Dirac notation for classic bits?

Question

I'm writing about Quantum Computing, and in my preliminary sections on classical computing, there are too many things going by the names 0 and 1. Would it be awful to denote the values of classical bits as $|0\rangle$ and $|1\rangle$? I realize that Dirac notation for classical bits might be overkill, but I'm finding it helpful to have specific notation that can't be confused with 0 probability and other such things.

Answer 1

You could stick to the variable names $b$ for bit and $|q\rangle$ for qubit, and then if you have to refer to their values, you can specify $b=0$ or $b=1$, and for qubits $|q\rangle = \alpha|0\rangle + \beta|1\rangle$. The text Computational Complexity by Arora and Barak do this. But as mentioned in the comments, that's totally up to you :)

Answer 2

Sometimes in quantum communication or quantum information there is no notational distinction made between classical and quantum bits, with $|x\rangle$ (for $x \in \{0,1\}$) describing either a classical or quantum bit (see Debbie Leung's course notes, for example). There are even cases where this makes more sense than using different notations, for example when expressing a so-called "classical-quantum state": \begin{align} \rho = \sum_{x \in \Sigma} p(x) |x \rangle \langle x| \otimes \rho_x \tag{1} \end{align} where the first register acts to express $\rho$ as a distribution over quantum states $\rho_x$ conditioned on some classical random variable $x$ drawn from an alphabet $\Sigma$. In this case, while you could interpret $|x \rangle \langle x|$ as a quantum state, its really functioning like the label of a classical random variable, e.g. a classical bit if we set $\Sigma = \{0,1\}$. This holds more generally: linear combinations of "states" of the form $|x \rangle \langle x|$ for $x \in \Sigma$ result in diagonal density operators and can be described as classical objects.

This notation will probably become confusing if any quantum computation is involved where states evolve according to the action of unitaries or channels - in that case, you would need to carefully clarify which operations would be valid on which registers in your computation (e.g. we are not allowed generate/remove entanglement between the registers in (1) because we treat the first register as purely classical).

Answer 3

QWorld have come up with a really neat convention that I like in their QBronze lectures e.g. for the set of classical bits $1011$, they use the notation $\lceil1011\rfloor$. That way you can write out combinations of probabilistic bits e.g. $0.5\lceil1011\rfloor + 0.5\lceil1010\rfloor$ where the coefficients here are probabilities instead of amplitudes (because we're using classcial bits).