# Correlation function late time decay and information loss

In the perturbative treatment of, let's say, a scalar field theory on AdS spacetime, the correlation functions decay exponentially at late times, indicating, as people say, information loss.

I guess there are two questions I'd like a clarification for:

1. How can I see that the exponential decay of correlations and information loss are related? Is there a mathematical treatment that links both notions together?

2. Is decay of correlation functions at late time ALWAYS indicating information loss, even if it is power-law like, or even slower than that?

Other than an informed answer, I would appreciate some references for further reading, that help connect the two notions together (they could be from different fields in Physics for example).

For the first question, the relation between an information measure and correlation function is given by: $$I(A : B) \geq \frac{\mathcal{C}\left(\mathcal{O}_{A}, \mathcal{O}_{B}\right)^{2}}{2\,\left\|\mathcal{O}_{A}\right\|^{2} \left\|\mathcal{O}_{B}\right\|^{2}}$$ where $$\mathcal{C}\left(\mathcal{O}_{A}, \mathcal{O}_{B}\right)$$ is the correlation function of observables $$\mathcal{O}_{A}$$ and $$\mathcal{O}_{B}$$ defined by: $$\mathcal{C}\left(\mathcal{O}_{A}, \mathcal{O}_{B}\right):=\left\langle \mathcal{O}_{A} \,\mathcal{O}_{B}\right\rangle-\left\langle \mathcal{O}_{A}\right\rangle\left\langle \mathcal{O}_{B}\right\rangle$$ and $$||\cdot||$$ denotes the operator norm , and $$I(A:B):= S\left(\rho_{A}\right)+S\left(\rho_{B}\right)-S\left(\rho_{A B}\right)$$ is the "mutual information" between two subsystems $$A$$ and $$B$$, which as you could see is a linear combination of their von Neumann entropies. Hence your "information loss" is due to this inequality. Now it might be interesting to know that in the case of a finite correlation length and exponentially decaying $$\mathcal{C}$$ , we have an "area law" behavior for the entanglement entropy which in the context of information theory implies that a quantum computation in the quantum circuit model can be simulated classically in polynomial time . In the holographic context, this exp. decay w.r.t. our mentioned inequality (together with Ryu-Takayanagi formula) was the basis of van Raamsdonk's argument for the emergence of spacetime . Also the mentioned behavior for the equal-time correlators is used as a good probe of thermalization process for strongly coupled field theories . My opinion for the second question is that since we have a polynomial decay of correlation functions for gapless (scale invariant) theories , then by (a) Zamolodchikov's c-theorem which states that the central charge along the RG flow (connecting two fixed points) decreases , and (b) entanglement entropy is proportional to the central charge and counts the number of d.o.f , we could also see the information loss as well. I hope this helps you.