zero when nothing is known and maximal when p(a) is peaked. Following Everett, but he set I_A^max = 0 (footnote 1).
  • I_{A:B} := H_A + H_B − H_AB = I_AB − I_A − I_B (eq. 3).
  • The conditional information I_{A|b} := I_A^max + Σ_a p(a|b) log p(a|b) (eq. 5). The relative information is its expectation, I_{A|B} := Σ_b p(b) I_{A|b} (eq. 6).
  • Key relation (7): I_{A|B} = I_A + I_{A:B}. “the mutual information is the expectation value of the increase in information about A after looking at B” (p. 2). It is the standard identity H(A|B) = H_A − I(A:B) rewritten.
  • App. A lists: I_{A:B} is symmetric and I_{A|B} is not; I_{A:B} ≤ min{log N_A, log N_B}, and at the maximum one of I_{A|B}, I_{B|A} is maximal; “I_{A:B} > 0 only when neither A nor B is a fact”; and for variables of different systems, I_{A:B} ≤ 𝓘, the quantum mutual information, with I_{A:B} ≤ 𝓘/2 for pure entangled states. No proofs are given.

Facts and relative facts (§III, pp. 2–3)

  • Fact: “the value of a variable A is a fact when the probability distribution p(a) is entirely concentrated on one value, namely when the information I_A is maximal” (p. 2).
  • Relative fact: A is a fact relative to B = b when I_{A|b} = I_A^max (eq. 9). A is a fact relative to B when that holds for every b with p(b) ≠ 0, equivalently I_{A|B} = I_A^max (eq. 10).
  • A can be a fact relative to B without being a fact, when I_A + I_{A:B} = I_A^max. Example: Robert’s location is a fact relative to Mary, or to his smartphone, but not to us (pp. 2–3). “nothing here hangs on Mary and Robert being self-reflecting humans.”
  • “B has information about A” when I_{A:B} ≠ 0, “in this general and nonmental sense” (p. 3). An airport panel and a geological layer are the examples.
  • “Every fact is a relative fact” (p. 3). The argument: information “is stored in physical variables and their correlations and can therefore be accessed by other systems”, so talk of information about a variable is “invariably, although implicitly” talk of it relative to some system, “generally us”.
  • Graded facts. Information is continuous and “there is simply no aspect of the world of which we can be totally certain”, so the authors use information “as a measure of the definiteness of a fact” and call something a fact “when the information is high enough for the level of accessible resolution or confidence needed” (p. 3).

Perspectives belong to commutative subalgebras (§§IV–V, pp. 3–4)

  • “only mutually commuting variables may have a joint probability distribution, so only these can simultaneously be a fact according to our definitions.” A collection of commuting observables is a classical subsystem. “Given a state on an algebra of observables and a commutative subalgebra A of a quantum system (a classical subsystem), the collection of facts relative to A is the perspective associated to A” (p. 3).
  • Defining facts relative to whole quantum systems “leads either to ambiguities as to which variables are a fact relative to a system or to complementary observables being facts simultaneously” (p. 4, citing [22–24]). A quantum system has information about another only through its commutative subalgebras, quantified by the quantum mutual information (p. 4).
  • Qubits are not observers, agreeing with Brukner [24]. A qubit has non-commuting variables, so no perspective. One of its variables has a perspective but “lacks the resources and structure needed to be called an observer” (p. 4). “‘perspective’ is a wider notion than ‘observer’”: any Copenhagen or QBist observer has a perspective, but not every perspective is an observer’s (p. 4).
  • The price: “there are no absolute facts: most statements that are true or false are true or false in a perspective and there are no perspective-independent facts” (p. 4).

Perspectives differ, merge, and can be disturbed (§VI, pp. 4–5)

  • Relative to us, |ψ0⟩ = |+⟩|a0⟩|b0⟩ (eq. 13). A correlates with Z of the qubit S: |ψ1⟩ = (|0⟩|a0⟩ + |1⟩|a1⟩)|b0⟩/√2 (eq. 14). Z is a fact relative to A, not to B. Then B correlates with A or Z: |ψ2⟩ = (|0⟩|a0⟩|b0⟩ + |1⟩|a1⟩|b1⟩)/√2 (eq. 15). Now I_{Z|A} = I_{Z|B} = I_Z^max and I_{A|B} = I_A^max. “The perspectives of A and B have merged.”
  • If B instead correlates with X of S, the result is eq. (16). X is a fact relative to B, A is not a fact relative to B, and “Z is not a fact relative to A anymore, even though A did nothing in this last timestep … because the interaction between B and S disturbed Z” (p. 5). I checked (16) from (14); it is right.
  • This is offered as “a solution to RQM’s combination problem” [26, 27] and as the basis of intersubjectivity. It also diagnoses Frauchiger–Renner: the superobservers “change the correlation structure of the variables of the friends” (p. 5). Each is a paragraph.

Measurement is continuous (§VII, pp. 5–6; App. B, pp. 8–9)

  • In von Neumann’s scheme Σ αₙ|aₙ⟩|b_r⟩ → Σ αₙ|aₙ⟩|bₙ⟩ (eqs. 17–18). I_{A:B} goes from 0 to H_A, and I_{A|B} reaches its maximum. “There is no special moment in which the information that the apparatus has about the system jumps from zero to a finite value” (p. 6). “A discontinuity only comes from the definition of ‘fact’ as something whose probability reaches a sufficiently high threshold” (p. 5).
  • The model (B3)–(B7): Ĥ = iω Σₙ |aₙ⟩⟨aₙ| ⊗ (|bₙ⟩⟨b_r| − |b_r⟩⟨bₙ|) for t ∈ [0, π/2ω]. Then |ψ(t)⟩ = cos ωt|ψ_i⟩ + sin ωt|ψ_f⟩ (B5), p_t(aₙ, b_k) = |αₙ|²(cos²ωt δ_kr + sin²ωt δ_nk) (B6), p_t(aₙ) = |αₙ|² unchanged, and I_{A:B}(t) = sin²(ωt) H_A (B7). I rederived (B7) from (B6).
  • (B8) has a sign error. It is printed as I_{A|B}(t) = I_A^max + cos²(ωt)H_A (p. 8, checked on the rendered page). By (7) and (B7) it is I_A^max − cos²(ωt)H_A, which is I_A at t = 0 and I_A^max at t = T. The printed form exceeds I_A^max for t < T. The H_A formula in App. B also lacks its minus sign.
  • The paper’s own caveat (p. 9). In this model A is a fact relative to B = bₙ for every t > 0, because p_{t>0}(a|bₙ) = δ_{a,aₙ}. That is a conditional relative fact appearing at once. The authors call it “an artefact of studying a too idealised system, where p_0(bₙ) is exactly vanishing”. A realistic state is mixed and full rank [56, 57], and then I_{A|bₙ} starts at I_A^max − H_A and “smoothly increases to close to maximal over a short time”. The full-rank case is described, not computed.
  • “All the above has actually little to do with quantum theory”. A classical register shows “the same apparent tension between discrete outcomes and continuous information acquisition” (p. 6).

Classical world, regress, Wigner (§§VIII–IX, pp. 6–7)

  • Classicality and intersubjective agreement come from the commutativity of the variables that make up “we” and from Zurek’s decoherence, einselection and quantum Darwinism [33, 34]. Pointer variables become facts relative to many environmental variables, and observers reading different fragments agree (p. 6). This is cited, not shown.
  • “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). “relativity iterates [37]”. The regress exists “only … if one wishes to exit relationality by demanding a truth beyond any context or perspective” (p. 7).
  • Questions about what the friend “really” saw are set aside as without empirical content, by analogy with absolute motion and absolute simultaneity (p. 7). Added ontologies (Bohmian trajectories, “the collection of all relative facts” [22]) are “unobservable and are there only to assuage our metaphysical unease”.
  • The conclusion says the reformulation “allows talk of relative facts also in-between interactions” (p. 7).

Appendices C–E (pp. 9–11)

  • EPR (C). For (|00⟩ + |11⟩)/√2, none of the variables is a fact relative to us, but Z₂ is a fact relative to Z₁ and X₂ relative to X₁ (eq. C2). Both cannot be facts in one perspective. In GHZ (C3) “the Z variable of any qubit is a fact relative to the Z variable of any other qubit, but the X variable is not a fact relative to anything”. This multipartite structure “is what solves the ‘preferred basis problem’” [34].
  • Wigner’s friend (D). Relative to Wigner, S–F goes from |+⟩S|ready⟩F (D1) to (|0⟩|saw Z=0⟩ + |1⟩|saw Z=1⟩)/√2 (D2). Wigner’s information about Z stays minimal, his information about F falls by log 2, and I{Z|F} goes from minimal to maximal. Wigner’s check M = |ψ1⟩⟨ψ1| (D3) does not commute with F, yet it yields M = 1 and “Nothing happens to F”, and Wigner “gains no information about Z or F” by it (p. 10). Relative to an outside O, after Wigner asks, H_Z = H_F = H_W = log 2 (D8) with I{Z|F} = I_{Z|W} = I_Z^max and I_{F|W} = I_F^max. “the value of Z relative to F is the same as the value of Z relative to W” (p. 10).
  • Extended Wigner’s friend (E). Absoluteness of observed events has two parts: Alice’s and Bob’s statistics are marginals of a distribution over Charlie’s outcome (E1), and opening the box reveals Charlie’s outcome (E2). The paper holds (E2) “true according to quantum theory” and (E1) false: “at no time will p_QM satisfy (E1)” for x ≠ 1 (E3). “Charlie’s outcome is not a stable fact with respect to Alice, in the RQM sense of [13]”, which is LIT-1666 (p. 11).

Claims

idclaimstrengthsupport
C1Relative information I_{AB} = I_A + I_{A:B} is the expected gain in information about A from learning Bstrong
C2A relative fact can be defined as maximal relative information, with no addition to quantum theory and no primitive observerstrongdefinitions, eqs. 9–12, pp. 2–3; the claim is that the definitions need nothing more, and they don’t
C3Facts and relative facts can coherently be held only by commuting sets of variables, so perspectives belong to commutative subalgebras and not to quantum systemsstrong as a consequence of the definitions; moderate as a repair of RQMjoint distributions exist only for commuting variables (p. 3). That this dissolves the earlier problems is argued in a paragraph (p. 4)
C4Every fact is a relative factmoderatean argument from the physicality of information (p. 3); it is a stance that fixes how “fact” is to be read
C5In a von Neumann measurement the measuring variable’s information about the measured one grows continuously; the apparent jump is a threshold on facthoodstrong for the expected quantities; weaker for conditional factsthe model, eqs. B3–B7; the authors’ own caveat that a conditional relative fact appears at once in the ideal case (p. 9); the full-rank case is described, not computed
C6Perspectives merge when classical subsystems couple to the same variable or to each other, and coupling to a complementary variable can remove relative facts from othersmoderateworked three-system example, eqs. 13–16, p. 5; no general statement or proof
C7This solves RQM’s combination problem and diagnoses Frauchiger–Rennerweakasserted, a paragraph each (p. 5)
C8Intersubjective agreement and classicality follow from commutativity plus decoherence and quantum Darwinismweak herecited to Zurek [33, 34], not shown (p. 6; App. C)
C9In the extended Wigner’s friend scenario RQM keeps (E2) and denies (E1)moderatestated with the local-friendliness argument it rests on cited [42] (pp. 10–11)
C10Questions about what “really” happened beyond any system’s information have no empirical content and can be droppedweak as argumentan analogy with absolute motion and simultaneity (p. 7); a philosophical stance

Concepts

  • Information about A: I_A = I_A^max − H_A, the distance of p(a) from uniform, in Everett’s sense but shifted to be non-negative.
  • Relative information I_{A|B}: the expectation over B of the information about A conditional on B’s value.
  • Fact: a variable whose distribution is concentrated on one value; in practice, one whose information is “high enough” for the resolution or confidence needed (p. 3).
  • Relative fact: A’s value is a fact relative to B (= b) when the conditional distribution of A is concentrated (eqs. 9, 10).
  • “Has information about”: B has information about A when I_{A:B} ≠ 0, in a “nonmental sense” (p. 3).
  • Classical subsystem: any collection of commuting observables of a quantum system. Classical in the paper’s sense, not macroscopic.
  • Perspective: given a state and a commutative subalgebra, the collection of facts relative to that subalgebra (p. 3).
  • Observer: a special case of a system with a perspective, with “additional special properties” such as many records, decoherence and communication (p. 4). Not a notion the account needs.
  • Merging of perspectives: two classical subsystems agree on a variable’s value and on each other’s values (p. 5).
  • Stable fact: used once (App. E, p. 11) in the sense of ref. [13], LIT-1666. It is not defined here.

Connections

  • Rovelli 1996, ref. [3] (LIT-1635, read). The 1996 paper made states and values relative to systems that are assumed given (THEORY-1068), and replaced “the state” with “the information that a system has about another system”. This paper keeps that and makes the information Shannon’s. The relatum moves from the system to its commutative subalgebras. The system is still given, now as a set of distinguished variables (§V, p. 4).
  • Rovelli 2021, ref. [1] (LIT-177, read in NOTE-174; THEORY-621). The chapter has facts “sparse, occurring only in interactions”. This paper “allows talk of relative facts also in-between interactions” (p. 7), because a relative fact is a property of a joint distribution at any time.
  • Di Biagio & Rovelli 2021, ref. [13] (LIT-1666, unread). It is cited for “relative fact” (p. 1) and for “stable fact” (p. 11). It is also listed among the earlier attempts “to define facts relative to entire quantum systems”, which “rendered the notion somewhat problematic” (p. 4).
  • Brukner, refs. [12, 24, 41]: “Facts are relative”, “Qubits are not observers”, and the no-go theorem for observer-independent facts. The acknowledgements credit his “insistence … on the unsatisfactoriness of associating facts with quantum systems” as the spur for the subalgebra move.
  • Adlam and Rovelli, ref. [22], introduced cross-perspective links and “the collection of all relative facts”. This paper calls the latter an unobservable addition (p. 7) and thanks Cavalcanti for “pressing” that it “is not an idea in line with the perspectival spirit of RQM”. That is a retreat from [22].
  • Zurek, refs. [33, 34] (Zurek 2009 is LIT-1640): the account of intersubjectivity is borrowed from quantum Darwinism (§VIII, App. C).
  • Everett, refs. [15, 16]: the information measure and relative states. The state here is relative “but it does not give ontological weight to the states themselves” (p. 4).
  • Local friendliness, Bong et al. [42] (App. E).

Bearing on the record

  • QUESTION-285: a quantitative criterion for the correlation rung, and a measurement rung that cannot be told apart from it. The paper defines “B has information about A” as I_{A:B} ≠ 0, “in this general and nonmental sense” (p. 3). A measurement is the same quantity driven to its maximum, I_{A|B} = I_A^max (§VII). So A19’s first rungs (relational event, correlation) get exact criteria that any commuting variables can meet, with no observer. But nothing marks off a measurement from any interaction that correlates. A19’s own definition, “an interaction organized to make particular differences consequential”, has no counterpart. “Organized”, “consequential”, record, use and revision are not treated, and stability is deferred to ref. [13] and Zurek. Facthood is graded, and the threshold is set by “the level of accessible resolution or confidence needed” (p. 3), which leaves open who or what needs it. The question should record that this paper answers the bottom of its ladder exactly, says nothing above it, and reintroduces a purpose-relative threshold at the point where the record would draw the measurement line. It does carry one thing upward: a perspective requires commuting variables (p. 3). A qubit has no perspective, and one of its variables has a perspective but is not an observer (p. 4). That is a necessary condition on any rung that keeps a fact, and a reason the question’s apparatus must be “classical” in the paper’s sense, not in the macroscopic one. A19’s “high-priority new prior art” holds for the definitions. It does not hold for the measurement–inquiry part of the question.
  • TERM-089: differentiation made quantitative, with a cost the term does not name. The term’s first function, coupling differences in one system to distinguishable outcomes in another, is the paper’s relative fact: a pointer variable whose values fix the measured variable’s, I_{A|B} = I_A^max (§VII). Two findings refine it. (a) The coupled outcome variables must commute among themselves to hold a perspective at all (p. 3). (b) Differentiating one variable can erase relative facts others held. When B couples to X, “Z is not a fact relative to A anymore, even though A did nothing” (eq. 16, p. 5). A coupling can also yield an outcome while carrying no information about the variable of interest. Wigner’s M returns 1 and “Wigner gains no information about Z or F” (App. D, p. 10). That fits the term’s “selectively”. Retention, enforcement and revision are not touched.
  • THEORY-621: a correction to one of its stated prices, and worked cases for its consistency gap. The THEORY’s title gives as a price that “values are not held at all times”. On this reformulation relative facts are features of the joint distribution at any time, and the conclusion says it “allows talk of relative facts also in-between interactions” (p. 7). That price belongs to the 2021 chapter’s sparse-facts version. A dated note should say the later formulation drops it. On consistency, the paper adds worked cases (§VI; App. D, D8: “the value of Z relative to F is the same as the value of Z relative to W”). Each is computed inside one joint distribution relative to one outside system, where agreement follows from probability theory. It does not derive the rule that amplitudes give probabilities only between facts relative to the same system. App. E restates its content: the interference terms are “a symptom of the relationality of facts”, and no joint distribution over Alice’s, Bob’s and Charlie’s outcomes exists (E3). The THEORY’s promote_when is not met.
  • THEORY-1080 (Rovelli 1996, consistency shown for one case). Its second half is still only shown in examples. The 2025 paper adds the three-system merge and Wigner’s friend relative to O. It also adds a qualification the 1996 text lacked: perspectives agree only when they “couple properly”, and coupling to a complementary variable can destroy a relative fact held elsewhere (p. 5). That is loss, not contradiction, so it neither proves nor refutes the THEORY. Its first half, that a relation between views is itself relative to a third system, is restated: “When we say that a variable B has information about another variable A, we are saying that this is the case relative to us … relativity iterates” (p. 7).
  • THEORY-1068 (Rovelli 1996, systems prior). The 2025 text keeps the relata prior. A system “is more appropriately thought of as consisting of a set of physical variables” (p. 4), assumed, as W(S) was. What changes is which relatum holds a perspective: a commutative subalgebra, not a system. A system as a whole can have none if its variables do not commute. The same supports CLAIM-477’s note that RQM is non-eliminative. The subalgebra is chosen, not constituted by the relations, so the analogy’s “partly fixed by the constraints” still finds no support in RQM.
  • LIT-1666 (the companion, read in parallel). This paper places its predecessor among the attempts to define facts relative to entire quantum systems, which “rendered the notion somewhat problematic” (p. 4). It uses “stable fact” once, in [13]‘s sense (p. 11), and does not restate the stable facts account. Whoever reads LIT-1666 should check whether its “relative to a system” survives this paper’s correction.
  • CLAIM-342 and the self-confirming loop. This bears on CLAIM-342 by analogy only. The paper says the sole route to another perspective’s facts is to “interact with either that variable or that system in an appropriate way”, and that interacting with the wrong variable can remove the fact (p. 5). Evidence that reaches a perspective is shaped by how it couples. The paper does not take that further.
  • No instruction for machine-learning practice.

Limitations

  • The consistency shown is internal to one perspective. Every agreement (§VI; D8) is computed from one joint state relative to one system (“us”, O). Within one classical joint distribution, if Z is a function of F and Z is a function of W, the two agree by probability theory. The question critics press is whether descriptions relative to different outside systems agree. The paper does not address it beyond (E3) and the dismissal of “really” questions.
  • Facthood is a threshold the paper does not fix. “high enough for the level of accessible resolution or confidence needed” (p. 3) brings back a purpose or a user at the point where the paper claims “no fundamentally special role for observers”. The authors do not discuss this.
  • “No special role for observers”, precisely. The paper gives no special role to observers, but every state is a state relative to a classical subsystem. The role is spread over every commuting set of variables.
  • Which variables are “distinguished” is assumed (§V). Choosing the commutative subalgebra is choosing the perspective, and the paper gives no rule for it beyond decoherence (§VIII), and that is cited.
  • Claimed resolutions are paragraphs. The combination problem, Frauchiger–Renner and the preferred-basis problem are each settled in a sentence or two (p. 5; App. C).
  • Errors in the text: (B8)‘s sign, and the missing minus sign in App. B’s H_A (p. 8). Also the GHZ statement that “the X variable is not a fact relative to anything” (App. C, p. 9). This is my check, not the paper’s: GHZ is a +1 eigenstate of X⊗X⊗X, so X₁ is a fact relative to the commuting product X₂X₃, a variable of the other two qubits. The statement holds for single-qubit variables only.
  • App. A’s properties are stated without proof, including the bound I_{A:B} ≤ 𝓘/2 for pure entangled states.
  • v1 was not read, so I cannot say which of these points the revision added.

Open questions

  • Do descriptions relative to two different outside systems, each a commutative subalgebra, ever disagree about a third pair’s relative facts? A general theorem, or a counterexample, would settle the consistency gap THEORY-621 and THEORY-1080 name (THEORY-1074).
  • What fixes the threshold at which information counts as a fact, and is it ever set without a user?
  • In a full-rank (realistic) measurement, how fast does the conditional I_{A|bₙ} approach maximum compared with I_{A|B}? The paper asserts it is smooth (p. 9) and does not compute it.
  • Is stability ([13]) needed for a relative fact to serve as a record, or does a relative fact at a time suffice? The paper allows facts “in-between interactions” and still calls on stable facts for the extended scenario.

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