I understand that a qubit is a quantum system with two basic states $|0\rangle$ and $|1\rangle$, so a general pure state of the qubit will be described by a linear combination $\lambda|0\rangle + \mu|1\rangle$ with $|\lambda|^2+|\mu|^2=1$ up to multiplication by an overall phase ($\kappa$ with $|\kappa|=1$), while a mixed state is described by a $2\times 2$ positive semidefinite matrix with trace $1$ (= density matrix). So far so good.
But it seems to me that qubits could then conceivably come in two variants: complex qubits in which $\lambda,\mu$ are complex numbers and the density matrix is Hermitian positive semidefinite with trace $1$, and real qubits in which $\lambda,\mu$ are reals and the density matrix is symmetric positive semidefinite with trace $1$.
But every explanation I've read (starting with the Wikipedia page on qubits) only seems to mention the complex case. The Bloch sphere (or ball, rather, with the sphere of pure states as its border), for example, specifically refers to the complex case, while in the real case we should get a disk with a circle of pure states as its border.
(I focus on the two-state system above, but of course the problem is the same, mutatis mutandis, for an $n$-state “qudit”.)
Is it that quantum mechanics always uses complex coefficients, complex vector spaces and Hermitian matrices? Or are there examples of (finite dimensional) quantum systems that are naturally described by real vector spaces¹? Since the fields in Maxwell's equations are real-valued, it seems somewhat miraculous that photon polarization states can also be described as a complex qubit, and I struggle to understand whether this is some convenience of description or a more profound fact of nature.
- I am well aware, of course, that any complex vector space can be considered as a real vector space (and the Hermitian scalar product degraded to a Euclidean one by keeping only its real part); but the specification that the overall phase is unmeasurable seems to make a genuine difference between the real and complex cases.
Edit: All the comments seem to say my question is stupid, so I won't insist if it gets closed for one reason or another; but the reason why I don't think they — or this other question which has been proposed as duplicate, completely answer it — is that while all this quite satisfactorily answers the question of why some (or most, or normal) quantum systems are described by complex Hilbert spaces (which I don't doubt for a minute), I don't see why this implies that real qubits can't exist.
