Very intelligent paper and interesting definition: "Two particles can be considered entangled when the energy of their interaction has been large at any time in the past, and neither of these particles are involved in the emission or absorption of other particles."

Clever solution by using virtual phase-particles that are not restricted by the speed of light.

Thank you.

Природа была права. А Бор с Эйнштейном построили только простые модели.

Nature was right. And Bohr and Einstein built only simple models.

Most of science accepts the Bohr conclusion. Einstein wanted determinacy, but in stead he got causality.

I take a different approach noting that QM always averages to the classical law.

Suppose the natural laws reside in vacuum space and everything that passes through it. Implied is that the laws require an amount of volume to contain a complete set of laws. If a liter of volume is tested the classical laws are found there. When the volume is subdivided the classical laws are found in both parts.

If subdivision continues far enough  eventually a fragment of volume will be encountered that does not contain a complete set of laws. It is the scale of QM. Any particle interaction that occurs in such a volume fragment will be unpredictable and described by statistics.

Experiments have shown that the classical law comes from an average of 30 or more quantum events of the same type. It puts some limit on how many types of volume fragments exist and what part of the natural laws they might contain. 

In this representation Bohr and Einstein both win the argument.

But the question is: how particles interact with each other at a distance (spooky action at a distance)?

Thr Feynman method Sum Over Paths allows particles to tunnel through time and space everywhere forward and backward. The entanglement is an example of the backward tunneling. In GR and 4D it isn't explained, but with higher dimensions and folding like TGD the effective distance may be much shorter than implied in 4D.

Beam splitting experiments have a similar feature. The laboratory frame sees spooky sequence, backward time, and distance. But in the frame of action all of the decisions are made at the same time in the same place. Much argument occurred, but the experiments gave consistent results.

Thus your action at a distance is an illusion created by relativistic transformation from the action to your measurement frame, but a stubbornly persistent illusion. In the frame of action the entangled particles are not separated by space or time, although it may seem so in 4D.

In the strict mathematical formalism, each separate system has its own Hilbert space.

The above mentionned feature that full state knowledge of the composite system is not reduced to the separate knowledge of each sub-system is encoded in the fact that the Hilbert space where the composite system lives in is NOT the direct sum of each sub-Hilbert space but rather the tensor product.

This represents the fact that the entangled states possible for the composite system are indeed states which cannot be represented by using only a direct sum of subsytem states (they don't "factorize" into sub-system states).

The space of states for the composite system can naturally, from QM point of view, contain superpositions of combined states which makes it a much bigger space than just the direct sum of sub-systems. The tensor product model exactly that fact.

It is also worth mentionning that the so-called "collapse" of wave functions which can mislead into thinking that information is lost is better explained by understanding that the measurement apparatus just get entangled with the system you measure. Any measure indeed mixes the system state with the measurement system and puts them in an entangled state of the system's outcome and the apparatus state exposing that outcome.

 Is it possible to give positive definition for "entanglement" in this case?

Entanglement is a shared probability, expressed as a wave function in QM, the chance of finding concentrated energies in locations or with momentums, in which the separation of probabilities does not occur without collapse of the wave function by taking a measurement or interfering with one of the entangled quantities. Information is not lost. Some potentials are not achieved, but their histories are not deleted. 

I think that this definition of entanglement is negative. That is, this definition describes what entanglement is not. It does not provide recipes for the preparation of entangled states.

The correlation energy in entanglement(s) is, usually, very weak. So, new light on the physical implications of the entanglement(s) might have an illustrative (and helpful) origin in helium-like systems[1,2], also[3].

1. Quantum entanglement of helium-like systems with varying-Z: compact state-of-the-art CI wave functions   http://iopscience.iop.org/article/10.1088/0953-4075/48/17/175002

2. https://www.researchgate.net/publication/224872168_Entanglement_in_helium

3. Entanglement of unstable atoms: modifications of the emission properties https://arxiv.org/pdf/0907.1062v1.pdf

Article Entanglement in helium

Paradoxes are supported by false assumptions and resolved when the assumptions are discovered. For the original question I demonstrated that both Bohr and Einstein had valid arguments and were correct to some extent in their own areas of expertise, but each was incorrect in the ways they related to the others work, Bohr is usually assumed to be correct because the scale of space in a particle interaction can never be explored or classified. Hidden variables are rejected by principle. I have demonstrated with the subdivided volume that the variables are elusive but not hidden, and the volume can be explored and classified. False assumptions are discovered by answering the question. "Where do physical laws and constants reside in nature?"

Entanglement of photons is routinely produced in down divider crystals where one photon is split making two photons that exit from different surfaces. A similar effect can be produced by circularly polarized photons encountering a still liquid surface at a shallow angle. Two streams of plane wave photons are produced polarized at right angles. Electrons can become entangled with photons in photo multiplier cascading devices. Also electrons can become entangled with electrons by close contact especially in the process of liberating electrons. Electron and positron pairs are created together and entangled until the positron or electron interacts with a third particle. This was not the original question, but it is a positive view of entanglement.

In other terms entanglement is an isolation of information shared by a set of particles. Decoherence is a mixing of that information with other information from outside of the set. It relates to the question of what constitutes a measurement in QM, and the answer is that the mixing of information constitutes a measurement.

If we don't know pre-history of interaction of two particles - can we conclude that they are entangled? 

Dear Alexey,

"If we don't know pre-history of interaction of two particles - can we conclude that they are entangled?"

I don't thick so, definitely. I rather think that the entangled particles still form *one* particle-structure for some of its properties, but can be detected separately by some other properties.

Especially with regard to its density structure, I think they still are *one* particle.

If we compare this with the speed of sound in materials, where the speed is larger in dense and structured materials than in air. For diamond, it is many times faster.

I don't say that information is structured the same way, but such behavior means that the value of the propagation speed of waves, v2=E/m (with "E" the energy per unity of volume and "m" the mass per unity of volume) can have a different "E" that is much higher than in vacuum, while "m" doesn't increase as much.

So, it is an interesting knowledge.

Additionally, I would say that because of the spreading of the entagled particles, the mass-density "m" will decrease accordingly. Even if the energy-density "E" remains the same, the speed will strongly increase.