The conservation of quantum information is equivalent to the time-reversibility, i.e.: unitarity of quantum dynamical evolution. There is no way of evolving from an electron and a positron to photons unitarily: a measurement must be performed. And indeed, if you are certain that you started with an electron and a positron, and at the end you are certain that you have only photons, then you must have performed a measurement along the way.
If you don't perform any measurement, what's going to happen is that the initial electron-positron state of the underlying quantum fields is going to (unitarily) evolve so as to produce a superposition of many different possible photon outcomes, as well as an outcome without interaction.
To elaborate, at every moment in time, the interaction term between the matter field and the photon field means that there's a certain probability rate for an interaction to occur. So once a small amount of time $\delta t$ elapses, the wave function will split into mostly a non-interacting term with still one electron and one positron, and a small term of order $\delta t$ where they have annihilated and produced photons. You then take another time step, and each of the two above terms splits further into every combination of possible interactions. The final state after an infinite time is a complicated and entangled superposition of many different field states.
That coherent superposition can be "run backwards" to recover the initial state, so it contains the full quantum information that you had to begin with. In practice, it will be measured, losing the coherent information.