As discussed in several parts of this study, the interpretation of the Stern-Gerlach experiment demonstrates that the incoming particle stream possesses two intrinsic quantum spin states. These are denoted as |↑⟩ and |↓⟩, using the Dirac ket notation, which we will elaborate on in future discussions. Considering the 50% outcome obtained post-measurement, the incoming spin state can be expressed as:

(1/√2) |↑⟩ + (1/√2) |↓⟩

Now, why do we refer to this as a 50% result? In the Stern-Gerlach experiment, the measurement of the particles’ spin states occurs with a 50%-50% distribution because the incoming particles do not have a predetermined spin orientation (for example, a vertical spin direction); they are distributed randomly. According to quantum mechanics, such states are measured with equal probability for the two possible spin orientations, “up” |↑⟩ and “down” |↓⟩. In the experiment, a magnetic field separates the particles into upward and downward directions, meaning that the particles must exist in one of these two states. If the initial spin state of the particles is in an unknown superposition, according to the principles of quantum mechanics, the measurement result will be:

50% |↑⟩, 50% |↓⟩

Here, we stated that this occurs with equal probability, indicating that this represents a pure quantum state. This is a state that expresses superposition. Generally, states in quantum mechanics may not be in pure states but can exist in mixed states. For instance, if the measurement of a two-state system shows a 60-20 distribution for states |a⟩ and |b⟩, we can express the initial state of the system as:

(3/5) |a⟩ + (2/5) |b⟩

Therefore, we can make the following generalization:

αₐ |a⟩ + αᵦ |b⟩, ve αₐ² + αᵦ² = 1

• The expression αₐ² represents the square of the state αₐ. Similarly, the state αₐ² represents the square of the state αᵦ. The sum of their squares being equal to 1 indicates that these two states are in superposition and that their probabilities are normalized. An important point here is that we do not always expect this to be 1; rather, this situation is a special case.

We can think of the expressions ∣a⟩∣a⟩ and ∣b⟩∣b⟩ as vectors. Specifically, these expressions represent vectors that correspond to two directions within a certain vector space.

Now, if ∣a⟩∣a⟩ and ∣b⟩∣b⟩ exist in this vector space, we can create new vectors through linear combinations of these two vectors in different proportions. This process is referred to as a linear combination.

Thus ;

• αₐ |a⟩ + αᵦ |b⟩. is a linear combination of the vectors ∣a⟩ and ∣b⟩. 
• Here, αaαa​ and αbαb​ are coefficients that determine the contribution of the vectors ∣a⟩ and ∣b⟩.

If these vectors are orthogonal to each other, it means that they do not intersect. In other words, there is no relationship between ∣a⟩ and ∣b⟩; one is independent of the other. This condition is referred to as orthogonality. If the lengths of these vectors are equal to 1, they are termed unit vectors. Vectors with these definitions are frequently used in quantum states because orthogonal and unit vectors are ideal for representing fundamental quantum states.

We previously discussed the condition ;

αₐ² + αᵦ² = 1 

When this special condition is satisfied, we define the vector as a unit vector. Not every vector meets this condition; it is a special case.

Now, let us discuss the special situation of the Stern-Gerlach experiment. In this particular scenario, we encounter particles with two distinct internal spin states. These two states are denoted as ∣↑⟩ and ∣↓⟩. The existence of these two states is analogous to the concept of a bit in classical digital computing. A bit is the fundamental unit of information in classical computers and can only take on two values (0 and 1).

Similarly, if we can keep two-state particles in quantum systems under control (i.e., we can fix them at a certain location or isolate them)—a term sometimes referred to as “trapping” in some sources—these two states, ∣↑⟩ and ∣↓⟩, can serve as fundamental units representing information in quantum information processing. These two quantum states correspond to the values 0 and 1 in digital computers. Thus, in quantum mechanical systems, these two states can function like the binary digits in classical digital computation, forming the basic building blocks of quantum computing.

In this way, the concept of the qubit (quantum bit), the fundamental unit of quantum information processing, emerges. Qubits can exist in two distinct states like classical bits, but they can also be in a superposition state, meaning they can be present in both ∣↑⟩ and ∣↓⟩ states “simultaneously.” This property underpins the greater power of quantum computers compared to classical computers.

When linear combinations like ∣a⟩+∣a⟩ satisfy the condition ₐ² +ₐ² +1 they form valid pure states. Such combinations give rise to new states that do not solely depend on the states ∣↑⟩ and ∣↓⟩ but represent a mixture of them. Consequently, it becomes possible to create a multitude of states in quantum computing.

In classical computing, information is expressed solely through two fixed states, such as 0 and 1. These two states create a vector space operating over F₂ (a field with 2 elements) when combined using operations such as ⊕ (addition) and ∧ (multiplication). This structure is closed, meaning it is limited exclusively to 0 and 1. This implies that classical computers can process information under specific rules.

In quantum computing, however, we can obtain many different pure states by performing linear combinations between fundamental states like ∣↑⟩ and ∣↓⟩. These states can be in superposition, allowing them to exist in multiple states simultaneously. This feature provides quantum computing with greater flexibility and power compared to classical computing, as it enables a vast number of combinations and operations.