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In the video here: https://www.youtube.com/watch?v=d_XuFkVdAYU (10:57), it is said that the loss of information when matter falls into a black hole is a paradox. He gives an example of throwing a chicken vs a duck and that information being lost once it evaporates.

But what about annihilation? If I take a chicken and an "anti-chicken" which is made of the corresponding anti-matter particles and make them collide and convert to energy. And then I do this with a duck and an "anti-duck". Doesn't this also destroy information on whether it was a chicken or a duck? Can you tell somehow from the radiation emitted weather it was a chicken or a duck?

Rohit Pandey
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Pair annihilation is reversible. If you have a positron and an electron collide and produce a pair of photons, then there is no arrow of time in the process, because the same pair of photons, time-reversed, could validly collide and produce the same positron/electron pair. Therefore, the photons contain the same information as the particles.

That's different from the black hole case, because the time reversed situation is the chicken and duck emerging from a white hole, and we do not currently believe that white holes are real astrophysical objects. Further, the no-hair theorems say that you can't add state to the black hole to specify the chicken or the duck. In some sense, the information really is lost, at least once ring-down and redshift have done their work.

Zo the Relativist
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Some clarification on the definition of information is needed here. Loosely, here, to conserve information means that the phenomenon is time reversible.

You are right that two particles, when annihilated, are lost from the Universe. But this is a Unitary process. In other words, their evolution can, in theory, be undone with the information at hand (or at least we can determine what their properties were).

The case of black hole evaporation: Now assume that a black hole evaporates over time by emitting Hawking radiation. Every particle that is emitted is entangled with the black hole. Now, we can measure this emitted particle and determine its properties. But we can never determine the full property of the entangled state as the other remaining entangled bit is behind the black hole horizon.

This is OK, but what if the entire black hole evaporates? Now, not only can we not measure the information of the state behind the horizon, but it does not even exist. It seems like we have lost the information of the other half of the entangled states permanently, which seems like a paradox because elsewhere in nature, information is never lost.

As a result we cannot now tell what fell into the black hole while it was created.

S.G
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