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Ideally, I'd be able to apply an operation to qubit 0 that would collapse the superposition to a set state (say 1) on qubit 0 & 1.

The operation applied to qubit 0 would be a one qubit operation (number of gates doesn't matter), and in this case qubit 1 cannot have any gates applied to it.

By 'collapse' I mean turning the superposition on both qubits into a binary number.

I'm not sure if this is even possible? Apologies if this comes off as a newbie question.

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    $\begingroup$ What are the allowed elementary operations? What do you mean by "collapse"? Is input fixed? Are auxiliary qubits allowed? If the input is fixed and equals $|00\rangle$ then you could just undo or remove the CNOT and the Hadamard and then apply XX on the resulting $|00\rangle$. Alternatively, you could use a constant channel if constant channels are allowed and input is arbitrary. Or you could SWAP both qubits with an auxiliary register initialized to $|11\rangle$ if auxiliary qubits are allowed. Or you could measure with post-selection if post-selection is allowed... $\endgroup$ Oct 15, 2021 at 6:53

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No, what you're asking for isn't possible. For example, it would violate the no communication theorem.

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