On the Hardness of Detecting Macroscopic Superpositions

When is decoherence "effectively irreversible"? Here we examine this central question of quantum foundations using the tools of quantum computational complexity. We prove that, if one had a quantum circuit to determine if a system was in an equal superposition of two orthogonal states (for example, the $|$Alive$\rangle$ and $|$Dead$\rangle$ states of Schr\"{o}dinger's cat), then with only a slightly larger circuit, one could also $\mathit{swap}$ the two states (e.g., bring a dead cat back to life). In other words, observing interference between the $|$Alive$\rangle$and $|$Dead$\rangle$ states is a "necromancy-hard" problem, technologically infeasible in any world where death is permanent. As for the converse statement (i.e., ability to swap implies ability to detect interference), we show that it holds modulo a single exception, involving unitaries that (for example) map $|$Alive$\rangle$ to $|$Dead$\rangle$ but $|$Dead$\rangle$ to -$|$Alive$\rangle$. We also show that these statements are robust---i.e., even a $\mathit{partial}$ ability to observe interference implies partial swapping ability, and vice versa. Finally, without relying on any unproved complexity conjectures, we show that all of these results are quantitatively tight. Our results have possible implications for the state dependence of observables in quantum gravity, the subject that originally motivated this study.