# Hayden-Preskill informational mirrors and decryption

I do have a question about an assumption made in the very interesting Hayden-Preskill paper of black holes as informational mirrors. Alice throws her top secret quantum diary which is $k$ qubits long into a black hole hoping to get rid of it. Unfortunately, the top forensic scientist Bob was already maximally entangled with the black hole, and in a sense, "knows" the full microstate of the black hole before Alice dumped her diary into it. If $c$ is the scrambling time, and Bob waits to collect $k+c$ qubits worth of Hawking radiation plus a few more to reduce the margin of error up to the desired level, then in principle, Alice's quantum diary can be read by Bob. This argument is purely quantum informational theoretic. What are the computational resources needed for Bob to decrypt Alice's diary? By assumption, the dynamics of black hole evolution are described by a reversible unitary transformation, and not by Krauss operators. So, if Bob is willing to wait until the black hole has completely evaporated away and reverse the computations, he can read Alice's diary in polynomial time. However, Bob is too impatient for that, and he wants to read it after only getting $k+c$ plus a few more qubits worth of information. If Bob is willing to wait exponentially long (and he's not), he can create exponentially many identical black holes one by one and go over all possible $k$ qubit combinations to dump into the identical black holes and wait until he finds one which gives Hawking radiation that matches the Hawking radiation he actually got from the original hole. Bob is also unable to reverse the data either because he doesn't know the microstate of the black hole after collecting $k+c$ qubits worth of Hawking radiation. And apparently, neither can Alice if she doesn't already have a copy of her diary.

There is an analogous situation in classical computation. A cryptographic hash function is a polynomially computable function which takes in as arguments a seed which can be publicly known, and some unknown string to be encrypted. It's a one-way function. It's not possible to compute to original string from the encrypted string using polynomial resources. However, if we are given another string and asked to test if it's identical to the original string, we can compute its hash function using the same seed and check if it matches. If it doesn't, they're not identical. If they do, there's a very high chance that they are indeed identical.

Are black holes quantum cryptographic hash functions? What is the best lower bound on the time it would take for Bob to read Alice's diary?

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I think a more precise question would be, Is black hole evaporation a quantum one-way function? And I'm not aware of a decisive argument one way or the other. But it's very suggestive that the simplest black holes in string theory, the black holes in AdS3, have an entropy based on representations of the monster group, whose construction features one of Marcel Golay's error-correcting codes. Sending information into a black hole might be like applying an error-correcting code to it and then generating a "remainder", from which the whole original state can in principle be reconstructed, but only in exponential time.

(Thanks to Vikram Dhillon for suggesting that one-way functions might play a role in quantum gravity, it's how I found this question.)

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