Justification of ignoring large set of entanglements If we can think about the universe as a wave function then many particles should be entangled with many other particles in the universe. The obvious question arises why we don't see those entanglements in everyday circumstances. One standard explanation given is those entanglements average out and cancel so we can ignore those. However, hardly any mathematical justification is given for them to cancel. My question is how much trust one should have on that particular assertion? Is there any mathematical arguments already put forward by anyone?
 A: Dear sb1, the assumptions implicitly contained in your question are upside down. Exactly the fact that the degrees of freedom of the Universe are becoming increasingly entangled means that we cannot observe the quantum phenomena in practice. 
The entanglement that emerges after a relatively short time is the entanglement among a huge number of degrees of freedom. So to detect that it is real, one would have to make an accurate and correlated measurement of all these entangled degrees of freedom that are relevant, and this is impossible in practice.
If one only studies a subsystem where some degrees of freedom is missing - because they become effectively undetectable - it is enough to "trace over" the unobservable degrees of freedom and to predict the observable system via its density matrix.
As Roy Simpson mentioned, the process of decoherence - which is extremely well understood as of today - guarantees that the density matrix for an observable system with many degrees of freedom - one that interacts with the environment (the unobservable degrees of freedom) - rapidly becomes diagonal, so observable interference goes to zero and quantum physics de facto reduces to classical physics.
What happens with the entanglement? Well, the entanglement between the observable and unobservable degrees of freedom is still there but it is unobservable because the unobservable degrees of freedom are unobservable. And when it comes to the entanglement between various observable degrees of freedom only, the part of it that has decohered disappears (and classical behavior takes over), and the part that hasn't decohered is still there (and quantum mechanics is still important).
But I think that instead of entanglement, you could have equally well used the term correlation because entanglement is nothing else than correlation that may use all the quantum information.
But even in classical physics, it would be true that the Universe - e.g. all its atoms - are "chaotically" evolving into a state in which the velocities and positions of all the particles are correlated in complicated ways (imagine a description using a distribution function on the phase space; as in reality, you can't quite measure and know the exact point on the phase space). Yes, it is true: evolution makes things complex. But exactly because they are so complex, the correlations become unobservable; your question incorrectly assumes that the complicated character should make the correlations more observable). The same is true for the entanglement: the convoluted character of entanglement makes it unobservable.
At the very end, let me emphasize that a diagonal density matrix is technically entangled, indeed. It's because this matrix is not a tensor product of a pure state vector with its conjugate: whenever this factorization is impossible, entanglement is nonzero. If the density matrix has many nonzero entries on the diagonal, it is a mixed state containing many independent contributions from pure states. That doesn't mean that this "entangled" character of the density matrix is "highly quantum" i.e. that it should exhibit many striking quantum properties: quite on the contrary, by getting a nearly diagonal density matrix, we get a highly classical situation!
A: The statement that the entanglement with many other objects "cancels out" is a little misleading, giving the impression that the various entanglements somehow undo each other, so they don't affect the final state. In fact, though, all of the various entanglements between a simple system of interest and the objects in its environment will play a role in determining the exact probability of the various possible states, and thus determining the outcome of the measurement.
What relieves us from having to track every single possible entanglement (which would be completely impractical) is that the effects of entanglement with the environment are random and fluctuating. On each repetition of the experiment, the system of interest will interact with a slightly different set of environmental objects, in slightly different ways, so the effects on the wavefunction will be different every time.
Since quantum mechanics only allows us to predict probabilities, the only way we can check theories by experiment is to make many repeated measurements of our system of interest, and use those repeated measurements to determine the probability of the various measurement outcomes. The random and fluctuating nature of the environmental entanglement means that while the details of the interaction with the environment plays a role in determining the exact probability for any given measurement, the effect of those interactions is different from one measurement to the next. Which means that, when you repeat the measurement many times to get a probability distribution, what you see is a kind of average over all the different ways that the environmental entanglement affects the various different runs of the experiment.
To give a highly imperfect analogy, it's a little like statistical mechanics or thermodynamics. While in principle the evolution of a thermodynamic system-- the canonical box full of ideal gas, say-- depends on the motion of every single particle making it up, in practice, the only thing we can measure is the average effect of uncountably huge numbers of moving particles. Thus, we have a well-established mathematical apparatus for figuring out the statistical behavior of all those particles, which gives us the highly successful fields of thermodynamics and statistical mechanics.
When people say that the effects of multiple entanglements "average away," the implied process is sort of similar. It assumes that the experiment will need to be run a very large number of times, and so what you calculate in the end is a sort of average over all the possible effects, in the same way that, say, the calculated temperature change of an expanding gas is an average over the change in speed of the umpty-zillion particles making up that gas.
(This is not a precise correspondence, so don't try to push it too far. It's just supposed to get the basic idea of the underlying process.)
A: Most of the time, we have decoherence, which suppresses interferences between configurations with different values for the pointer basis. On the other hand, showing a pointer basis always exists is an open problem. According to the Everett interpretation, each pointer basis splits into a parallel world of its own. Or at least, it can be treated as if it did.
But if some day in the future, we construct a quantum computer implementing Shor's factorization algorithm for integers with a huge number of digits, we can have a complicated multipartite entanglement between many qubits which eventually interfere constructively and destructively to get a nontrivial result.
A: In practice, except in very particular circumstances, you can't do experiments to observe highly entangled states. The results of any feasible experiment will be the same as if it was done on an equivalent mixed state. It's the same reason that nobody has observed a cat in the state $\frac{1}{\sqrt{2}}(|$alive$\rangle+|$dead$\rangle)$. What experiment would distinguish between a superposition of an alive and a dead cat and a mixture of these two states?
