Reading from Matuschak & Nielsen Quantum computing for the very curious]:

A fundamental fact about this measurement process is that it disturbs the state of the quantum system. In particular, it doesn’t just leave the quantum state alone. After the measurement, if you get the outcome 0 then the state of the qubit afterwards (the “posterior state”) is the computational basis state ∣0>. On the other hand, if you get the outcome 11 then the posterior state of the qubit is the computational basis state ∣1>.

Summing all this up: if we measure a qubit with state α∣0>+β∣1> in the computational basis, then the outcome is a classical bit: either 0, with probability ∣α∣^2, or 1, with probability ∣β∣^2. The corresponding state of the qubit after the measurement is ∣0> or ∣1>.

A key point to note is that after the measurement, no matter what the outcome, α and β are gone. No matter whether the posterior state is ∣0> or ∣1>, there is no trace of α or β. And so you can’t get any more information about them. In that sense, α and β are a kind of hidden information – the measurement doesn’t tell you what they were.

One reason this is important is because it means you can’t store an infinite amount of classical information in a qubit. After all, α is a complex number, and you could imagine storing lots of classical bits in the binary expansion of the real component of α. If there was some experimental way you could measure the value of αα exactly, then you could extract that classical information. But without a way of measuring αα that’s not possible.

Here is a naive idea/question.

This gating of quantum information when measured into a classic computation result looks like a functor from

  • a category representing computation for one quantum bit
  • a category representing computation for one classical bit

Here, category is not well formalized.

The questions are:

  • How do you formalize this notion of computation as a category? The quantum computation diagrams in the book provide, perhaps, a way to formalize a category for quantum bits. The diagrams represent composable graphs. Maybe the objects are the quantum wires, and the maps are the quantum operators X, H, etc - essentially, the Pauli matrices.

  • Are there other computation theories different from classic computation? More specifically: we can view computation theories as a type of categories with some structure - but what is that structure?

  • Is, then, a way in which the quantum bit is a universal computation theory?

To get to that, we’d have to define a notion of functor of computation theories. Taking a measurement an example of such functor.

This has got to be an interesting concept, because measurements of quantum bits, while formally understood, does not have a well understood interpretation.

Then, the quantum bit should be universal in some way among computation theories. Or, if not universal, then simplest in some way - the idea being that physics is a geometric realization of computation theories, and simplest computation theories are more likely to ‘come into being’.