Unitarity is represented by a complex function e^{iHt} and so forth, which is analytic. The loss of unitarity does not mean there is a complete loss of everything; in particular quantum information can still be conserved. A simple analytic function of this sort describes standard quantum physics. Gravity as we know is given by a hyperbolic group, such as SO(3, 1) ~ SL(2,C), where the latter has a map to SL(2,R)^2. The functions over these groups have posed difficulties for quantum gravity, for they are explicitly nonunitary. The trick of performing a Wick rotation on time or with τ = it is a way of recovering the compact groups we know in quantum physics.
It does turn out I think that we can think directly about quantum gravity by realizing that the SL(2,R) is related to a braid group with Z --- > B --- > PSL(2,Z), and that the braid group is contained in SL(2,R). Braid groups have correspondence with Yang-Baxter relations and quantum groups. The group SL(2,Z) is the linear fractional group, which is an elementary modular form. An elementary modular function is
f(z) = sum_{n=-∞}^{n=∞}c(n)e^{-2πi nz}
which in this case is a Fourier transform. In this case we are safely in the domain of standard QM and QFT. In general modular functions are meromorphic (analytic everywhere but infinity) and analytic condition is held on the upper half of the complex plane.
Of particular interest to me are the Eisenstein series of modular functions or forms. These define an integer partition function, which is an acceptable partition function or path integral for a stringy black hole. I include a graphic here illustrating an Eisenstein function. This has a certain self-similar structure to it, or what might be called an elementary form of a fractal. In this picture unitarity is replaced with modularity. In this more general setting the transformation do no promote a field through time by some operator, but that the operator simply computes the number of states or degrees of freedom in a way that is consistent. Unitarity is then a special case of this, which happens to fit into our standard ideas of causality.
Gravity is as you say entropy increasing with the concentration of degrees of freedom. Gravity must then of course have some quantum aspect, for it is ultimately an accounting machine for degrees of freedom. These degrees of freedom are quantum field states, which means for gravity to be a "counter" of quantum states it must also be quantum.
I did not bring this part of my work in the paper I submitted here, except only with some sparse mention. My main thrust has been to argue how locality of quantum fields and unitarity are emergent.
Cheers LCAttachment #1: Eisenstein_14.jpg