A macroscopic quantum three-box paradox: finding consistency with weak macroscopic realism

Abstract

The quantum three-box paradox considers a ball prepared in a superposition of being in one of three Boxes. Bob makes measurements by opening either Box 1 or Box 2. After performing some unitary operations (shuffling), Alice can infer with certainty that the ball was detected by Bob, regardless of which box he opened, if she detects the ball after opening Box 3. The paradox is that the ball would have been found with certainty in either box, if that box had been opened. Resolutions of the paradox include that Bob's measurement cannot be made non-invasively, or else that realism cannot be assumed at the quantum level. Here, we strengthen the case for the former argument, by constructing macroscopic versions of the paradox. Macroscopic realism implies that the ball is in one of the boxes, prior to Bob or Alice opening any boxes. We demonstrate consistency of the paradox with macroscopic realism, if carefully defined (as weak macroscopic realism, wMR) to apply to the system at the times prior to Alice or Bob opening any Boxes, but after the unitary operations associated with preparation or shuffling. By solving for the dynamics of the unitary operations, and comparing with mixed states, we demonstrate agreement between the predictions of wMR and quantum mechanics: The paradox only manifests if Alice's shuffling combines both local operations (on Box 3) and nonlocal operations, on the other Boxes. Following previous work, the macroscopic paradox is shown to correspond to a violation of a Leggett-Garg inequality, which implies non-invasive measurability, if wMR holds.