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# 3.5 Entanglement

Entanglement is a consequence of quantum theory and leads to a correlation between the outcomes of measurements which cannot be explained by classical physics. Two-particle states which show such non-classical correlations are known as entangled states.

Consider the following thought experiment. Suppose you have a two-particle system in the spin state as given in Equation 10. Before the experiment you know that particle 1 can be either spin-up or spin-down with equal probability. However, if you measure particle 1 to be spin-up then you know that particle 2 is spin-down as the measurement means that the two-particle state has collapsed into the arrangement of spins. In contrast, if you measure particle 1 to be spin-down then you know that particle 2 is spin-up as the two-particle state has collapsed into the arrangement of spins. This type of prediction is quite puzzling because the two entangled particles can be as far apart as possible and when a measurement is made on particle 1 then it is known simultaneously what the outcome of a measurement on particle 2 will be.

Entanglement is essential for quantum computing. Entangled states are generated as part of the workings of a quantum computer, as you will see later.

## Exercise 11

Confirm that the following states are normalised and determine whether the states are entangled.

### Answer

The normalisation condition for a general two-particle state is that the sum of the squares of the probability amplitudes is equal to 1.

The states are entangled if the two-particle state cannot be factorised into the product of a particle 1 state multiplied by a particle 2 state.

1. Checking the normalisation of state :

   showing state is normalised.

   can also be factorised:

   which is a particle 1 state multiplied by a particle 2 state so state is not entangled.

2. Checking the normalisation of state :

   showing state is normalised.

   State cannot be factorised and so is an entangled state.

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