Quantum Mechanics of a Single Bit

We will begin our discussion of QM with the simplest possible system, one which has only two states, corresponding classically to a single Yes/No question. As we will see, the molecule of ammonia, NH₃ can be approximated by such a system, in a certain energy regime. We will see that all the machinery of QM, the mathematics of linear algebra, can be introduced in a purely classical discussion of this system, as if we were computer scientists, discussing a single bit. Quantities which a classical physicist/logician would think of as measurable can be modeled as diagonal matrices, while classical operations which change the state of the system are off diagonal matrices. A general classical probability distribution is a diagonal matrix, and the usual formula for the expectation value of a quantity is written as a trace of the product of the matrix for the quantity with that for the distribution. This trace formula immediately generalizes to all matrices, even if they are not diagonal. For those matrices which are diagonal in some orthonormal basis, the trace can be interpreted as a probability distribution for that matrix to take on one of its eigenvalues.
This discussion will show that QM is, in a certain sense, inevitable. That is, for any system, even one that we think of classically, we can introduce quantum variables, which have uncertainty even when we have completely fixed the values of the classical variables.
The choice of which variables are definite corresponds, for a two state system, to a choice of basis in a two-dimensional vector space. From the point of view of linear algebra, this choice seems arbitrary. The difference between classical and quantum mechanics is in the nature of their equations of motion. Classical mechanics has initial conditions which can all be definite at the same time, while in more general quantum mechanical systems, only some of the initial variables are compatible with each other. The collective coordinates of macroscopic quantum systems do not obey classical mechanics, but they do, up to fantastically small corrections of size < e−10²⁰ , obey a classical stochastic theory, in which the uncertainties are very small (of order 10⁻¹⁰ or smaller). These mathematical facts about the quantum theory are sufficient to explain why our intuitions about the logic of the world are incorrect.
The present author is among those who believe that we will never find a consistent interpretation of the facts of the quantum world, which admits the concept of an underlying reality with exact probabilities for histories. This is not a settled question, and we will try to avoid injecting our prejudices into most of this text.