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nal results, we will take V and N to infinity while holding the photon density N/V constant. 

The primary quantity investigated will be the quan- tum mutual information I$_{S}$$_{:}$$_{F}$ = H$_{S}$ + H$_{F}$ − H$_{S}$$_{,}$$_{F}$ be- tween the system S and fragment F , where H denotes the von Neumann entropy. From this we will calculate the redundancy R$_{δ}$ , which is the number of distinct frag- ments in the environment that supply, up to an informa- tion deficit δ , the classical information about the state of the system. More precisely, R$_{δ}$ = 1 /f$_{δ}$ , where f$_{δ}$ is the smallest fragment such that I$_{S}$$_{:}$$_{F}$$_{f}$$_{δ}$ = (1 − δ ) H$_{S}$ . (Only very large fragments f ≥ 0 . 5 will have complete classical information about the object [4].) 

The sphere and the photons in the environment are assumed to be initially unentangled: ρ 0 = ρ 0 S ⊗ ρ 0 e ⊗ · · · ⊗ ρ 0 $_{e}$, where ρ$_{S}$ and ρ$_{e}$ are the density matrices of the system and of a single photon, respectively, and a superscript “0” denotes prescattering states. The photon momenta are distributed according to ρ 0 e = ∫ ∞ 0 d k p ( k ) k $^{2}$| ⃗ k 〉〈 ⃗ k | for p ( k ) ∝ k $^{2}$/ [exp( kc/k$_{B}$T ) − 1] and ˆ n = ˆ k$_{i}$ a unique direction. 

The decoherence of the superposition is governed by 

where 

and S$_{⃗}$ x$_{p}$ is the scattering matrix acting on the single pho- ton state when the particle is located at ⃗ x$_{p}$ . Because γ controls the suppression of the off-diagonal terms of the object’s density matrix in the position basis, γ and Γ ≡ γ N are the decoherence factors attributable to a sin- gle scattering photon and the environment as a whole, respectively. The two-dimensional ρ$_{S}$ can be diagonal- ized and its entropy is 

We use the classical cross section of a dielectric sphere [15] in the dipole approximation ( λ ≫ a ) and assume the photons are not sufficiently energetic to resolve the su- perposition individually ( λ ≫ Δ x ). We further assume that the object is heavy enough to have negligible recoil and that photon energy is conserved. Under these con- ditions, the key matrix element (which coincides with γ in the case of monochromatic radiation) is 

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to leading order in 1 /V . Above, ˜ a ≡ a [( ϵ − 1) / ( ϵ − 2)] 1 / 3 is the effective radius of the object and t is the elapsed 

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time. The states | ⃗ k ( λ ) 〉 are photon momentum eigen- states with wavelength λ making an angle θ with the separation vector ⃗ Δ x . 

For increasing V , photon momentum eigenstates be- come diffuse so individual photons decohere the state less and less (i.e. γ → 1). This is balanced, of course, by an increasing number of photons in the box, which will lead to a finite decoherence factor for the whole environment, Γ = γ $^{N}$. In the V → ∞ limit we use e = lim$_{q}$$_{→∞}$(1+1 /q ) q to get Γ = exp( − t/τ$_{D}$ ), where [16] 

and C$_{Γ}$ = 161280 ζ (9) /π 3 ≈ 5210 is a numerical con- stant. We have replaced the photon density N/V with the more physical irradiance I (radiative power per unit area). Given Eq. (1), we identify τ$_{D}$ as the decoherence time . Although the rate of decoherence (and, as we shall see, the redundancy) depends on the angle of illumination θ , decoherence is usually so rapid that it hardly matters. 

To get the mutual information, we can avoid calculat- ing H$_{SF}$ by using the identity [Eq. (8) of [6]] 

where H$_{S}$$_{d}$$_{E}$ = H$_{S}$ is the entropy of the system as de- cohered by the entire environment E and H$_{S}$$_{d}$$_{E}$$_{/}$$_{F}$ is the entropy of the system if it were decohered by only E / F . We obtain H$_{S}$$_{d}$$_{E}$$_{/}$$_{F}$ from H$_{S}$ , Eq. (3), by making the re- placement Γ → Γ 1 − $^{f}$. Despite the mixedness of the envi- ronment, it is possible to diagonalize the post-scattering state ρ$_{F}$ to get H$_{F}$ because of the special form of our model; the photons are of mixed energy but are in direc- tional eigenstates, while the elastic scattering conserves energy but mixes photon direction. This allows us to write 

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where we have broken the momentum eigenstates into a tensor product | ⃗ k$_{i}$ 〉 = | k$_{i}$ 〉| ˆ n 〉 /k$_{i}$ of magnitude and di- rectional eigenstates. Above, χ$_{F}$ = ( k$_{1}$, . . . , k$_{fN}$ ) is the vector of the magnitudes of the photon momenta of F , p ( χ$_{F}$ ) = ∏ fN i =1 p ( k$_{i}$ ) is the momentum probability dis- tribution, and | χ$_{F}$ 〉〈 χ$_{F}$ | = ⊗ fN i =1 | k$_{i}$ 〉〈 k$_{i}$ | . S k$_{i}$ ⃗ x$_{p}$ is defined by 

We then have 

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