Shutters, Boxes, but No Paradoxes: Time Symmetry Puzzles in Quantum Theory

Abstract

The “N‐box experiment” is a much‐discussed thought experiment in quantum mechanics. It is claimed by some authors that a single particle prepared in a superposition of N+1 box locations and which is subject to a final “post‐selection” measurement corresponding to a different superposition can be said to have occupied “with certainty” N boxes during the intervening time. However, others have argued that under closer inspection, this surprising claim fails to hold. Aharonov and Vaidman have continued their advocacy of the claim in question by proposing a variation on the N‐box experiment, in which the boxes are replaced by shutters and the pre‐ and post‐selected particle is entangled with a photon. These authors argue that the resulting “N‐shutter experiment” strengthens their original claim regarding the N‐box experiment. It is argued in this article that the apparently surprising features of this variation are no more robust than those of the N‐box experiment and that it is not accurate to say that the particle is “with certainty” in all N shutters at any given time.

Figure 1 Hilbert Space of the Shutter Particle.

Notes

Correspondence: Department of Philosophy, University of Maryland, College Park, MD 20742, USA. E‐mail: [email protected]

The Born Rule tells us that the probability for outcome q _k when observable Q is measured on a system prepared in state |Ψ⟩ is given by |<q _k |ψ⟩|².

Aharonov and Vaidman (2002, 3).

Of course, adhering strictly to the name “N‐box experiment” would require us to call this the “two‐shutter experiment”.

The “reduced” density operator corresponding to photon reflection in the Hilbert space of the shutter particle is W _r,sh=Tr _ph[½(|a′⟩|a⟩⟨a|⟨a′| + |a′⟩|a⟩⟨b|⟨b′| + |b′⟩|b⟩⟨a|⟨a′| + |b′|⟩b⟩⟨b|⟨b′|)]=½(|a⟩⟨a| + |b⟩⟨b|); the probability that a shutter particle in this state will be post‐selected is Tr[|Ψ ₂⟩⟨Ψ ₂|W _r,sh]=⅓.

As opposed to a “weak measurement” in which the interaction Hamiltonian between the system and measuring device is weakened; cf. Aharonov and Vaidman (1990).