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# Quantum Computing and Markets, from Jonathan Bowers

September 21, 2014 |

This is a short video about quantum computing. I was drawn to the simple explanation of a complex idea and there seemed to be many potential market lessons or at least gaining a new vantage point.

1. There is more than one complimentary way to view something.

2. If you participate you can't know what would have happened if you didn't and vice versa, particularly relevant for size.

3. Intrinsic randomness of superposition allows observations without a probability distribution. Still wrapping my brain around this. An example of ever changing cycles? Or something else entirely? If there is no probability distribution how do you define outcomes of events?

4. Quantum correlations are richer in describing interactions. This idea seems ripe for trying to understand the complexity of market interactions.

5. Attaining perfection is hard, maintaining it is impossible.

6. Viewing quantum computing affects the results, perhaps in the same way talking a position does.

7. How did you get that result? I don't know is the right answer in quantum computing. Do it yourself so you can understand. Don't trade what you don't understand. don't blindly follow someone else's "system" perhaps there are others…

## anonymous writes:

I agree that the quantum-collapse idea has broader applications to life in general. Danilov & Mogiliansky, many years ago now, used the collapse concept to talk about Tversky's famous quotation, "Preferences are not read off of a master list, but are constructed in the elicitation process." QM discussions have a tendency to go off the rails very quickly, and to attract sophists & cranks. But superposition and measurement-disturbances are, in my opinion, relevant and common to quotidian life.

1 = duality

2 = contravariance (the math.DG kind, not the math.CT kind), on trees

3 = I don't understand what this means either, but we can clearly reason about the unknown without attaching PRECISE probability numbers to it.

4 = I would guess no. Quantum correlations can have negative intersections. But someone here could find out: just allow your covariance matrices to take on complex values.

5 = set of measure zero

6 = talking or taking a position?

P.S. If someone finds complex numbers arcane or silly, I'm happy to share a few bits of perspective.

## Mr. Isomorphisms writes:

Just checked the video and revised on Wikipedia and a little googling. I don't think 3 is an accurate statement. Quantum states are expressed as waves which can be added together. Lots of things in the normal world can also be expressed as finite-energy functional spaces. But I don't see where the video said there is no probability distribution. Nor is superposition "intrinsically random". Superposition is just adding things together.

If you check the Wikipedia page for quantum probability it says that outcomes of events are basically defined in the normal way, except since it's a complex space the i=-i equivalence shows up and screws with things.

OK, someone asked for my ideas on clarifying complex numbers. Here's my attempt for those interested:

- First the physical intuition. Electricity travels along power lines. If the transmission efficiency is 1 then the consumer gets 100% of the power produced at the generating station. If the transmission efficiency is sqrt(-1) then all of the energy from the station goes to heating the power line and does no useful work.

Now the maths.

- The usual numbers are annoying because signs "jump" from + to - with nothing in between. google.com/search?q=klein+j+invariant+whirling+upon the top video shows something else with only very specific points matching up, but instead of showing us only the actually equivalent points it shows us the whole movement, as it were — which is only posible in a smooth space. Complex numbers let us watch what's happening "in between" negative and positive.

- That's why e^i pi = -1. Nothing more special than that pi is halfway around a circle that goes from positive to negative, circling back to positive.

- View the complex line as a line and an angle âŸ³. The angle replaces the usual concept of sign. All of the arithmetic on complex numbers works basically the same as regular arithmetic, except that you also take the "angle" (the "amount negative or positive") into account. If you multiply two numbers you need to add their "angles". (In finance, "angle" means "correlation".)

- So I generally think of the complex multiplication happening just on the unit disc, i.e. everything is equal magnitude but has different signs (and fractions of signs). Then I do regular multiplication second.

- However it's impossible to tell -i from +i. (-i)^3 == (—)i == -i So that has to be taken into account.

- Complex numbers don't require any changes to your metaphysical worldview, because they can be represented with real numbers. The matrix representation is on Wikipedia and it looks just as "half-negative" as (num)*(num)=-1 does.

- You can derive trigonometry from complex numbers. In other words the arithmetic induced by adding sqrt(-1) to a number system, matches what people figured out from plane triangles.

- For those who can use R (or don't mind logging into cloud.sagemath.org, opening a cloud terminal and typing R), run the functions in https://gist.github.com/5a30e61fb305ee52cff . That instantiates a "plat" function, basically like python mpmath's cplot except Cielab colours are psychologically superior to RGB.

You can "plat" polynomials like this: plat( Z, function(x) (x^2+1) * (x-2.3) )

Then red is positive, green is negative, and other colours are somewhere in between. Size is indicated with brightness, but there's nothing special going on here that you can't see in a regular plot.

I believe playing around with simple (and not-simple, if you can think of any) functions in these "plats" will give anyone (a) a better understanding of those functions, and (b) the feeling that the complex numbers are not scary or weird.

tl;dr. The complex numbers just let us look in-between positive and negative. More advanced: if you want to envisage a complex variety, then an idea I had recently is to animate the plat with a parameter that traces around the unit circle (all possible "signs" fed into the function).

The "worse" version of a parabola moving from +1 to -1 to +1 would be it "flaps its wings" with in-between being a flatline at 0–not very parabola-like.

## Jonathan Bowers writes:

I'll agree that I may not have articulated my understanding of superposition very well. In the video they show a coin flip since it also has two outcomes, but emphasized that while the coin flip has a probability distribution quantum computing does not. Perhaps that means that the distribution is always changing or unstable or when or how you measure it changes it.

## Mr. Isomorphisms writes:

I think it's just a superposition (convex combination) of 0 and 1. Same concept as a screwdriver if it could be any mixture of orange juice and vodka. You probably have a small range that you consider "screwdriver" but what if there was a word that meant "anything that's 100% orange juice, 100% vodka, or anything in between". That's a convex combination of oj and v.

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