Observing a Quantum Measurement

As a model quantum detector system, place a qubit in each path of a Stern-Gerlach apparatus,
so that each qubit registers the passage (or not) of the atom along its path. Now, prepare a
spin-1/2 atom with spin-up along X, send it through, and bring the paths back together after
passing the quantum detectors. The spin qubit and the two detector qubits are now in a GHZ
state. Of course to see this, one must (irreversibly) read out the individual detector states and
measure the atomic spin with an ordinary “downstream” Stern-Gerlach system. While these all
produce random outcomes, the appropriate products are definite, revealing the state. The
products correspond to tensor product observables of which the GHZ state is an eigenstate.
We can choose three independent, compatible observables to characterize the state. Two of
our choices recover the usual Stern-Gerlach collapse scenario that one would get without the
quantum detectors. The third detects the coherent superposition of both scenarios, finding
the two detectors in a Bell state conditioned on the X-state of the atom’s spin. Thus, when
observing the detectors along with the atom, “which path” is compatible with “both paths.”
Without the quantum detectors, one loses the option of measuring the third observable.