What Is Nonlocal in Counterfactual Quantum Communication?

We revisit the “counterfactual quantum communication” of Salih et al. [1], who claim that an observer “Bob” can send one bit of information to a second observer “Alice” without any physical particle traveling between them. We show that a locally conserved, massless current—specifically, a current of modular angular momentum, $L_z$ mod $2\hslash$—carries the one bit of information. We integrate the flux of $L_z$ mod $2\hslash$ from Bob to Alice and show that it equals one of the two eigenvalues of $L_z$ mod $2\hslash$, either 0 or $\hslash$, thus precisely accounting for the one bit of information he sends her. We previously [2] obtained this result using weak values of $L_z$ mod $\hslash$; here we do not use weak values.

Figure 1

Alice and Bob at opposite ends of a single cavity with a barrier placed symmetrically between them. A particle wave packet is initially at Alice’s end.

Figure 2

Alice and Bob at opposite ends of an asymmetrical cavity with two barriers between them. A particle wave packet is initially at Alice’s end.

Figure 3

Alice and Bob at opposite ends of two cavities, symmetrically about the $z$ axis. The potential (excluding the mirror) is invariant under a $\pi$ rotation about the $z$ axis, as is the initial particle superposition.

Figure 4

Alice and Bob again share a single cavity, with the mirror at Bob’s end in a superposition of two orthogonal states.