The first to introduce the term "entanglement", was Erwin Schrödinger in 1935, who used it to describe a phenomenon found in experiments, but at the time no relevance was found for theoretical physics, this phenomenon happened when trying to describe individually a set of particles that had interacted.

Albert Einstein that year along with Boris Podolsky and Nathan Rosen publishes an article where he discusses the famous phenomenon, and does so by formulating the well-known paradox EPR, which is a form of mental experiment. In this experiment they postulate entanglement as a statistical property of the physical system formed by a pair of electrons, which come from the same source, so they are highly correlated due to the law of conservation of the linear moment, according to this, if we We tried to make a measurement on one of the interlaced particles. The collapse of the state inferred by the measurement would instantly affect the other. Because quantum mechanics can not explain the root of this phenomenon, it is concluded that the quantum theory should be incomplete and exclude the possibility of action at a distance (Principle of locality). Quoting their words, they concluded that "The description of physical reality through quantum theory based on a wave function is incomplete".

In 1964 Bell accepts the conclusions of the EPR paradox and with this proposes a possible explanation to the phenomenon through his theory of local hidden variables or Bell's theorem, this theorem supposed a consecration for quantum physics, since it allowed quantify mathematically the EPR paradox in terms of inequalities, and with it its experimental realization. This theorem has strong philosophical implications in modern physics because it is due to an impossibility theorem, it says that "No physical theory of local hidden variables can reproduce all the predictions of quantum mechanics". Bell concludes that

- The results obtained by a measurement are determined by the properties of the particles and are independent of the measurement (realism).

- The results obtained in a certain position are independent of any action taken outside of their immediate surroundings (locality).

Unfortunately the most important aspects of entanglement were not very well understood at the time, and only in the mid-90s was it formalized in terms of entropy inequalities based on von Neumann's entropy, a violation of these inequalities meant a sign that the quantum states were intertwined, although the physical meaning was not yet clear. Later, many explanations were proposed, all of which were in the area of quantum information theory.

Quantum entanglement: A property of compound systems.

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Paradoxically, entanglement is considered one of the most interesting quantum phenomena and also has no equal in classical physics. It is said that a system is intertwined if and only if, its quantum states can not be expressed as the tensor product of its local constituents. In other words, this implies that the existence of global states of compound systems can not be written as a product of states of each of the subsystems. A simple way to explain this, is to suppose the existence of two quantum systems, one called A and the other B, then these systems are put in contact, if these systems are intertwined, then the values of certain properties of system A are correlated with the values that the properties of system B will assume, these correlations will persist even if these systems are spatially separated. [1] Mathematically this means that

Here is the global state or composite system, and , they are the subsystems that compose it.

States of maximum entanglement.

The states of maximum entanglement are called Bell states or EPR pairs. Two particles reach the maximum entanglement as their direct interaction increases, that is, the greater the interaction, the greater the correlation. In this case the EPR pair is composed of two qubits, which is an analog of the classical bit, so it represents the basic unit of quantum information. A state is maximally interlaced if from local operations (Operation that only acts on one qubit, that is, it does not act on two qubit simultaneously) any other state of the base can be constructed. The states of Bell form an orthogonal base of the quantum state space of a system composed of two qubits, these bases are called Bell bases[1].

Quantum teleportation

Quantum teleportation was theoretically proposed by several scientists in 1993, and was achieved for the first time in 1997 through optical means in the laboratory, in 2006 it was possible to transport information from a light beam to a macroscopic system at a distance of 50 cm, in 2007 it is possible to maintain the entanglement between a pair of photons at a distance of 144 km, this opened the possibility of transporting information over long distances.

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Classically the intuitive idea of teleportation is that a body that is in a spatial area X at a time t, dematerializes and appears in a spatial area Y at a time t + T. Quantumly this is a bit different, since what is transported is the quantum state of the particle, but because the particles that are spaced in the different positions are indistinguishable, the result is equivalent. Quantum transportation does not involve transport of matter or energy, what is transported is the information, in addition the quantum state to be transported does not need to be known, and it is important to mention that there is no contradiction as to what special relativity imposes, with This means that there is no information transmission at speeds greater than the speed of light, since in order to transport the state, a classic channel is needed.[2]. The teleportation protocol is as follows:

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Be Alice (A) and Bob (B), two individuals who are far away from each other and wish to do teleportation, they to do this before separating have shared a state of maximum entanglement of two qubits or state of Bell (Red dots), then Alice receives an unknown status (Blue color dot) of two levels, which wants to send it or transport it to Bob, if the state was known it would be enough for Alice to call Bob through a channel classic (eg phone) and tell you the details of the state so that he can recreate it in the particle I owned. Since it is not like that, Alice resorts to the state of Bell and places her unknown state in contact with it, thus creating a correlation, so that now she has a state composed of three qubits, so Alice can reconstruct her state in terms of the bases of Bell and the qubits that belongs to Bob, this is done by observing that each state of Alice belongs to Bob.[3].

Quantum entanglement gave rise to several related discoveries with information and quantum computing, among these we have Quantum Cryptography through Bell's theorem, and dense coding or others. Unfortunately, quantum entanglement has a very complex structure, is very fragile to the environment and can not be increased if the systems are not in direct contact.

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For more information you can consult the following bibliography.

[1]. Michael A. Nielsen e Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge.

[2]. “Fundamentos matemáticos de la mecánica cuántica” de John von Neumann – Instituto de Matemáticas “Jorge Juan” Madrid.

[3]. “Entrelazamiento” de Amir D. Aczel - Editorial Crítica SA