What experiments can distinguish between mixed and pure states?

Question

To distinguish between a coherent and de-cohered stage of the same system what experiments can provide the answer? The term Experiment is used here in the Bohr-Einstein-debate sense, a realizable physically sensible procedure, not necessarily practical in current labs.

Assumptions: Suppose the system initially is in a true superposition state |ψ⟩ = 1/√2(|0⟩+|1⟩), thus its density matrix initially has coherences (off diagonal) terms.

Also assume that If we decide to switch-on an external auxiliary apparatus, and bring it into interaction with our system then, due to the interaction with that auxiliary apparatus its density matrix becomes \begin{bmatrix}0.5 & 0 \\ 0 & 0.5\end{bmatrix} We'll call that new stage 'the system has decohered'.

The question is - what experimental test can determine whether the system has decohered or not.

(*As the model-system any convenient physical 2- state model can be used i.e. spins, energy levels etc.)

Answer 1

So basically you want to distinguish the state $| + \rangle \langle + | $ from the dephased state $\frac{1}{2}(| 0 \rangle \langle 0 | + | 1 \rangle \langle 1 | ) = \frac{\mathbb{I}}{2}$.

Here's a simple experiment: apply a Hadamard to both states and then measure in the $\sigma_{z}$ basis. For the ``true superposition'', this transforms it into the state $| 0 \rangle \langle 0 | $ and so we get the output $0$ with probability 1 (when measured in the $\sigma_{z}$ basis). The mixed state on the other hand is unitarily invariant and therefore yields probabilities $(\frac{1}{2}, \frac{1}{2})$ in any basis (inclusing the $\sigma_{z}$ basis).

Update: This process can be generalized to other states and bases -- the idea is to transform to a basis where the coherence in the pure state manifests itself (while the incoherent state transforms into something else; allowing them to be distinguished).