This is the second part of a post on garden hoses, quantum cryptography, and schemes to verify position. The first part, which details the first proposed scheme to verify your position--and exactly how you can cheat it with the help of two friends--is here.
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Having been cheated once (see Figures 1 and 2, in part I of this post), the two verifiers, to try to foil you, might ask you to perform a task that requires information from both of them (Figure 3). For instance, they each send you a number and demand that you calculate the sum, on the principle that you need to be where you say you are in order to receive the two numbers and add them up.
But this system, too, is fool-prone (Figure 4). Each of your comrades can intercept the signal, make a copy, and relay it to the other. That way, both can perform the calculation and send back the result just as if you were really there.
Because this fakery requires your comrades to copy the information, the verifiers figure they can ensnare you by exploiting ideas from quantum cryptography--namely, that a quantum state cannot be reliably copied (Figure 5). The verifier on the left sends you a photon in a certain quantum state and the one on the right sends you a single ordinary bit. To confirm your position, you need to return the photon to the left if the bit is 0 and send it on the right if the bit is 1. Unable to make a copy of the photon, your comrades are at a loss for how to cover for you.
But being good quantum physicists, not to mention good friends, they soon realize a trick. In advance, they prepare entangled pairs of particles (Figure 6 and Figure 7). These pairs can transmit the quantum state - and therefore, in effect, the photon itself -- between them using the process known as quantum teleportation. This allows them to set up a game of photon hot potato. The friend on the left side of your purported position intercepts the photon and teleports it to your friend on the right. If the bit is 1, the friend keeps it and sends it to the right verifier; if it is 0, the friend on the right teleports the photon back to the left.
Last year Buhrman and his colleagues proved that in general, no quantum cryptographic scheme is guaranteed to expose a location cheater (see arXiv:1009.2490v4) and recently they laid out a general recipe for evasion. That's where the hoses come in (see arXiv:1109.2563v2). Teleporting a particle is like pouring water in a hose--it comes out the other side. Each of your friends connect the hoses in a certain way so that the photon flows back and forth and ends up on the side where it's supposed to be. Buhrman demonstrated this on stage with hoses he'd bought at a garden supply store. We were all grateful he brought a towel.
The whole thing sounded very reminiscent of the holographic principle. (For more on the holographic principle, see this Scientific American article, "The Illusion of Gravity.) One of the deepest principles in quantum gravity, it holds that the amount of information within a region of space does not scale with the volume but with the area. It is always possible to simulate the goings-on within a region of space perfectly by manipulating its boundary. You really can read a book by its cover.
That is precisely what is happening in position evasion. Your two comrades are the boundary in one dimension, and they can connive to pretend that you're in between them when you're not. "By being on the surface, you can simulate any behavior on the inside," Buhrman told me. So there is a very general reason why cheaters can always succeed. And this is in a system with no gravity.
It is mildly disturbing that it is always possible to fake a position. The verifiers do have the last laugh, though. They can place such exorbitant demands on your friendship, requiring them to share an impractically large number of entangled pairs of particles and execute elaborate schemes for tossing the photon back and forth between them, that even your BFF would probably sell you out.
I am grateful to Theiss Research, under whose auspices I received an FQXi mini-grant to help pay my way in Singapore.
Diagrams adapted from slides by Harry Buhrman.