Quantum counterfactual communication without a weak trace

The classical theories of communication rely on the assumption that there has to be a flow of particles from Bob to Alice in order for him to send a message to her. We develop a quantum protocol that allows Alice to perceive Bob's message “counterfactually”; that is, without Alice receiving any particles that have interacted with Bob. By utilizing a setup built on results from interaction-free measurements, we outline a communication protocol whereby the information travels in the opposite direction of the emitted particles. In comparison to previous attempts on such protocols, this one is such that a weak measurement at the message source would not leave a weak trace that could be detected by Alice's receiver. While some interaction-free schemes require a large number of carefully aligned beam splitters, our protocol is realizable with two or more beam splitters. We demonstrate this protocol by numerically solving the time-dependent Schrödinger equation for a Hamiltonian that implements this quantum counterfactual phenomenon.

Figure 1

A stacked MZI IFM device with $N$ beam splitters. If the upper path is free (a) the particle always exits through the upper slot. If the upper path is blocked by detectors (b) the particle exits through the lower path with probability $cos{(\pi /2N)}^{2N}$.

Figure 2

Optical setup of a stacked IFM device showing the spatial occupations of Alice, Bob, and the transmission line.

Figure 3

Quantum evolution of the probability density distribution (solid red curve) of the (a) 0-bit and (b) 1-bit processes. The yellow line (at $x=3.5$) indicates the beam splitter, the dotted green line shows the potential, and the black dashed lines show the spatial divisions.

Figure 4

Flow of particles and information in (a) classical communication schemes and (b) weak-trace-free quantum counterfactual communication. The two parts of the transmission line are separated by a beam splitter. In both cases information flows from Bob to Alice.

Figure 5

Bit error rate (solid) and interaction-free violation rate (dotted) as functions of the process encoding number $M$. The beam-splitter angle, $\theta =\pi /2N$, was given a standard deviation of ${\sigma }_{\theta }=0.01$ rad. (a) and (b) show simulations of devices with $N=2$ and $N=7$ beam splitters, respectively.