Source
Di Biagio and Rovelli (2025), LIT-1676, arXiv 2510.11349v2, §VII (pp. 5–6), Fig. 1 and App. B (pp. 8–9); read in NOTE-1214.
What was actually shown
The general claim. In von Neumann’s scheme the coupled system goes from Σ αₙ|aₙ⟩|b_r⟩ to Σ αₙ|aₙ⟩|bₙ⟩ (eqs. 17–18) under the Schrödinger equation. I_{A:B} goes from 0 to H_A, and the relative information I_{A|B} reaches I_A^max. “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. With Ĥ = iω Σₙ |aₙ⟩⟨aₙ| ⊗ (|bₙ⟩⟨b_r| − |b_r⟩⟨bₙ|) on t ∈ [0, π/2ω] (B3), |ψ(t)⟩ = cos ωt|ψ_i⟩ + sin ωt|ψ_f⟩ (B5), and p_t(aₙ, b_k) = |αₙ|²(cos²ωt δ_kr + sin²ωt δ_nk) (B6). The marginal p_t(aₙ) = |αₙ|² does not change, so the outside system’s information about A is constant, while I_{A:B}(t) = sin²(ωt) H_A (B7). The reading checked (B7) from (B6): H_B and H_AB each carry the binary entropy of cos²ωt, which cancels. By eq. (7), I_{A|B}(t) = I_A^max − cos²(ωt)H_A. The paper prints ”+” (B8, p. 8), which would exceed the maximum for t < T. That is a sign error.
What could have come out otherwise, and did, in the ideal case. The conditional quantity behaves differently. “according to the state |ψ(t)⟩, A is a fact relative to B = bₙ for all values of t > 0, because p_{t>0}(a|bₙ) = δ_{a,aₙ}” (p. 9). As soon as the pointer reads bₙ with any probability, that reading fixes A. The authors call this “an artefact of studying a too idealised system, where p_0(bₙ) is exactly vanishing”. In a realistic, full-rank state I_{A|bₙ} starts at I_A^max − H_A “and smoothly increases to close to maximal over a short time” (p. 9). That case is described, not computed.
Not specifically quantum. “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).
What this does not say
- That every relative fact is reached continuously. The expected relative information is continuous. A fact conditional on a pointer value is reached at once in the ideal model, and the paper’s remedy, full rank, is asserted with citations [56, 57] and not worked out.
- When a measurement ends. That is set by a threshold on information, “an ‘acceptable’ level” (p. 6), and the paper does not fix it.
- That continuity solves the measurement problem. Which outcome occurs is outside the account. The paper’s position is that asking what “really” happened has no empirical content (§IX).
- That a measurement differs from any correlating interaction. Nothing in the definitions separates the two. A measurement is a coupling that takes I_{A|B} to its maximum.
- That this is new as mathematics. It is Shannon quantities evaluated on the standard von Neumann model, and the authors say as much for the classical case.