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).


1 Answer 1


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 [1], 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 [2]. 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 [3]. Also the mentioned behavior for the equal-time correlators is used as a good probe of thermalization process for strongly coupled field theories [4]. My opinion for the second question is that since we have a polynomial decay of correlation functions for gapless (scale invariant) theories [5], then by (a) Zamolodchikov's c-theorem which states that the central charge along the RG flow (connecting two fixed points) decreases [6], and (b) entanglement entropy is proportional to the central charge and counts the number of d.o.f [7], we could also see the information loss as well. I hope this helps you.

[1] https://arxiv.org/abs/0704.3906v2

[2] https://arxiv.org/abs/1309.3789

[3] https://arxiv.org/abs/1005.3035

[4] https://arxiv.org/abs/1103.2683

[5] https://arxiv.org/abs/1409.1231v4

[6] http://adsabs.harvard.edu/abs/1986JETPL..43..730Z

[7] https://arxiv.org/abs/quant-ph/0505193


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