A marginal but, I believe important remark. The expression "quantum scale" has no definite and precise meaning (this is one of the most fundamental difficulties of the theory). Sometimes one claims that the limit is defined by macroscopicity, but this is sensless, because there many many macro-processes which require that quantum linearity holds  in order to be understood (think of superconducting devices, or even the internal structure of a macroscopic crystal etc.). John Bell has been quite sharp on this point: Quantum mechanics is absolutely ununderstandable and logically inconsistent because nobody knows where the border line between quantum objects and classical objects has to placed, and the theory does not help at all in this sense.

All electrons in atoms are entangled. The symplest electronic entangled system is  two electron  system  in Helium. All electrons in atomic cores are entangled. The quantum states of multielectron systems  have been widely studied in atomic physics,  all these states are entangled.

In general, it is very difficult to find solid state systems where electrons are NOT entangled. Complex interactions such as Coulomb, spin-spin, spin-orbit produce some amount of entanglement in ground & excited states of condensed matter systems. Here I am talking about large von Neumann entropy of multi particle systems (generalization of the two-particle entanglement). 

Even if the ground and low energy states have not too much entanglement in general (see Area Law of entanglement), as soon as a delocalized environment interacts with the system the entanglement is boosted. As one of many examples, consider a series of atoms inside a cavity formed by two mirrors, so that the photons in the cavity act as a delocalized environment. In this situation the natural states are light-matter hybrids called polaritons, where light and matter are entangled. Moreover, the 'matter part' of these polaritons is in general a maximally entangled state. This system of course can only be achieved in laboratory conditions, but in general any system of discrete levels connected with a bosonic delocalized environment will have non-trivial entanglement structure.

So, at least in condensed matter physics and quantum optics, entanglement is very present, and in most cases it has a non-trivial effect on the system structure and dynamics.

Thank you. Entanglement is the rule then.

So now what are the consequences of this general entenglements?

The simple consequence is that natural systems are and behave as entanglement requires. Without entanglement the complex atoms will have completely different properties. So what do you mean with your question?

OK Giancarlo. My question is more about the dynamics of entanglement of particles. Anytime some measure is performed one way or another sets of particles are not entangled anymore so that not everything is entangled in nature and I am wondering about this coexistence of local and non local particles.

Dear Charles, you are right, but what you assume is that measurement processes reduce the statevector, something which conflicts with quantum mechanics. So you are aiming at the measurement problem. My answer cannot avoid to make reference to the only consistent way which I know about accounting for reduction processes, i.e. collapse theories, which are stochastic and nonlinear modifications of the standard evolution equation of quantum mechanics. If you adhere to this point of vie, the entanglement, but only the one corresponding to the positions of macro-objects is actually forbidden, but other kinds of entanglement can remain.

In any case, your problem acquires a different and much more general status if you accept that reduction has to be consistently accounted for.

Charles:

- Does entanglement have any known consequence at a macroscopic scale.

I am fond of the idea that all particles emerge from some indivisible fundamental substance of the universe which pervades the entire space. The entanglement is possible due to this indivisibility of the fundamental substance through Newton's second law of motion which states that action and reaction are equal in magnitude and opposite in direction. The reason particle and anti-particle fly away in opposite direction at creation should also be seen as due to entanglement. Wave particle duality may also have something to do with entanglement. If you drop a stone on the surface of a lake, the resulting waves are all entangled. Lastly, this concept of indivisible substance can also explain the instantaneous action of gravity. In general relativity the gravitational waves travel at speed of light but no such waves have been detected yet. As an alternative explanation if we allow the instantaneous action than it is comparable to the generation of tsunami wave in the middle of the ocean which makes the waters at the distant shores recede instantaneously. 

Dear Giancarlo. Isn't it the Bohr's point of view that measurement reduces the state vector? Do you know of any text about collapse teories for beginners?

Dear Vikram.

Interesting point of view. Is it shared by many?

Charles,

Your question touches several fundamental problems as already said by others above. As being a chemical physicist with a focus on quantum chemistry, entanglement is one essential reason why the theoretical chemical field to describe the fundamentals of atoms, molecules etc, becomes technologically a very complex and difficult enterprise.

Although the utilization of the word entanglement was not generally promoted in quantum chemistry originally, we used the more common word separability, which in connection with many electron quantum systems are almost the same . 

In addition to the measurement problem, the process of decoherence and the holistic character of quantum mechanics it is not entirely clear how entanglement works in connection with e.g. photons, i.e. the parametric down conversion or in connection with "quantum jumps" and the emission or absorptiion of photons.

Even less is known in the case of beta emission, except to provide the necessary particles in order to fullfil the appropriate statistics of the reaction.

An interesting and fundamental field is the quite recent advance of the concept of decoherence scenarios, which promote envigorating interdisciplinary interactiions.

Interesting answer Erkki.

I agree, little is known. The superposition principle applies at the quantum scale but little is known about the consequences.

A marginal but, I believe important remark. The expression "quantum scale" has no definite and precise meaning (this is one of the most fundamental difficulties of the theory). Sometimes one claims that the limit is defined by macroscopicity, but this is sensless, because there many many macro-processes which require that quantum linearity holds  in order to be understood (think of superconducting devices, or even the internal structure of a macroscopic crystal etc.). John Bell has been quite sharp on this point: Quantum mechanics is absolutely ununderstandable and logically inconsistent because nobody knows where the border line between quantum objects and classical objects has to placed, and the theory does not help at all in this sense.

Dear Mirjana,

The electrons in atoms are not entangled. They are correlated but not entangled. The singlet state of helium or one of the triplet state can be entangled state iff coulomb potential can be switched off, which is impossible.  so the correlation provides us an aspect of entanglement but not entanglement itself. Thanks...

Entanglement is not correlations,I perfectly agree on this. In a statistical mixture of factorized states there are correlations but not entanglement. However entanglement can be defined mathematically as the fact that the state of two systems is not the product of a state of one of them and one of the other, i.e. it is not factorized. Electrons in atoms are actually entangled because the total state is not the direct product of two states. In such a case, the basic features of entanglement are all there, apart from the puzzle of the instantaneous collapse produced on one system by the measurement of the other, i.e. nonlocality. But this is another story ....

Dear Charles:

With reference to your post on 1st page.

Indivisible fundamental substance pervading the universe is a well known concept in India with a large following.It is mentioned in Bhagavad Gita as the Unmanifest. Sanskrit word "Dvaita" means duality and "Advaita" means non-duality also explained as one without a second. Non-duality is possible only if the one without a second is indivisible. So the multiplicity seen in the universe is the apparent multiplicity. There is a famous Advaita Vedanta philosophy of Sankaracharya with followers all around the world. Vivekachudamani of Sankaracharya describes the rules and procedure for verifying the non-dual existence (also called Brahman or Atman).

https://en.wikipedia.org/wiki/Advaita_Vedanta

Vivekachudamani of Sri Sankaracharya and Swami Madhavananda

http://advaitaashrama.org/Book/Detail/218

srimad-bhagavad-gita

http://www.chennaimath.org/istore/product/srimad-bhagavad-gita-sridhara-ordinary/