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For the classical Poison and Rat puzzle, we need at least $\lceil\log_2({\rm bottles})\rceil$ rats to figure out the poisoned bottle.

If we have Schrödinger’s quantum rats, can we use fewer rats(qubits) to determine which bottle is poisonous?

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No, the lower bound for the number of rats you need is still the same. Let $N$ be the number of bottles, so our experiment must return $\log_2(N)$ bits of information. For each rat, we know if it lives or dies, so we get a maximum of 1 bit of information from it (even though it's a qubit, you still have to measure it, and that reveals at most 1 bit of information). Hence, you need at least $\lceil\log_2(N)\rceil$ rats.

However, quantum rats do provide other advantages. For example, you can tell if a bottle contains poison without ever killing a rat! (https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester)

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