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Added a paragraph about the effect of the planet's own internal complexity
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Chiral Anomaly
Chiral Anomaly
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Edit

As pointed out in comments, my answer above only considered one example, aptly described in a comment as $|\text{here}\rangle+|\text{there}\rangle$. My answer was also specialized in another respect: it only considered decoherence resulting from the planet's interaction with external entities. That specialization is relatively easy to handle mathematically, because it ignores the object's internal structure. But a real planet has a rich internal structure on the molecular level, and the molecular complexity of the planet itself is more than sufficient to account for decoherence of its various parts. This implies, once again, that the planet's interior is not protected from decoherence, and this assertion is not restricted to examples of the form $|\text{here}\rangle+|\text{there}\rangle$.

Edit

As pointed out in comments, my answer above only considered one example, aptly described in a comment as $|\text{here}\rangle+|\text{there}\rangle$. My answer was also specialized in another respect: it only considered decoherence resulting from the planet's interaction with external entities. That specialization is relatively easy to handle mathematically, because it ignores the object's internal structure. But a real planet has a rich internal structure on the molecular level, and the molecular complexity of the planet itself is more than sufficient to account for decoherence of its various parts. This implies, once again, that the planet's interior is not protected from decoherence, and this assertion is not restricted to examples of the form $|\text{here}\rangle+|\text{there}\rangle$.

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Chiral Anomaly
Chiral Anomaly
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If this planet started as a pure quantum state, how fast would that state lose its coherence?

In this answer, I'm interpreting "started as a pure quantum state" to mean "started in a quantum superposition of different locations".

The speed of decoherence for objects of various sizes in various environments was analyzed by Tegmark in "Apparent wave function collapse caused by scattering", https://arxiv.org/abs/gr-qc/9310032. One of the environments considered in that paper is the Cosmic Microwave Background (CMB), and the largest object considered in that paper is a bowling ball. Here, I'll apply Tegmark's equation to an earth-sized planet, assuming that the only source of decoherence is the CMB.

This answer uses some numbers from Tables 1 and 2 in that paper, along with an equation from section 4.1 in that paper.

Tegmark starts with a quantum model of ordinary scattering processes, such as the scattering of CMB from the object, and derives this equation: $$ \rho(\mathbf{x},\mathbf{y},t)\approx \rho(\mathbf{x},\mathbf{y},0) \exp\left(-\Lambda t\left[ 1-e^{-r^2/2\lambda^2} \right]\right) \tag{1} $$ with $r\equiv |\mathbf{x}-\mathbf{y}|$. The quantity $\rho(\mathbf{x},\mathbf{y},t)$ is called the reduced state or the reduced density matrix. (In this case, the "matrix" has continuous "indices" $\mathbf{x}$ and $\mathbf{y}$.) It represents the degree to which a quantum superposition of different object-locations $\mathbf{x}$ and $\mathbf{y}$ remains coherent in the presence of environmental effects. If $\rho(\mathbf{x},\mathbf{y},t)$ is zero for $|\mathbf{x}- \mathbf{y}|>R$, then coherence over distances $> R$ is lost. Equation (1) tells us how quickly this happens.

Equation (1) involves two constants, $\Lambda$ and $\lambda$, with the following meanings:

  • $\Lambda=\sigma\Phi$, where $\sigma$ is the total scattering cross-section (of the bowling ball or the earth) and, in Tegmark's words, $\Phi$ is the "average particle flux per unit area per unit time." In the present example, the "particle flux" is the flux of the CMB.

  • $\lambda$ is a length scale that characterizes the effect of the environment in equation (1). According to Tegmark's Table 1, the CMB is characterized by $\lambda\sim$ 2 millimeters.

According to Tegmark's Table 2, a bowling ball has $\Lambda\sim 4\times 10^{15}\text{ s}^{-1}$. To scale this up to the earth, we can assume that the scattering cross-section is proportional to the object's cross-sectional area. Using $\sim 10$ cm for the radius of a bowling ball and $\sim 6,000$ km for the radius of the earth, the ratio of their areas is $\sim 4\times 10^{15}$. Therefore, for the earth, we have $\Lambda\sim 10^{31}\text{ s}^{-1}$. We can use this value of $\Lambda$, together with $\lambda\sim 2$ millimeters for the CMB, to draw the following conclusions from equation (1):

  • If the earth starts in a quantum superposition of two different locations separated from each other by much more than $2$ millimeters, then it decoheres (effectively "collapses" into just one of the locations) in $\sim 10^{-31}$ seconds.

  • For smaller separations, decoherence is slower, but still very fast. If the earth starts in a quantum superposition of two different locations separated from each other by roughly the width of a single atom, then it decoheres in $\sim 10^{-15}$ seconds — roughly one hundreth of the time required for light to cross the diameter of a typical hair folicle.

So...

If this planet started as a [quantum superposition of different locations], how fast would that state lose its coherence?

Answer: Very, very quickly.

In such a system, clearly there are many more degrees of freedom that are isolated from the environment compared with those coupled to the outside. So I want to know if those isolated DOF somehow protect the purity of the quantum state.

Equation (1) depends only on the size of the object (more accurately, on its scattering cross-section), not on its internal composition. Since the earth is bound together as a single object, decoherence caused by scattering of CMB at the surface causes decoherence of the whole thing. You can't hide from decoherence by tunneling underground.

By the way, a planet-sized object always has a significant gravitational field, which influences the motion of all interplanetary molecules that are anywhere nearby. Since that influence depends on the planet's location, this is presumably sufficient to cause rapid decoherence even if the CMB were absent. But since we don't yet have an empirically-established theory of quantum gravity, I'll leave that fun thought on the shelf for now.