Source
Zurek (2009), LIT-1640, section II (pp. 2–4), Figs. 2–3 and section III (p. 5) of the arXiv preprint 0903.5082; read in NOTE-1229.
What was actually shown
The measures. ρ_SF = Tr_{E/F}|Ψ_SE⟩⟨Ψ_SE| (eq. 2) and I(S : F) = H_S + H_F − H_{S,F} (eq. 3), for fragments F that are disjoint collections of environment subsystems, each “individually accessible to measurements” with commuting observables (Fig. 1). Redundancy is the number of disjoint fragments supplying (1 − δ)H_S: R_δ = 1/f_δ (eq. 4, p. 3).
What could have come out otherwise. For Haar-random pure states of SE, fragments smaller than half the environment hold almost nothing, and “such random states … exhibit no redundancy” (Fig. 2). States produced by decoherence instead rise “quickly” to H_S at small f_δ (Fig. 2, schematic). In spin-environment models “the same number of spins suffices to reach the plateau”, so redundancy grows with the number of environment subsystems interacting with S; in quantum Brownian motion R_δ ≈ s^{2δ} for squeeze factor s (p. 3). These results are cited from refs. [12]–[16].
Selectivity. With Shannon entropies of a specific observable σ(μ) = cos μ σ_z + sin μ σ_x in a spin model with N = 50, the whole environment holds information about many observables, but R_{δ=0.1} “is sharply peaked around the pointer observable: Redundancy is a very selective criterion” (Fig. 3). A cited theorem [12, 24]: “only pointer states can be discovered from fractions of E. Other observables … can be deduced only to the extent they are correlated with the pointer observable” (p. 5). THEORY-1053 gives the reason only orthogonal sets can be copied at all.
The identification. “Large redundancy implies objectivity: The state of the system can be found out indirectly and independently by many observers, who will agree about their conclusions” (Fig. 2 caption). Independence is argued from commuting observables: “Observables of different F’s commute – such measurements are independent. Yet, underlying correlations mean that their outcomes imply the same state of the system, as if S were classical” (p. 3).
What this does not say
- That a mutual-information plateau guarantees independently accessible classical records. The paper infers redundancy from the plateau (“Rapid rise and gradual leveling of I(S : F_f), Fig. 2, implies redundancy”, p. 3) and objectivity from redundancy. It gives no condition on the joint state of S and several fragments under which each fragment’s record determines the same outcome. Its closest step is to note that von Neumann mutual information can exceed H_S through global correlations (up to 2H_S, Fig. 2; p. 4) and to switch to observable-specific Shannon quantities for Fig. 3 (p. 5). The caveat that mutual information alone is not enough is the later literature’s, not this paper’s.
- That redundancy is derived in general. It is shown in models cited from elsewhere; Fig. 2 is schematic; the mixed-environment claim rests on work “in preparation” (p. 4).
- That the parts are found rather than chosen. The system–environment split and the subdivision of E into accessible subsystems are assumed (Fig. 1). Redundancy is relative to that choice.
- That decoherence suffices for records. The paper says it does not: a completely mixed environment “can still induce decoherence” but is “useless” as a channel, and air molecules’ records become “effectively undecipherable” (p. 4).
- That objectivity here is the same as fact-hood across observers in a relational sense. Agreement is for pointer observables read from a shared medium, with the Born rule assumed in computing the measures (p. 6). It is not a consistency result of the kind THEORY-621’s promote_when asks for.
- That the measurement problem is solved. That is the author’s verdict (p. 9); this finding is the redundancy result and its reading.