CPT symmetry and quantum frameness
I just finished a paper with Barry Sanders (the quantum guy, not the football guy) that develops a quantum resource for overcoming a superselection rule created by a matter–anti-matter dissimilarity. (OK, whose head just exploded?) Maybe I should just post the abstract:
Due to Lorentz invariance and locality, physical laws are invariant under simultaneous Charge-Parity-Time (CPT) inversion, which makes this symmetry one of the most fundamental in the universe. We show that CPT symmetry leads to superselection, which can be circumvented with CPT frameness resources. Our frameness resource is applicable to quantum communication between superselection sectors of the universe that can be identical or inverted under CPT inversion. Whether the sectors are identical or inverted need not be known.
Here’s the problem in a nutshell: suppose Alice lives in a part of the universe that is entirely made of normal matter and Bob lives in a part of the universe that is made entirely of anti-matter (yes, I know there is an imbalance of matter and anti-matter in the universe, but bear with me). Can Alice and Bob communicate with one another? If so, can they tell if they are in different sectors of the universe or not?
It turns out this is a trickier question than it might first appear. First of all, it does rely a bit on the nature of what it means for something to be a “particle” versus an “anti-particle.” For most particles, their corresponding anti-particles have the opposite sign for all internal symmetries: electric charge, baryon number, lepton number, strangeness, charm, bottomness (beauty), topness (truth). Neutrinos and anti0neutrinos, which have opposite lepton number, also happen to have opposite parity – neutrinos are all left-handed while anti-neutrinos are all right-handed (see, also, chirality and helicity). It also turns out that certain meson decay processes suggest that we must adhere to the Feynman-Stueckelberg interpretation of anti-particles as particles moving backward in time. Thus to switch from a particle state to an anti-particle state, we need to apply the full CPT operator and swap the internal symmetries, parity, and time.
Suppose Alice sent Bob a message encoded in, say, electrons, to be measured with some Stern-Gerlach-type device. The electrons actually look like “positrons” to Bob (Bob’s “electrons” would be Alice’s positrons). In other words, he’ll measure them (assuming he doesn’t annihilate them in the process) as having the opposite spin from what Alice encoded them with, i.e. a |0> state to Alice looks like a |1> state to Bob. Now, if Bob knows he’s in a different sector of the universe, all he has to do is flip each qubit state he gets and he’ll be able to decode the message. But what if he doesn’t know?
Ah, there’s the rub. What Alice and Bob need is a common reference frame. Something that is invariant under the CPT transformation should look the same to Alice and Bob. As it turns out, circularly polarized photon states are invariant under the CPT operation. To understand why, consider a particle spinning on its axis that is moving away from you and further suppose its direction of motion is parallel to its axis of spin. Suppose it moves in a “right-handed” manner, i.e. it is spinning clockwise as seen from your viewpoint. Now suppose you hop onto something that is moving faster than this particle. Once you pass it, if you now look back on the particle (which is behind you), it will appear to be spinning counter-clockwise (it’s just like looking at a clock from behind). The “rotation” of photons corresponds to circular polarization. The thing about photons, however, is that you can’t move faster than light so you can never get into a frame that makes the rotation look different!
In the paper, we show that you can implement that BB84 protocol with this concept (in addition to the usual straightforward communication). In the process, it also turns out that Alice and Bob can use this information to figure out if they are in the same or in different sectors (i.e. whether they are both made of matter, for example, or one is matter and the other anti-matter). They can’t, of course, tell which is which. In fact the only way to tell is for them to communicate with all other sectors of the universe and look for an imbalance.