# What does it mean that a qubit is a triple $(H,X,Z)$ with $H$ Hilbert space and $X,Z$ Pauli operators?

In this paper, http://users.cms.caltech.edu/~vidick/teaching/fsmp/fsmp.pdf, it gives the definition of a qubit as follows:

A qubit is a triple $$(H, X, Z)$$ consisting of a separable Hilbert space H and a pair of Hermitian operators $$X, Z$$ acting on a H such that $$X^2 = Z^2 = Id$$ and $${X, Z} = XZ + ZX = 0$$.

What does this mean and why does this work? Can I choose any two pauli observables?

• Section 1.2.2 of that paper, where the definition you mention is introduced, later explains the intuition and formality behind the definition. Is there any particular part of that explanation that you have trouble with? Jun 10, 2021 at 5:38
• And to what two observables do you refer? X and Z are chosen because they make up the Hadamard and computational basis, respectively. This is also mentioned in the explanation of the definition in the same paper. Jun 10, 2021 at 5:39
• That sounds like a very mathematical definition of a qubit ... Jun 10, 2021 at 8:52
• Hi @epelaaez, thank you! It seems to me that this is key: Informally, we will distinguish a quantum degree of freedom from a classical one by requiring that the quantum degree of freedom can be measured (observed) in two mutually incompatible ways. Jun 10, 2021 at 15:27
• Essentially, we are requiring that the two operators we chose are as incompatible as possible. However, what I don't understand is why this provides us with two degrees of freedom. Is it because one can't be constructed from the other. The relation between the classical and quantum definition seems a bit imprecise to me. Jun 10, 2021 at 15:29