Source
Schindler, Strasberg, Galke, Winter & Jabbour (2025), “Unification of observational entropy with maximum entropy principles”, arXiv 2503.15612v1, LIT-082, read in full in NOTE-043. The simulation code was not run.
What was actually shown
The definition (Eqs. 10–13). Any state τ defines a linear constraint in informational form, S(ρ;τ) ≤ S(τ), on which τ is the MaxEnt state. Then S^τ_M(ρ) = S(τ) − D_M(ρ‖τ) = −Σ p_x log(p_x/V_x), with prior-weighted volumes V_x = Tr(τM_x)e
Bounds (Eqs. 17–18, Proof 2). S(τ) ≥ S^τ_M(ρ) ≥ S(ρ) for ρ in the constraint set; the upper bound is tight iff M cannot tell ρ from τ. S(τ) is the most information any measurement can extract from a constrained state (Proof 1, with a saturating state).
Special cases (§V, Table II). Twenty-four numbered items: Jaynes, von Neumann, traditional observational entropy (τ ∝ 1), Boltzmann, Gibbs, canonical, microcanonical, Wehrl, free energy, the H-theorem entropies and others. Item 21 recasts von Neumann’s 1929 H-theorem entropy difference as D_M(ψ‖Uψ).
Infinite dimensions. Traditional observational entropy is infinite for almost all states there; the prior makes it finite whenever S(τ) < ∞ (§VIII.7). The continuity bound (Eq. 39) needs finitely many non-zero volumes.
Second laws (Eq. 27, Proof 7). The time-averaged D_M(ρ(t)‖τ) splits into an equilibration term, D_M(ρ(t)‖ρ̄) averaged, and a prior-fit term, D_M(ρ̄‖τ). The first is small when the number of outcomes m ≪ √d₂(ρ̄), under nondegenerate energies and gaps (Eq. 28). The second is small under the ETH (Eq. 31). Under local detailed balance the entropy is non-decreasing (Eq. 35, Proof 11).
What could have come out otherwise: the lower bound could have failed for some measurement, and the prior could have left the infinite-dimensional pathology in place. Neither happens, under the stated constraint.
What this does not say
- That entropy increase is proved. Only sufficient conditions are. The mesoscopic regime of the paper’s own Fig. 4 falls between them, and the equilibration time is shown to exist but is not estimated.
- That fluctuation probability decays as Δ/δ. Proof 5 gives only the Markov upper bound Pr ≤ 2Δ/δ; the main text’s ”∝” overstates it.
- That the prior-weighted volumes are necessary for generic increase. Argued from §VI and Figs. 4–5, with no theorem (C11).
- That integrable systems differ in kind. Four hand-chosen initial conditions in one 2-D gas (C12).
- Which measurement and prior are right. The entropy is objective once (M, τ) are fixed; choosing them is a modelling decision (§X).
Connections
- THEORY-463 holds that a strictly irreversible approach to equilibrium cannot come from isolated Hamiltonian dynamics alone, and names this paper as the constructive side of the coarse-graining and maximum-entropy threads. The two agree: here the increase comes from the choice of M and τ, and is proved only under extra conditions.
- The algebra “prior entropy minus measured divergence from the prior” restates cross-entropy and KL identities; the paper draws no connection to learning.