To begin, let us first discuss superposition states in C² and formalize this concept mathematically. A superposition state in C² is defined as:

– ψ⟩ = a₀|0⟩ + a₁|1⟩,

where a₀ and a₁ are elements of C², meaning the coefficients are members of the set of complex numbers. For non-superposed states in C², there are only two measurable states: |0⟩ or |1⟩.

What about in a superposition state? 

This question introduces complexity. In superposition states, a quantum state is not in a definite classical state but is expressed as a combination of multiple potential states. Thus, a superposition state in C² is represented as:

– ψ⟩ = a₀|0⟩ + a₁|1⟩.

We know that this representation indicates a superposition state. Additionally, we understand that a₀ and a₁ are complex numbers. These coefficients represent the probability amplitudes for the state being in |0⟩ and |1⟩. This implies that, before measurement, the system exists in both |0⟩ and |1⟩ states simultaneously. When a measurement is performed, the system collapses to one of these states with a certain probability, yielding a definite outcome.

Now, turning to the non-superposed case mentioned above, let us discuss the matrix representation of the observable. In quantum mechanics, a matrix representation is a mathematical expression that represents the measurement of a physical quantity. The matrix representation expresses the self-adjoint operator that represents this physical quantity in matrix form. This operator describes how measurements affect quantum states.

In simpler terms, the observable matrix:

Is an operator that acts on the state vectors of a quantum system and determines the outcomes obtained during measurement.

For a non-superposed observable state, the matrix A is represented as:

A

(𝟎 𝟎)
(𝟎 𝟏)
which is a self-adjoint operator.

The eigenvalues of the matrix A defined above are 0 and 1, with the corresponding eigenvectors being |0⟩ = (1,0)ᵀ and |1⟩ = (0,1)ᵀ, respectively. According to the measurement axiom in quantum mechanics:

• If A is measured and the system is in the state ψ⟩ = |0⟩, the measurement result will definitively be 0, and the system’s post-measurement state will remain |0⟩.
• Similarly, if A is measured when the system is in the state |ψ⟩ = |1⟩, the measurement result will definitively be 1, and the post-measurement state will be |1⟩. This example illustrates that quantum measurement in this case is deterministic.

Now, let us discuss some properties of the operator A and how it behaves during the quantum measurement process.

The matrix A defined above indicates that the operator A only acts on the |1⟩ state and does not affect the |0⟩ state. In Dirac notation, this operator is expressed as:

– A = |1⟩⟨1|

This representation shows that the operator A only measures the |1⟩ state and yields an eigenvalue of 1 for this state.

To better understand the effect of the operator on vectors, we can perform the following calculations:

1. Effect of A on the |0⟩ state:
   A|0⟩ = |1⟩⟨1|0⟩ = 0|0⟩. This shows that when the |0⟩ state is measured by the operator A, the result is 0, and the post-measurement state remains |0⟩. This indicates that the |0⟩ vector lies in the null space (ineffective space) of the operator A.
   
2. Effect of A on the |1⟩ state:
   A|1⟩ = |1⟩⟨1|1⟩ = |1⟩ = 1|1⟩. In this case, when the operator A measures the |1⟩ state, the result is an eigenvalue of 1, and the post-measurement state remains |1⟩. This shows that the |1⟩ vector is an eigenvector affected by the operator A.

In conclusion, this analysis explains how a quantum mechanical measurement affects a specific eigenvector and what outcomes are produced. From a physical perspective, the method by which the measurement is performed is not important here; what is significant is that the operator A is self-adjoint (hermitian). This property implies that the associated observable is physically realizable. Self-adjoint operators correspond to measurable observables in quantum mechanics, and thus it is assumed that such measurements can be performed in practice.

---

MATHEMATICAL ADDITIONAL EXPLANATION

Let us discuss the mathematical meaning of the expression |a⟩⟨b|. For example, A = |1⟩⟨1| is the Dirac notation form of a quantum operator. This expression represents a type of operator known as a projection operator, which selects or “projects onto” a state associated with a specific eigenvector.

|1⟩⟨1| is the tensor product of these two vectors, resulting in a matrix. This matrix projects only onto the |1⟩|1⟩ state in the vector space. In other words, when this operator is applied to a quantum system, the system remains in or is projected onto the |1⟩|1⟩ state.

Let’s go through an example:

⟨1| = [0 1],
|1⟩ =

[0]
[1]

Therefore, |1⟩⟨1| is represented as:

|1⟩⟨1| =

[0 0]
[0 1]

This type of operator maintains the system in the |1⟩ state and is referred to as a Hermitian conjugate when expressed as ⟨1|. It is represented as a row vector.

---