Information paradox with regard to quantum entanglement when width approach Planck length Context:

Suppose I have a 3 d particle placed in a spherical box of infinite well from [-r,r] in all three directions, it's expectation value of position thus equaled to 0.

Thus its surface area equaled to $4 \pi r^2$

From quantum gravity, (existence) https://arxiv.org/abs/gr-qc/9403008 and (a fairly good numerical approximation) https://en.wikipedia.org/wiki/Planck_length we knew that space and time were quantized fractions. Suppose the minimum length equal to ds, then the maximum partition of the surface of the spherical ball equal to $N= 4 \pi r^2/ds^2$.

Which meant that, as the increase of r, the number of possible segment that our probe could be placed will increase.

In analogy, suppose I have a particle of spin 1/2. Where we place the particle at the center of the ball and measure its spin. Then the ball with larger r could have more "segment" area for observation.

Question 1

Was these analysis true? If not, why? Further what's its implication?

Question 2

Suppose I created a pair of such spin 1/2 particle entangled together. One placed in a ball of $r_a$ the other placed in a ball or $r_b$. If $r_b>r_a$, then our measurement could be more "precise" about the ball b than ball a. In an imaginary extreme case where $N=4 \pi r_a^2/ds^2=2 $, measurement for ball a thus could only be up or down.

What's happened to the information here? Were they still consist?


1 in question 2, since it's an infinite well(although it was not possible in real), It did not had to be exact in the "center" of $(0,0,0)$. By the fact that the particle was not at the boundary, and, since it's spherical coordinates, by symmetry, position expectation value was at the center. In fact, it didn't even have to be at the center position. Wave was good. The encoding was based on the probability of $T_{funning}$ was selected such that it equaled to 0 or $<<1$. Thus could be ignored regard to numerical calculation.

2 the imaginary extrem was based on the fact that electron's classical readius was in e-16 and plunk length was in e-32 thus $N=2$ could not happen, but just to demonstrate the idea. Feel free to ask question about the context.


When the quantum gravity was assumed, it was essentially ignite the continuous of the space-time.

Especially, when taking the length comparable to $ds$(in this case, the plank length). Quote my professor: the superposition of states were destroyed. In the extreme case of $ds^3$, the wave function became a Dirac Delta function.

Thus, by taking the length comparable to $ds$, we essentially altered the particles and thus information was no longer valid, especially, the entanglement was altered. (Also, in another sense, when we confined the particle in such small fixed region, we had indirectly made an observation about to its position.)


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