How do you know if the signal is a pure or mixed state when doing state reconstruction (quantum tomography)? You're trying to reconstruct the density matrix by sampling a signal.
If you measure spins coming from a source along the Z axis, and 50% of the time they are spin up and 50% spin down, how do you know if the signal is a mixed state or a pure state? Do you have to also measure along an orthogonal axis, such as the X axis?
 A: You can't infer the purity by observing a single axis (e.g., $\hat{Z}$), so you'd indeed have to measure a non-commuting observable (e.g. $\hat{X}$). Only then will you be able to gain (some) phase information which will in turn tell you about the quantum coherence (i.e., purity) of the state.
PS: I'm not sure if you can construct the whole density matrix with only two measurement axes, though. More von Neumann projections may be needed.
A: Doing tomography involves using several different measurement bases. If we restrict to the simplest type of measurements, projective measurements, you need at least $N+1$ of them to fully characterise an arbitrary state.
To see this, note that any fixed measurement basis $\{\lvert u_i\rangle\}_{i=1}^N$ pinpoints $N-1$ parameters: the probabilities $\lvert\langle u_i\vert \rho\rangle\rvert^2$ associated to each possible outcome. Due to the normalisation constraint, these are $N-1$ independent parameters. Therefore, if you find $N+1$ measurement bases which all give you information independent from the others, you manage to retrieve $(N+1)(N-1)=N^2-1$ independent parameters, which is the number of dimensions of the space of $N$-dimensional states.
For two measurement bases to give "independent information" in this sense, they need to be mutually unbiased bases (MUBs). The exact number of MUBs for a given dimension is still an open question, so it is not ensured, as far as I know, that you can find $N+1$ measurements that make the above argument work. Nonetheless, $N+1$ works as a lower bound: if the bases are not MUBs, then there is some redundancy in the information they provide, and therefore you need more than $N+1$ of them to get your $N^2-1$ independent parameters.
I should also note that if $N$ is the power of a prime, $N=p^M$ for some prime $p$, then we can always find $N+1$ MUBs. This is, in particular, the case for qubit systems, in which you have $N=2^m$ with $m$ the number of qubits.  You can have a look at (Durt et al. 2010) for more information about this (in particular section 1.2).
Once you reconstructed the state, you know its density matrix $\rho$, so you can simply compute the associated purity as $\operatorname{Tr}(\rho^2)$.
On a more concrete note, in the case of a qubit, $N=2$, doing tomography provides you with the expectation values $\langle \sigma_x\rangle,\langle \sigma_y\rangle,\langle \sigma_z\rangle$, with which you can represent the state on the Bloch sphere, and then the state is pure if and only if sits on the border of the sphere.
