3 where H χ ˆ F is the entropy of ρ χ ˆ F and H 0 e = H 0 $_{F}$/fN is the initial entropy of a single thermal photon (which di- verges since the photon Hilbert space is infinite dimen- sional). Although the conditional state ρ χ ˆ F lives in an infinite-dimensional vector space, it has only two nonzero eigenvalues, After plugging these into the formula for entropy, we can perform the integral in Eq. (11). The divergent pieces cancel in the mutual information and we are left with The mutual information is plotted in Fig. 2(a) as a function of f for different values of t . This is a partial information plot [2, 4, 7]. It charts how much information about the object is available to an observer depending on the size of the fragment captured. For times t ≫ τ$_{D}$ , the mutual information has a distinctive plateau that indicates redundancy. The summations in (13) can be written in a closed form analogous to Eq. (4), but the power series is more useful for calculating the redundancy. For large times, Γ = exp( − t/τ$_{D}$ ) is exponentially small and the sum is dominated by the lowest power of Γ. If f < 1 / 2, then f < (1 − f ) < 1 and This allows us to estimate the redundancy (for δ < 0 . 5) in the limit t ≫ τ$_{D}$ : The key points are these: First, the redundancy de- pends only weakly (logarithmically) on the information deficit δ , which is consistent with previous results [6, 17]. Second, the redundancy increases linearly with time at a rate given by the inverse of the decoherence time. This is intuitive because (1) photons scatter off the object at a constant rate and (2) it is precisely the depen- dence of photon out states on the position of the object (roughly corresponding to a record) that causes decoher- ence. Since even very tiny objects have extremely short decoherence times [1, 9], the redundancy quickly becomes enormous. It should be noted that there will not be strong re- dundancy if the object is illuminated uniformly from all directions—despite the fact that the rate of decoherence is simply given by averaging Eq. (6) over the solid angle. (b) For isotropic illumination, the same time slicing is used but there is greatly decreased mutual information because the directional photon states are already “full” and cannot store more information about the state of the object. Zero redundant copies are produced and the mutual information approaches 0 as t → ∞ for all f < 1. (a) For point-source illumination, individual curves are la- beled by the time t in units of the characteristic time τ$_{D}$ , Eq. (6). For t ≤ τ$_{D}$ (red dashed lines), the information about the system available in the environment is low. The linear- ity in f means each piece of the environment contains new, independent information. For t > τ$_{D}$ (blue solid lines), the plateau shape of the curve indicates redundancy; the first few pieces of the environment give a large amount of information, but additional pieces just confirm what is already known. On the plateau, the mutual information approaches it’s maximum classical value, I$_{S}$$_{:}$$_{F}$$_{f}$ = 1 bit = ln 2 nats ≈ 0 . 69 nats. The remaining information (i.e., above the plateau) is highly en- coded in the global state, in the sense that it can only read by capturing almost all of E . I$_{S}$$_{:}$$_{F}$$_{f}$ ≈ ln 2 − 1 2 Γ $^{f}$. (14) R$_{δ}$ = 1 ln[(2 δ ln 2) − $^{1}$] t τ$_{D}$ . (15) λ χ ˆ F = 1 2 ± 1 2 fN ∏ i =1 ∣ $_{∣}$〈 ˆ n | S k$_{i}$ ⃗ x$_{1}$ $^{†}$S k$_{i}$ ⃗ $_{x$_{2}$}$| ˆ n 〉 ∣ $_{∣}$. (12) I$_{S}$$_{:}$$_{F}$$_{f}$ = ln 2 + ∞ ∑ n =1 Γ (1 − f ) n − Γ fn − Γ n 2 n (2 n − 1) , (13) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 f I S:F$^{f}$ t = 10 - $^{1}$t$_{D}$ 10 - 1 GLYPH<144> 2 10 0 10 1 GLYPH<144> 2 10 1 10 3 GLYPH<144> 2 10 2 (a)Point-source illumination 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 f I S:F$^{f}$ (b)Isotropic illumination FIG. 2. The quantum mutual information I$_{S}$$_{:}$$_{F}$$_{f}$ versus frag- ment size f at different elapsed times for an object illumi- nated by (a) point-source blackbody radiation, Eq. (13), and (b) isotropic blackbody radiation, Eq. (17).