The Necromancy-Hardness of the Schrödinger's Cat Experiment

Motivated by puzzles in quantum gravity AdS/CFT, Lenny Susskind posed the following question: supposing one had the technological ability to distinguish a macroscopic superposition of two given states |v> and |w> from incoherent mixture of those states, would one also have the technological ability to map |v> to |w> and vice versa?  More precisely, how does the quantum circuit complexity of the one task relate to the quantum circuit complexity of the other?  Here we resolve Susskind's question -- showing that the two complexities are essentially identical, even for approximate versions of these tasks, with the one caveat that a unitary transformation that maps |v> to |w> and |w> to -|v> need not imply any distinguishing ability.  Informally, "if you had the ability to prove Schrödinger's cat was in superposition, you'd necessarily also have the ability to bring a dead cat back to life."  I'll also discuss the optimality of this little result and some of its implications.

Paper (with Yosi Atia) in preparation