# Information Loss in annihilation

The concept of information loss is usually discussed with respect to a black hole. My understanding is that whatever matter you put into the black hole, it has only 3 "hairs" and so one doesn't know, just by determining the properties of the black hole, the mechanism by which the black hole was formed. There have been many developments with many people now believing that information is not really lost but gets mangled, etc.

Why is this loss of information not discussed in a far more pedestrian context? If you have a particle and an anti-particle annihilating into two photons, say; by observing the photons, you cannot reconstruct the velocities of the two particles. Have you lost information in this case? Is this concept of information identical to the Shannon definition? If annihilation is unitary, with entropy being conserved, I understand that Shannon information is also conserved. But, we cannot reverse-evolve to a unique initial state, can we? (Velocity isn't Lorentz-invariant, but, let us say that everything is carried out in a single inertial frame.)

More generally, I don't understand how information is not lost in so many processes that are many-to-one because of the nature of particle physics and why this is different from the scenario with black holes.

QFT is time reversible and unitary. That the outcome of a particular scattering experiment (in this case $e^{+} + e^{-}\rightarrow 2\gamma$) is random doesn't mean that you couldn't construct the initial state assuming that you reproduced the experiment many times, and did nothing but measure the final photon states.
In this link the cross-section for this collision is found in terms of the Mandelstam variables commonly used to encode the momenta of particles in particle physics. Note that the answer depends on $s,t,$ and $u$. Since the differential cross section is measurable, this means that this experiment lets us measure our mandelstam variables, and therefore, we can determine information about the momentum of the electron, independently of the momentum of the positron.
(As an aside, here is a mechanism how this could happen. We have our particle in state $\rho_1$ and another particle in state $\rho_2$. They interact with some unitary $U$, so that we have $U (\rho_1 \otimes \rho_2) U^\dagger$. Subsequently, particle one escapes from the black hole, while the particle 2 stays inside. Upon exiting, the first particle's state is $F(\rho_1) = \text{Tr}_2[U (\rho_1 \otimes \rho_2) U^\dagger]$. This is very much like decoherence and it is usually (but not necessarily) the case that $S(F(\rho_1)) \geq S(\rho_1)$. The kind of operations where entropy is increased are common when interacting with the environment and it is precisely entangling operations with the environment - also called decoherence - that cause entropy increase in states and the emergence of the classical world. For a more detailed read on this, see this execellent review by Zurek: Decoherence, einselection and the quantum origins of the classical)