# What is quantum entanglement?

In this post I try to figure out, how entanglement of a two qubit system can be seen and how it can be realized with quantum gates.

In a two qubit system, maximum entanglement is reached in one of the four Bell states:

(1)

Ok, so where is this entanglement now really apparent?

Each of the two qubit can be described by:

(2)

(3)

The overall state of a two qubit system is then:

(4)

Let’s try to find the values for the single coefficients (e.g. ⍺, β, ɣ, …) for the Bell state above (1) which satisfy:

(5)

Bad luck! We will not find values for the single coefficients, which means, that we’re not able to describe the entangled state by each qubit itself. They’re entangled!

Such an entangled state for a two qubit system can be achieved by a Hadamard gate on qubit 1 followed by a CNOT gate with qubit 1 as control qubit. (see it in Quirk)