# To make 2-qubits entanglement on quantum circuit

### First

To calculate on quantum computer we think about state vector as a simulator. Now we see the 2-qubits entanglement.

The state vector of 2-qubits superposition is,

`[1 1 1 1]`

These 4 element of vector denote state probability of 00,01,10,11. If we can control these probability we can make variety of applications using superposition and entanglement expected for quantum speedup.

### Basic of quantum entanglement

The famous quantum circuit of entanglement is,

`---H---*----       |-------X----`

And the state vector is,

`[1 0 0 1]`

Now we have probabilty of only 00 and 11. This circuit select 50% of 00 and 11 from 4 possibility of 00,01,10 and 11.

### To control state vector

For example, when we have,

`[1 0 0 0]`

This is the initial state of 2-qubits of 00. We can get this state vector from 2 |0> qubits.

`[1 ⊗[1 0]  0]`

And with X pauli gate we can change the state vector changing the shape of state vector like,

`[0  or [0  or [0 1      0      0 0      1      0 0]     0]     1]`

Next if we want to have just 2 probability from 4 state like,

`[1 0 1 0]`

We can get this state vector by applying H gate for just 1 qubit.

And if we want all probability for all 4 state vector,

`[1 1 1 1]`

We can get this state by applying H gate for both 2-qubits.

### How to get 3 from 4

To get 3 probability from 4 state vector element like,

`[1 0 1 1]`

Now we realize it using RY gate. First we have,

`[1 0 0 0]`

By rotating the first qubits 1/3 of degree we get,

`[1 0 √2 0]`

And then we apply Controlled-RY gate for this state vector we can get the expected state vector. The unitary matrix of Controlled-RY gate is,

`[1    1      1/√2 -1/√2     1/√2  1/√2 ]`

And we finally get,

`[1 0 1 1]`

as expected state vector.

### Conclusion

To get 3 probability from 4 state vector, we can use RY gate and Controlled-RY gate to realize what we want.