Qubits and P-bits

[gwillen]

I have heard the refrain "everyone would understand quantum mechanics, if only it were explained thusly!" a surprising number of times recently, and I want to explore one such value of "thusly": The idea that quantum mechanics is just what falls out if you allow probabilities to be negative. (See e.g.I like the idea behind the Scott Aaronson link

Comments

[gwillen]

Oops, I guess I never addressed the "nice pattern" from paragraph 4. In brief: you can represent a probability distribution over N bits with 2**N reals between 0 and 1, in the same way that you can represent a quantum state with N bits using 2**N complex numbers between 0 and 1. In the case of probability, the normalization constraint -- that all the values have to add to 1 -- saves you one of those reals, giving 2**N - 1. Similarly, in quantum mechanics, the normalization constraint saves you one real, and invariance under change of global phase gives you another one, making 2**N - 1 complex numbers.

[edanaher]

Cute. I'm not convinced that it's actually tremendously enlightening, but it'd definitely at the very least be useful in the first day or two of a class to get people used to some of the math/notation.

[gwillen]

Awesome, glad someone at least read it. ;-) I guess it does seem like it would be most useful to an online or other informal presentation, rather than a course (where one might expect enough mathematical maturity from the students that complex-valued Hilbert spaces should be a safe place to start.) I definitely have a lot of trouble visualizing said spaces, especially once you get more than one bit, so I think contemplating the contrast between p-bits and qubits will help me get a better handle on the properties of the latter.

[quartzpebble]

o.O

Talking about bits (P- or Qu-) is confusing. Wavefunctions and spatial probability densities make much more sense to me.

[gwillen]

Welllll, I like quantum information better than quantum physics.... I don't really possess the mathematical maturity to deal with infinite-dimensional Hilbert spaces, I just pretend to. Bits are nice because they have very, very few degrees of freedom, as compared to, say, the wavefunction of a photon or something.

[gwillen]

Anyway if you can deal with wavefunctions you can definitely deal with qubits. I recommend Nielsen and Chuang, /Quantum Computation and Quantum Information/, if you are interested.