What does it mean when we say we want to quantize economics? Well, it doesn’t mean that we’re emulating (or somehow abusing) quantum physics. Quantum mathematics is about information and probabilities and observables and how they relate to each other, as Scott Aaronson said, and money is a form of information that does not behave like a classical object.
Now when we try to apply quantum ideas to areas outside of physics we encounter a lot of obstacles. One is that we’ve constantly been told by famous physicists and mathematicians that quantum mechanics is fundamentally incomprehensible, so we’ll never really understand it. We have these weird phenomena like Schrodinger’s cat which can be alive or dead at the same time. We’re often told that concepts such as superposition and entanglement only apply to the tiniest quantum particles and you’ll never encounter them in your everyday life. And then finally there’s this impenetrable mathematics. But let’s just forget physics for the time being, and think of something much simpler like a coin toss.
If we wanted to model the state of a coin toss where we don’t know the outcome, there are two possible outcomes heads or tails, so we would want a two-dimensional space. And we could represent the state of the coin, if it was a balanced coin, with a diagonal line (see screenshot of video below).
That balance between the two states is really the idea of quantum probability. We can represent heads as an up arrow 
In order to get the probabilities we’ll just take the projections. The projection on the horizontal axis, when squared, is going to give the probability of heads, and the probability of tails is going to be the square of the projection onto the vertical axis. The reason we are squaring them is because we want the projections to always add to one, and by the Pythagorean theorem that will be the case.
So the basic difference between classical and quantum probability is that classical probability uses what’s called the 1-norm, so the options are heads or tails, while quantum uses this 2-norm which involves the square. Making this change from a 1-norm to a 2-norm leads to all these different quantum concepts such as superposition, negative probability, interference, and entanglement.
So what do we mean by negative probability. Well if we just flip our state over, mirroring it around the vertical axis, then now we’ve got a negative probability for heads. The size of the probability when we take the norm, because we’re squaring it, is obviously positive again so nothing has really changed, except that when we add probabilities together we can get a plus and a minus canceling out in what are called interference effects.
We can also imagine rotating the state by acting on it by a matrix such as the so-called Hadamard transformation. It has the effect here of rotating clockwise by 45 degrees. Our state is now perfectly aligned with the heads axis, so this is like a case where a coin is in the heads up state and it’s going to stay there. Our projections which were originally on the axes have also rotated around so each of these projections now have a tails component. One is positive and one is negative, so when they add together they cancel out. That’s an example of interference, so there’s now no tails component.
We can also think of more complicated systems. Imagine for example that we had two coins, so now there are four different things that we need to keep track of: there’s heads and tails for one coin, and heads and tells for the other coin. That’s four dimensions which is obviously not easy to draw. But quantum coins can be entangled so for example you might have the only possibilities as heads-heads or tails-tails. So what this means is that just by making this switch from a one norm to a two norm we’re allowing all of these quantum phenomena such as superposition, interference, entanglement and so on.
Further reading:
Aaronson S (2013) Quantum Computing Since Democritus. Cambridge: Cambridge University Press.
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