Non-superposed states can be defined as states that do not overlap. With this definition in mind, we can discuss the 𝐶² ⊗ 𝐶² space. In this defined space, the expression essentially represents the tensor product of two vectors, which results in four fundamental states. These states are expressed as follows:

– |0⟩|0⟩, |0⟩|1⟩, |1⟩|0⟩, and |1⟩|1⟩.

Now, let us delve into a more detailed discussion of this space. We begin by defining the concept of a self-adjoint operator. If an operator is defined in a Hilbert space and is equal to its Hermitian conjugate, it is referred to as a self-adjoint or Hermitian operator. The eigenvalues of Hermitian operators are real numbers. In quantum mechanics, measurable physical observables are represented by self-adjoint operators.

With this definition in place, we can proceed to the process of tensor products. We know that the tensor product combines two vector spaces to form a larger vector space.

Now, let us define operators 𝐴 and 𝐵, which are self-adjoint operators in the 𝐶² space. Since these operators are Hermitian, they are equal to their Hermitian conjugates. Mathematically, this is expressed as follows:

– 𝐴 = 𝐴† and 𝐵 = 𝐵†

Since 𝐴 and 𝐵 are self-adjoint, if we apply them to the vectors |𝑥⟩ ⊗ |𝑦⟩, we obtain the following mathematical expression:

–(𝐴 ⊗ 𝐵) (|𝑥⟩ ⊗ |𝑦⟩) = 𝐴|𝑥⟩ ⊗ 𝐵|𝑦⟩

The operator 𝐴 acts on the first qubit |𝑥⟩, and the operator 𝐵 acts on the second qubit |𝑦⟩. The tensor product of the two operators enables the application of these effects simultaneously.

A two-qubit quantum system is defined in two separate 𝐶² spaces, and specialized operators are used to perform operations on individual qubits without affecting others. These operators allow for measurements or operations on a specific qubit within the system. Let us now discuss the measurement of qubits below.

1. Measurement of the First Qubit:

We now know that to perform a qubit measurement, it is necessary to define the appropriate operator. It is important to note that the operator defined here will not affect the second qubit. Let us denote the operator applied to the first qubit as 𝐼. If we adjust the expression accordingly, we get:

-𝐴 ⊗ 𝐼

Here, 𝐼 is the identity operator and does not alter the vector it acts on. It represents the measurement of the first qubit in a two-qubit system. While 𝐴 is applied to the first qubit, the second qubit remains unchanged. Mathematically, we can express this as follows:

– (𝐴 ⊗ 𝐼)(|𝑥⟩ ⊗ |𝑦⟩) = 𝐴|𝑥⟩ ⊗ |𝑦⟩

In this expression, 𝐴 acts only on the first qubit and does not affect |𝑦⟩ in any way.

2. Measurement of the Second Qubit:

Now, let us again use the identity operator 𝐼 and discuss how to measure the second qubit. Since we need to act on the second qubit instead of the first, we must adjust our mathematical expression as follows:

– 𝐼 ⊗ 𝐴

Rearranging the expression, we get:

– (𝐼 ⊗ 𝐴)(|𝑥⟩ ⊗ |𝑦⟩) = |𝑥⟩ ⊗ (𝐴|𝑦⟩)