After discussing the theory so far, let us define an operator 𝐴 and explore how tensor products and operators work in two-qubit systems.

Let the operator 𝐴 be a 2 x 2 matrix defined as follows:

𝐴 =
(00)
(01)

Now, let us talk about this operator. We have mentioned this in many places, but let’s reiterate. Defined operators act on and affect specific quantum states.

For this matrix:

-The first row [0, 0] means that the 𝐴 operator turns the |0⟩ state into zero.

-The second row [0, 1] means that the 𝐴 operator leaves the |1⟩ state unchanged. Thus, if this matrix is applied to the |1⟩ state, nothing changes.

-In other words, this operator nullifies the first component and preserves the second component as it is.

Now, let us define the identity operator 𝐼. This operator leaves vectors unchanged:

𝐼 = 
(10)
(01)

With these operators defined, let us now pose the following question:

What will the matrix be if we want to measure the first qubit?

We know that if we want to measure the first vector, our formula will be 𝐴 ⊗ 𝐼.

𝐴 ⊗ 𝐼 = 
(0 0 0 0)
(0 0 0 1)
(0 0 0 0)
(0 0 0 1)
These matrices are obtained as a result of the Kronecker product, which will be discussed in detail in the mathematics section.

What will the matrix be if we want to measure the second qubit?

We know that if we want to measure the second vector, our formula will be 𝐼 ⊗ 𝐴.

I ⊗ 𝐴 = 
(0 0 0 0)
(0 0 1 0)
(0 0 0 0)
(0 0 1 0)

Now, after calculating the vectors, we need to calculate the quantum states. In the 𝐶² space, two qubits have a total of 4 states. These states are as follows:

– |00⟩, |01⟩, |10⟩, |11⟩

These states are calculated using the Kronecker product. It is essential to know the matrix representations of |0⟩ and |1⟩:

|0⟩ = 
(1)
(0)

1⟩ = 
(0)
(1)

After the Kronecker product for each state, the resulting matrix expressions are shown below.

1. ∣0⟩ ⊗ ∣0⟩:
1
0
0
0

2. ∣0⟩⊗∣1⟩:
0
1
0
0

3. 1⟩⊗∣0⟩∣
0
0
1
0

4. ∣1⟩⊗∣1⟩
0
0
0
1

After expressing the states of the qubits in matrix form, let us discuss the following:

How do we apply the 𝐴 operator to these matrices? Here, an intensive mathematical process is required.

Now, let us also discuss the measurement of the qubits.

a) Measurement with the 𝐴 ⊗ 𝐼 Operator:

We know that this operator affects the measurement of the first qubit and only evaluates the state of the first qubit.

– If the measurement result is 0, the state of the system is either |0⟩|0⟩ or |0⟩|1⟩. This means that the first qubit is |0⟩.

– If the measurement result is 1, the state of the system is either |1⟩|0⟩ or |1⟩|1⟩. This means that the first qubit is |1⟩.

b) Measurement with the 𝐼 ⊗ 𝐴 Operator:

We know that this operator affects the measurement of the second qubit and only evaluates the state of the second qubit.

– If the measurement result is 0, the state of the system is either |0⟩|0⟩ or |1⟩|0⟩. This means that the second qubit is |0⟩.

-If the measurement result is 1, the state of the system is either |0⟩|1⟩ or |1⟩|1⟩. This means that the second qubit is |1⟩.

Can we perform this operation with matrices? 
Yes. The above generalizations can indeed be derived through matrix operations.