# In $|x_1,...,x_n\rangle$, can the individual constituents be called qubits?

A qubit is a quantum system in which the Boolean states 0 and 1 are rep- resented by a prescribed pair of normalised and mutually orthogonal quantum states labeled as $${|0⟩, |1⟩}$$

According to [1]. Then a quantum register $$\mid x_1x_2...x_n\rangle, x_i\in\{0,1\}$$ is defined to be collection of n qubits.

Now I often see expressions like $$\mid x_1, ... x_n \rangle$$ where the $$x_i$$ belong to some $$S \subset \mathbf{Z}$$.

1. Can the individual constituents $$\mid x_i \rangle$$ be called qubits even though they are non-binary?
2. Would it be appropriate to call $$\mid x_1, ... x_n \rangle$$ a qubit register in this case?
3. What is the physical interpretation of such a register?
• I think qubits are binary by definition, but there may be another to represent them I don't know of. Aug 27 '20 at 5:47
• "Now I often see..." -- Where? Aug 30 '20 at 9:39
• please remember that each post should contain a single, focused question
– glS
Aug 31 '20 at 7:16

## 1 Answer

The $$|x_i\rangle$$ you mention here are qudits, they are the generalization of qubits to base $$d$$ with $$|S| = d$$. It is categorized by a superposition of $$d$$ states, same way a qubit is described by the superposition of 2 states.
In base 3 it has a specific name as well, this is called qutrit.

• I see that it is a generalization in syntax, but what does it mean? What's the physical meaning of $\mid -12\rangle$?
– gen
Aug 27 '20 at 13:31
• Also how can you consider the basis vector $\mid 0 \rangle$ to be a superposition of two states? Which two?
– gen
Aug 27 '20 at 13:50
• @gen: First question: -12 is not natural number, hence it cannot appear in your definition. Second question: in $|0\rangle$ and $|1\rangle$ basis any qubit can be writen as $|q\rangle = \alpha|0\rangle + \beta|1\rangle$, where $\alpha, \beta \in \mathbb{C}$. So, $|0\rangle$ is superposition where $\alpha = 1$ and $\beta = 0$. This seems strange but the qubit is considered to be in superposition until it is measured. In this case, zero is always returned. Aug 28 '20 at 6:53
• @MartinVesely thanks for your comment. First question: Yes, the definition was mistyped. I meant to write $S \subset \mathbf{Z}$. So I'm still interested what $\mid -12 \rangle$ means. Second: That's fair enough I guess.
– gen
Aug 29 '20 at 2:33