Are these two quantum systems distinguishable? Suppose Stanford Research Systems starts selling a two-level atom factory. Your grad student pushes a button, and bang, he gets a two level atom. Half the time the atom is produced in the ground state, and half the time the atom is produced in the excited state, but other than that you get the exact same atom every time.
National Instruments sells a cheap knockoff two-level atom factory that looks the same, but doesn't have the same output. In the NI machine, if your grad student pushes a button, he gets the same two-level atom the SRS machine makes, but the atom is always in a 50/50 superposition of ground and excited states with a random relative phase between the two states.
The "random relative phase between the two states" of the NI knockoff varies from atom to atom, and is unknown to the device's user.
Are these two machines distinguishable? What experiment would you do to distinguish their outputs?
 A: Let me give some reference that might be useful to make things clear.
It's Landau-Lifshitz, book 5, chapter 5:

The averaging by means of the statisitcal matrix ... has a twofold nature. 
  It comprises both the averaging due to the probalistic nature of the quantum description (even when as complete as possible) 
  and the statistical averaging necessiated by the incompleteness of our information concerning the object considered....
  It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried 
  out as a single operation, and cannot be represented as the result of succesive averagings, 
  one purely quantum-mechanical and the other purely statistical.

This "twofold averaging" is exactly the reason why the two states cannot be distinguished in any way.
Let me add another nice citation:

It must be emphasised that the averaging over various $\psi$ states, which we have used in order to 
  illustrate the transition from a complete to an incomplete quantum-mechanical description has only 
  a very formal significance. In particular, it would be quite incorrect to suppose that the description 
  by means of the density matrix signifies that the subsystem can be in various $\psi$ states with various 
  probabilities and that the average is over these probabilities. Such a treatment would be in conflict 
  with the basic pronciples of quantum mechanics.

A: These systems are not distiguishable. The average density matrix is the same, and the probability distribution obtained by performing any measurement depends only on the average density matrix.
For the first system, the density matrix is
$$\frac{1}{2} \left[\left(\begin{array}{cc}1&0\cr 0&0\end{array}\right)+ \left(\begin{array}{cc}0&0\cr 0&1\end{array}\right)\right].$$
For the second system, the density matrix is
$$\frac{1}{2\pi} \int_\theta \frac{1}{2}\left(\begin{array}{cc}1&e^{-i\theta}\cr e^{i \theta}&1\end{array}\right) d \theta.$$
It is easily checked that these are the same.
A: Case 1: $\frac{1}{2}\left[\left|0\right>\left<0\right|+\left|1\right>\left<1\right|\right]$.
Case 2, average over phases $0$ to $2\pi$: $$\frac{\int\left[(\left|0\right>+e^{i\theta}\left|1\right>)
                         (\left<0\right|+e^{-i\theta}\left<1\right|)\right]d\theta}
          {\int\left[(\left<0\right|+e^{-i\theta}\left<1\right|)
                         (\left|0\right>+e^{i\theta}\left|1\right>)\right]d\theta}.$$
The cross terms average to zero because $\int\limits_0^{2\pi} e^{i\theta}d\theta=0$, so it's the same density matrix. If this is really what the different manufacturers deliver, it's not a cheap knock-off.
A: The density matrices in both cases are identical. If quantum mechanics is exactly linear, both states ought to be indistinguishable. But if there are some slight nonlinearities in the time evolution, we ought to be able to distinguish between them in principle. But you have to realize nonlinearities in quantum mechanics lead to all sorts of problems, which is why most people assume quantum mechanics is exactly linear.
A: Perhaps the machine described here can actually be built. Let me propose a heated flask containing a 50-50 mixture of gaseous, monatomic carbon-14 and nitrogen-14. When you push a button, a pinhole opens up and allows exactly one atom to escape. Is it: either a carbon atom or a nitrogen atom, with 50% probability, or is it an atom in a 50-50 superposition of the carbon/nitrogen states?

EDIT: Let's do this just a little differently. Let's prepare a bottle
  of pure, gaseous monotomic carbon-14 and then wait for 7000 years.
  Now let's let an atom out of the bottle. Is it a carbon atom, a
  nitrogen atom, or a 50-50 superposition of both?

The weight of expert opinion in the answers posted thus far seems to indicate that these two descriptions are experimentally indistinguishable. I suspect this is correct, although it's a funny conclusion that flies in the face of the common-sense belief that an atom is either carbon or nitrogen, but not both.  However I think two stipulations ought to be noted:


*

*I can see no reason why a given atom ought not to appear in an 80/20 superposition, so long as the long-term average is 50/50.

*I don't believe the machine is actually constructible because I don't think there is a mechanism which can reliably produce exactly one atom at a time. You never quite know just how many atoms you've let out, and that introduces enough uncertainty in the measurement to avoid any of the glaring contradictions that seem to be present.
EDIT: When Andrew posted this question, he promised a follow-up question. It's six months later and I haven't seen the follow-up. So here's what I think the follow up was going to be:
Suppose you have a gas in equilibrium. According to thermodynamics, the probability of an atom being in a given state is given by an exponential function of the energy. So, according to Copenhagen, we have atoms in different energy eigenstates which make random transitions from one state to another, emitting or absorbing photons when they make transitions. Question: Is there a way to experimentally distinguish this model from an alternate model where all the atoms are in continuously varying superpositions of states, radiating or absorbing continuously as the charge distribution of those superpositions oscillates like tiny antennas?
If Andrew is out there, I wonder if this was his follow-up question?
A: This is such a nice philosophical question with such a neat resolution that I can't resist dropping a comment. The reduced density matrices of the atom are the same for Stanford and National, but quantum mechanics is irreducibly holistic. The wave function describes the entire universe. If the atom was prepared by Stanford, it will be entangled with traces of the environmental record in Stanford in a particular way, but if it was prepared at National, it will be entangled with traces at National in a different but still specific way. Holistically, there is indeed a difference. To suppose the atom can be considered in isolation from the rest of the world is a major fallacy in quantum mechanics.
