# About the relation between entropy and Fisher information matrix

It's well known that the Fisher information metric can be given by $$g_{i,j}=-E\left[\frac{\partial \ln(p(x,\theta))}{\partial \theta_{i}}\frac{\partial \ln(p(x,\theta))}{\partial \theta_{j}}\right],$$ where $p(x,\theta)$ is a pdf of random variable x and $\theta$ is a parameter vector.

In the paper Derivation of Gravitational Field Equation from Entanglement Entropy, instead, the authors claimed that $$g_{i,j}=-\partial_{\theta_{i}}\partial_{\theta_{j}}(S(\theta)),$$ where $S(\theta)=-\sum_{x} p(x,\theta)\ln (p(x,\theta))$ is the entanglement entropy with $p(x,\theta)$ the spectrum of the density matrix. They then regard the entanglement entropy as a Hessian potential so that they can derive gravitational metric from entanglement entropy.

But my derivative result (by a direct calculation) is $-\partial_{\theta_{i}}\partial_{\theta_{j}}(S(\theta))=g_{i,j}-\sum_{x} \ln(p(x,\theta))*\partial_{\theta_{i}}\partial_{\theta_{j}} p(x,\theta)dx$, where the second term does not vanish. Then it seems that the entanglement entropy is not exactly a Hessian potential.

Can anybody help to clarify this? Is there an error somewhere in my or their results?

• Is that the entropic gravity theory with results in the news recently? Dec 25, 2016 at 1:13
• Wikipedia (en.wikipedia.org/wiki/Fisher_information#Matrix_form) mentions "certain regularity conditions" under which something, which seems the same as you ask, is true. Try having a look. I'm also interested in the answer... Apr 26, 2017 at 20:45