I think pop science has a lot to answer for with it sacrificing the truth for the sake of hype. Although quantum entanglement cannot be understood in classical terms, it is not as strange as commonly portrayed. In particular, quantum entanglement cannot be used to communicate information. That is, if I have a collection of quantum entities, and my partner who is very far from me has the corresponding quantum entangled partners of my quantum entities, then there is nothing I can do to my quantum entities that will have any influence on the results of any measurements my partner performs on her quantum entities. This is the crucial limitation of quantum entanglement that prevents it from being the magical phenomenon suggested by the phrase "spooky action at a distance".

 

In the Einstein-Podolsky-Rosen thought experiment (or Aspect's real experiment), there is no actual communication between the entangled particles. Instead, one has an uncanny correlation between a pair of random particle states that cannot be explained in terms of classical physics. It might seem that the outcome of a measurement of one particle has somehow communicated itself to the other particle, and the use of the phrase "spooky action at a distance" promotes this notion, but this is merely an assumption based on thinking in classical terms. The impossibility of the experimenter to affect the state of one particle by manipulating its entangled partner represents the true limitation of quantum entanglement compared to the hype. It is often said that correlation is not causation. Quantum entanglement is correlation, communication is causation.

 

Nevertheless, quantum entanglement cannot be understood in classical terms. It requires the concept of a multi-particle quantum state, in particular the quantum superposition of multi-particle states. In the absence of quantum superposition, the correlation between multiple particles that occurs as a result of interaction between the particles can be understood in classical terms. But it is the quantum superposition of correlated multi-particle states that is quantum entanglement, and it is the quantum superposition that is why quantum entanglement can't be understood in classical terms. It should be also noted that a quantum superposition of uncorrelated multi-particle states is not entangled. The point is that measurement of a correlated multi-particle state is the measurement of all the particle states by the measurement of any one of the particle states. And once this measurement has been performed, the multi-particle state is no longer entangled because it is no longer a quantum superposition.

 

In my explanation, I avoided the mathematics. But I can say that if a single-particle quantum state is a sum of basis vectors in Hilbert space, then a multi-particle quantum state is a sum of tensor products of the basis vectors of the corresponding single-particle states. Then a multi-particle quantum state is entangled if it is not separable as the tensor product of single-particle quantum states. A measurement of a single particle of a multi-particle system corresponds to forming the inner product of the multi-particle state with the basis vector corresponding to the outcome of the single-particle measurement. If the multi-particle quantum state is a tensor product of single-particle quantum states (ie not entangled), then the quantum state of the remaining unmeasured particles will be unchanged compared to their original quantum state. But if the multi-particle quantum state is a non-separable sum of tensor products (ie entangled), then the quantum state of the remaining unmeasured particles will depend on the outcome of the single-particle measurement. Thus, we have in the case of entangled quantum states the manifestation of correlation and the ability to determine the state of unmeasured particles from a measured particle, in contrast to the non-entangled quantum state without correlation and without the ability to determine the state of unmeasured particles.

 

I should also mention that quantum mechanics has pure states and mixed states, and these together complicate the above discussion. I have restricted my discussion to pure states.