I have a very basic understanding of entanglement and radioactivity. But say 2 uranium atoms are entangled and then 1 of them decays, what would happen? Would the other atom decay as well? Or if not would this break the entangled state?
They are entirely unrelated concepts. Two entangled particles are not "clones" of each other which magically do everything the same way; they merely have been put into a state which displays a strange statistical correlation when you "bring both parts back together."
So for example a free neutron and an electron both have the same spin-1/2 structure; you can certainly entangle the spins of the neutron and the electron, even though they can never be identical to each other. All that this entanglement means is that if you do thousands of experiments you will see strange correlations between the two particles which no theory based on classical probability can explain. If the free neutron decays (as free neutrons will!) into a proton, electron, and electron-antineutrino, the other electron will not "decay" (how could it?). If you instead entangle two neutrons' spins, then their decays will be independent of each other regardless. (But, there may be some interesting entanglement, say, between the emitted electron or proton spins and the other neutron.)
The "no theory based on classical probability" idea can be explained in a couple ways, my favorite is to imagine a game that we call "betrayal" where a team of 3 people tries to beat several "challenges" set before them as a team. We put them all in different rooms, make sure they can't classically communicate, and each room has two buttons labeled 0 and 1, plus a computer screen that will flash an objective. To get the teammates to "betray" each other, first, one quarter of the time we do a "control experiment" where we flash the objective "make the sum of your button presses even" and the team wins if the sum of their 3 chosen numbers is even; second, three quarters of the time we choose one to be a "traitor" and flash on their screen "make the sum of your button presses even", but the other two get "make the sum of your button presses odd" and the team wins if the sum of their 3 chosen numbers is odd.
Classical methods have to fail these tests with at least probability 1/4, while quantum methods can have a probability arbitrarily close to 1 as the quantum entanglement of an initial state is pristinely preserved. (If it interacts with other things around it, it entangles with those things too, and then it's harder to detect the entanglement bringing together only 3 of the entangled pieces.)
Old-ish question, but someone bumped this to the homepage and since the other answers are not mentioning coherent collective phenomena at all I would like to provide another answer.
I agree with CRDrost's answer in that you will hardly ever see such effects for nuclear decay and also that if one particle decays, the other one will not magically do so. However I would like to stress a point about the "strange correlations" that he mentions: Entanglement or coherence between multiple particles can certainly cause enhanced decay rates.
In fact this is a well known and observed effect in a variety of systems. It is also known as the phenomenon of superradiance.
For nuclei it will be a bit hard to observe this since establishing the required entanglement between them is hard. However the effect can easily be seen e.g. in atomic lattices or confined atomic vapors. The coherence is established by shining a coherent laser on the sample. Then the enhanced decay rates can be seen as a broadening in the spectral line shape.
- For a good review on the topic see: M. Gross, S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Physics Reports, Volume 93, Issue 5, 1982, Pages 301-396, ISSN 0370-1573, http://dx.doi.org/10.1016/0370-1573(82)90102-8. (http://www.sciencedirect.com/science/article/pii/0370157382901028)
- The above also contains a good list of observations of the phenomenon.
Entanglement can be thought of as a correlation between states. When you say that 2 uranium atoms are entangled, you need to say how.
For example, two electrons could be entangled in such a way that one is spin up, and the other spin down. This occurs when we look at a 2-electron system as a whole and determine that the expectation value (mean value) of the spin of the system is 0. This corresponds to a superposition of states: one where the 1st electron is spin up and the other is spin down, and the converse state.
Now when we pull these electrons away from one another and measure the spin of an individual electron, we know that the other electron has the opposite spin without actually measuring it. This is because the 2-electron system has now become one of the two states of which it was a superposition.
The exact nature of why this occurs (i.e. when/if the wavefunction 'collapses'; at which point the transition from superposition to eigenstate occurs) is a topic of contention.
To entangle a system of two particles with two states each, not-yet-decayed and already-decayed, you need to put them into a superposition of these two states and have their states at least partially depend on each other. Basically, you can have states of the type that either both or none have decayed yet (but you do not know if either has), or of the type that one (but not both and not neither) has already decayed. Any combination (superposition) of these and of the unentangled states are also possible.
If you then wait until you get a decay out of that system there can be a change in the state (or, if you like to look upon it that way, in your information about the state) of the other particle as well: The first one must have previously been in the undecayed state, so the other one is in the state that is compatible with that information after the first one does decay.
If you do not like the informational interpretation, another way of looking at this problem is that you (somewhat incidentally) performed a measurement on the state of the particle that decayed. In the two-particle system, that collapses the wavefunction to only those states where that particle was able to decay (was in the undecayed state).