Indistinguishability and Nonlocality

The title refers to two features of quantum systems that are sometimes linked together to (mistakenly, as I will explain) claim that quantum physics allows for a faster-than-light communication.

In quantum physics, identical particles, such as photons or electrons, lose their individuality and become truly indistinguishable. This means, among other things that we will explore here, that when a photon with certain properties (such as a particular colour, momentum and polarization) gets destroyed and another photon gets created with exactly the same properties, we cannot claim that the old photon has now reappeared. Our classical intuition lets us down here because if a glass of wine disappeared from our kitchen, we would think that someone had just taken it. If the same glass then reappeared, we would assume that someone had just brought it back. Yes, it could be that the original glass broke and someone just brought back an identically looking glass with the same amount of wine in it, but which of the two options (or any other) it was is a question that can, in principle, be ascertained without any ambiguity. This is what our classical intuition tells us and it is this intuition that fails in the quantum world.

In the quantum world, it is the underlying fields that maintain their identity. There is the electromagnetic field (whose excitations are photon), there is the Dirac field (whose excitations are electrons and positrons) and so on. But the excitations themselves are completely indistinguishable. When we create a photon with certain properties and then another one with the same properties, we have no way of telling which is which. All we can say is that there are now two photons, but we cannot say that there is the first and the second photon, the latter is a meaningless statement quantum mechanically speaking.

Now, in the quantum world, we have two kinds of particles, bosonic and fermionic. Photons are bosons and we can create as many of them as we like with the same properties. Photons can all go into the same physical state is how physicists like to put it. Fermions on the other hand obey something known as the Pauli Exclusion Principle: only one fermion can occupy one and the same state. If we create another fermion, this can only be done if it has different properties to the first one.

Schrödinger presented a beautiful analogy to illustrate the difference between the classical and bosonic and fermionic behaviour. Suppose we have three children, Tom, Dick and Harry and a teacher wants to give them two awards. The teacher is interested in how many different ways the awards can be distributed. We now discuss 3 different types of awards:

1. A teacher has a one-pound coin and a two-pound coin. She can clearly distribute them in 9 different ways. She can give one coin to any of the 3 children and then the same with the next coin, therefore 3 times 3 = 9.

2. A teacher now has two identical one-pound coins. She can either give one pound each to two boys (3 different ways) or two pounds to any of the three boys (3 different ways). So, in total: 3+3=6 different ways to distribute the awards.

3. The reward now is a membership to a football club. This can be given to any of the two boys, the third boy being left out. Therefore, there are only 3 ways of doing so because each of the 3 boys could be left out.

The case A) corresponds to how classical particles behave (leading to the Maxwell-Boltzmann statistics). The case B) encapsulates bosonic particles, while the case C) are fermions. The twist, however, is that it’s not the boys who represent the particles! The particles are the rewards and the boys represent different physical states that the particles can take up. So with bosons, both coins (particles) can end up being rewarded to the same boy (state). In the lab physicists now routinely make Bose condensates in which a billion atoms are put into the same low energy state. With fermions, however, this is impossible because it is meaningless to give two club memberships to one and the same person. With electrons, this explains why some materials are insulators while others are conductors (an explanation first presented by Arnold Sommerfeld). One last thing I’d like to add for completeness is that only the counting of the B) and C) kinds exist in nature. The case A) happens to be approximately true in some regimes, but only approximately.

Now for the other feature called nonlocality. Nonlocality is sometimes used as a synonym for entanglement. My readers will know that I believe that the phrase itself is deeply misleading, as it leads us to conclude that there is something spooky regarding entanglement (it’s not for nothing that Einstein called entanglement “a spooky action at a distance”). Instead, quantum physics is local, but the elements of reality that obey locality are the so-called q-numbers (where “q” is for quantum) and not the ordinary c-numbers (basically real numbers, and “c” stands for classical). The fact that a position and a momentum of a particle are both q-numbers leads directly to one of the key features of quantum physics, Heisenberg’s uncertainty principle.

Locality is encoded in the following mathematical fact. When we change a q-number in one place (say, where one particle is) no other q-numbers can change anywhere else in the universe (say, the q-numbers of another particle far away from the first one). It is this mathematical property that lets quantum physics comfortably comply with special relativity and stops us from generating any spooky action at a distance. My latest book, “Portals to A New Reality”, describes all this in great detail if you are interested in learning more.

But now, I’d like to show you how we can put together entanglement and indistinguishability to seemingly violate special relativity by communicating faster than the speed of light. Suppose that Alice and Bob share a pair of entangled photons in the state HH+VV. In other words, both photons are polarized the same way, either horizontally (H) or vertically (V), and the two state are in a quantum superposition. Alice now splits her photon spatially so that if it’s H, it goes one way and if it’s V is goes another (a device for doing so is called a “polarization dependent beam splitter”). She then does one of two possible things: 1) Alice rotates V into H and brings the two spatial locations together, leading to the overall state H(H+V); 2) she just leaves the photon as it is. But, because photons are indistinguishable, the state in case 1) becomes H(H+V)+(H+V)H = 2HH+HV+VH, while in the case 2) the state remains HH+VV. The final result is that in case 1) Bob has an unequal probability of detection for H and V states of the photon, while in case 2) the probabilities are the same. Therefore, Bob can make a measurement to tell if Alice has implemented the protocol in case 1) or not (which is case 2) and thus Alice can communicate to Bob instantaneously.

And this is exactly what I said earlier should be impossible in quantum physics: by changing a q-number (polarization) of her photon, Alice can instantaneously change the q-number (polarization) of Bob’s photon (which could be very very far from Alice’s photon). Clearly there is a mistake in the protocol for otherwise either quantum physics, or relativity, (or both!) would be wrong. The mistake in the above protocol is that we started by treating the photons as distinguishable all the way until, at the very end, we imposed indistinguishability (by forcing the state of the two photons to be the same).

We can do this with Tom, Dick and Harry too! Suppose that the teacher distributes awards according to A) and then separates the children to different spatial locations such that Tom is in one place and Dick and Harry are together in another. Now, it is possible that Tom has both coins, only one of the two or none and correspondingly for Dick and Harry. But if the teacher now magically switches to identical coins on Tom’s side, Dick and Harry will know about this as they too will end up with identical coins (either one each or each having both coins).

The inconsistency, therefore, lies in suddenly changing the rules of the game half-way through the game. If we play hide and seek and, just as I am about to find you, you run out of your hiding place, show me four aces and claim that you won a game of poker, then anything is possible.

I’ve brushed aside the fact that the position of Alice’s photon is at some stage of the protocol entangled to the polarization of Bob’s photon. The above protocol 1) actually swaps entanglement from polarization to position and polarization. But entanglement between Alice and Bob remains the same in 1) and 2) which is why the state of Bob’s photon does not change no matter what Alice does.

Let me end with a sweeping, yet in my eyes, true generalisation. Anytime anyone claims that there is a paradox in quantum mechanics, I maintain that it is because they’ve changed the rules of the game somewhere in their analysis. Either a magical classical domain is invoked at some point, or a principle that sounds plausible (again, most likely due to our classically underpinned intuition) is postulated that then ends up being violated in quantum mechanics. Either way, to get consistent answers, our best theory needs to be applied to everything equally and all the time. And quantum mechanics is certainly consistent as well as complete (that’s why Heisenberg called it a “closed theory”). This will be true even if one day an experiment is done to violate it. When that happens, we will find a new theory that supersedes quantum physics, but quantum physics will still be consistent and complete in its own domain, just as classical physics is in its own.

So, stop trying to find paradoxes in quantum physics. You are barking up the wrong tree. Concentrate instead on designing and performing an experiment that genuinely contradicts it. Not many people are doing this because it’s hard work. It’s also risky as you might not succeed - quantum physics has been giving us impeccably good predictions without any deviations for over 100 years. On the other hand, Newtonian mechanics survived for about 200 years, so we might have to wait a while longer to falsify quantum physics. But the logic of scientific discovery reassures us that it is bound to happen sooner or later.

Take care of yourselves,

Vlatko