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Composite systems i.e. quantum systems that are made of several particles, can also be in a superposition of states, like single particles do.

The strange fact is that we can achieve states were the several particles are strongly correlated:

We know that in this case if particle a is measured in state \(\ket{a_1}\), then particle b will always be measured in \(\ket{b_1}\). This entangled state would be described by this formula

\[\ket{\psi} = \ket{a_1} \ket{b_1} + \ket{a_2} \ket{b_2}\]

The result of the measurement of a cannot be known a priori, but the result of the measurement of b will be certain. This property is called entanglement, and occurs when the wave function cannot be written as a product state in terms of the states of the individual particles.

The concept of measurement in quantum mechanics can be interpreted as a collapse of the state (Copenhagen interpretation): one of the terms in the superposition will be randomly “choosen”. This leads to a paradox in the sense that for separated entangled particles, this “choice” will be done at the same time at arbitrary far locations! Einstein called this the “spooky action at a distance”.

How the correlation occurs in more complex settings cannot be explained by classical physics and statistics. A thorough and definitive experimental proof of the validity of the rules of quantum mechanics wrt. entanglement was achieved with the so-called EPR Experiments based on the Bell Inequalities.

A few examples of realization of entanglement are listed in the physical implementation section: This is real!