Buniy and Hsu also seem to be confused about the topics that have been covered hundreds of times on this blog. In particular, the right interpretation of the state is a subjective one. Consequently, all the properties of a state – e.g. its being entangled – are subjective as well. They depend on what the observer just knows at a given moment. Once he knows the detailed state of objects or observables, their previous entanglement becomes irrelevant.
... When I read papers such as one by Buniy and Hsu, I constantly see the wrong assumption written everything in between the lines – and sometimes inside the lines – that the wave function is an objective wave and one may objectively discuss its properties. Moreover, they really deny that the state vector should be updated when an observable is changed. But that's exactly what you should do. The state vector is a collection of complex numbers that describe the probabilistic knowledge about a physical system available to an observer and when the observer measures an observable, the state instantly changes because the state is his knowledge and the knowledge changes!In the section of our paper on Schmidt decomposition, we write
A measurement of subsystem A which determines it to be in state ψ^(n)_A implies that the rest of the universe must be in state ψ^(n)_B. For example, A might consist of a few spins ; it is interesting, and perhaps unexpected, that a measurement of these spins places the rest of the universe into a particular state ψ^(n)_B. As we will see below, in the cosmological context these modes are spread throughout the universe, mostly beyond our horizon. Because we do not have access to these modes, they do not necessarily prevent us from detecting A in a superposition of two or more of the ψ^(n)_A. However, if we had sufficient access to B degrees of freedom (for example, if the relevant information differentiating between ψ^(n)_A states is readily accessible in our local environment or in our memory records), then the A system would decohere into one of the ψ^(n)_A.This discussion makes it clear that ψ describes all possible branches of the wavefunction, including those that may have already decohered from each other: it describes not just the subjective experience of one observer, but of all possible observers. If we insist on removing decohered branches from the wavefunction (e.g., via collapse or von Neumann projection), then much of the entanglement we discuss in the paper is also excised. However, if we only remove branches that are inconsistent with the observations of a specific single observer, most of it will remain. Note decoherence is a continuous and (in principle) reversible phenomenon, so (at least within a unitary framework) there is no point at which one can say two outcomes have entirely decohered -- one can merely cite the smallness of overlap between the two branches or the level of improbability of interference between them.
I don't think Lubos disagrees with the mathematical statements we make about the entanglement properties of ψ. He may claim that these entanglement properties are not subject to experimental test. At least in principle, one can test whether systems A and B, which are in two different horizon volumes at cosmological time t1, are entangled. We have to wait until some later time t2, when there has been enough time for classical communication between A and B, but otherwise the protocol for determining entanglement is the usual one.
If we leave aside cosmology and consider, for example, the atoms or photons in a box, the same formalism we employ shows that there is likely to be widespread entanglement among the particles. In principle, an experimentalist who is outside the box can test whether the state ψ describing the box is "typical" (i.e., highly entangled) by making very precise measurements.
See stackexchange for more discussion.