# Dirac notation for classic bits?

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.

• This is just a notational thing, so its more or less up to you. I've seen authors use a round bracket $|1)$ to differentiate classical bits Jun 14, 2022 at 21:59
• Thanks, Danny. Can you point me to an author that uses a round bracket? Jun 14, 2022 at 23:14
• I have seen more of angle brackets |0> rather than |0) Jun 15, 2022 at 1:47
• depends on what you plan to do with it. A ket is a way to denote a vector. Is it convenient for you to describe your bits as vectors? Most likely not. Of course, you can use whatever notation you want, if it's for your own fruition it's up to you. You can decide that "$2$" means "$17$" if you want. But doing so is not standard in any way, and probably not a great idea if you want to understand and be understood by others
– glS
Jun 15, 2022 at 8:32
• @sohamb172 Where have you seen angle brackets used for classical bits? Jun 15, 2022 at 15:49

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 :)
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.
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).