Contemporary physics is home to many Soufflé Subjects. These are subjects born in ignorance, grown on enthusiasm and destined to die on insight.
Quantum Information Theory is one such subject. It is born of a misunderstanding, based on ignorance, which led to the enthusiasm, it is new and great, which will ultimately lead to the insight, a widespread understanding of a simpler unifying stance.
Let us start at the beginning, the misunderstanding.
Physicists believe they understand what a wave function is – it describes the probability of finding a point particle here or there. That is the essence of the Born Interpretation and the surrounding philosophy of Niels Bohr comprising the Copenhagen Interpretation.
It seems fair to say, most physicists continue to believe in the Born Interpretation of wave functions. However, it seems they no longer believe in the Copenhagen Interpretation, as it describes Quantum Measurement. In their view, that bit must be wrong somehow while the other bit, the Born interpretation, must be right. This thought process is natural. In the common view, it is best to change one thing at at time. Why change two things at once?
Of course, there is a fairly elementary problem with that attitude. Scientific theories tend to involve a series of assumptions which collectively are necessary to ensure consistency. The careless thinker assumes that it is perfectly okay to negate one of these without changing any of the others. Er, good luck with that Eugene!
So, where in this soup do we find clear evidence of a misunderstanding?
On my reading, physicists have already abandoned the idea that wave functions describe the probability of finding a point like particle here or there. They are busy constructing theories to say how accurately particles are found. Very clearly they do not believe in the Copenhagen Interpretation but they still believe in the Born Interpretation.
Curious, huh? How is that logical?
Clearly, they are mighty confused. On the one hand, they think a wave function describes the probability of a particle popping up somewhere. On the other hand, they want to describe the explicit dynamics of measurement to avoid this.
It is a perfectly confused amalgam of what Schrödinger said versus what Bohr and Heisenberg said. Terribly safe to pass the exam, but perfectly crap as a theory.
How does this relate to Quantum Information Theory?
Well, in that theory we suppose there are two kinds of information. We are supposed to believe that there is Classical Information and Quantum Information. However, nobody ever really defines what this means.
Answer: It is a Soufflé Subject.
When you prick Quantum Information Theory the subject deflates instantly leaving a ton of hot air. The reason is simple. The wave function itself is actually an unknown parameter, and you cannot determine it directly. Thus the probability densities we define on unknown wave functions are only known indirectly through inference of observations at the classical level. The information theory continues to be couched in probability terms, but there is absolutely no need for some special quantum probability or quantum information.
Sadly, this has eluded most workers. They continue to suppose that some “special” theory for probability and information is required. This is where the enthusiasm comes from:
Wow! Maybe we (physicists) get to re-invent everything!
Heroic physicists can replace: probability theory; information theory; control theory etc etc.
The jury will be out for some time, but I think the outcome for physics is just embarrassing.
Physicists simply lack insight. They are struggling to reconcile their view of Probability as Intrinsic (the Copenhagen Interpretation) with that of Probability as a State of Knowledge (Statistical Inference and Information Theory). This is a tension between two incompatible views, with the second one in the ascendant.
In short, the physics community is just taking the long way around. Physics is essentially the very last science to comprehend statistical inference in any level of depth. This is an uncomfortable position for physicists, since they like to think they are first to everything.
However, in this case Physics is definitely last with a bullet.
When you consider the possibility that probability is a state of knowledge, then many of the conundrums of Quantum Theory simply melt away. In particular, one can then view the wave function as an important latent variable, or to use the more common physical parlance, a hidden variable.
Let’s see: Engineers, Economists, Computer Scientists, Sociologists and even, the God Particle Forbid, Parapsychologists know what a latent variable is. Latent variables are the things you can only indirectly observe which you posit influence those variables that you can directly observe. In short, their values are inferred, a statistical concept.
Physicists, as a community, have the most immense problem with this concept. The idea of a latent variable, a thing which cannot be directly observed, is acid to their very soul. How could anything be hidden to the all seeing and all knowing God-Like eye of the Physicist? That is pure sacrilege! It cannot, it must not be. Resist to the bitter end!
This is such a soul destroying idea to one in the grip of the Mind Projection Fallacy, that it is a genuinely unthinkable thought. The simplest possible explanation, that the wave function is the hidden variable, is the one no practicing physicist can think.
Hence they do not, to their very great cost.
This idea, although simple and productive, is beyond their ken.
When you take this line of thought, things look very very different.
You are entitled to introduce probability densities on wave functions to describe your knowledge of them. You can sharpen your statement of physical theories by stating priors: such as de-coherent systems prefer to be in eigenstates of the pointer basis selected by the environment.
In short, the confusion simply melts away. The physics community cannot conceive of a probability density over a wave function, since that would be a probability of a probability. This is very confusing if you only ever learned that probabilities were frequencies.
However, if the wave function is simply a hidden parameter:
1) there is no problem with a probability density over its value
2) the hidden nature of the variable leads to an explanation for in-determinism
3) we do not need any new Quantum Information Theory or Quantum Probability
In short, the wave function is a dynamical parameter, an initial condition, so a probability density of that makes perfect sense. Further, the wave function is the natural non-local hidden variable underpinning observed stochastic behavior at the classical level.
That is the insight that deflates the soufflé.