Before delving into the topic, let us begin by asking the question: What is an Orthonormal Basis? An Orthonormal Basis is a special set of vectors in a vector space that possesses two fundamental properties: Orthogonality and Normalization. Now, let’s discuss these concepts one by one.
- Orthogonality: The vectors forming the basis must be orthogonal, meaning they are perpendicular to each other. Two vectors are considered orthogonal if their inner product is zero. But what does “vectors forming the basis” mean? In a vector space, these are independent vectors used to express any vector within that space. These vectors determine the dimension of the space, and every vector can be represented as a linear combination of these basis vectors. For instance, in a three-dimensional space (ℝ³), the basis usually consists of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1). In quantum mechanics, the orthonormal basis for a qubit is typically expressed as:
|0⟩ and |1⟩. These vectors can also be represented as:
– |0⟩ = (1, 0)
– |1⟩ = (0, 1)
2. Normalization: Each vector forming the basis must have a length (norm) of one. The norm of a vector is found by taking the square root of its inner product with itself. By combining these properties, an orthonormal basis provides a structure in which all vectors in the vector space can be expressed as a linear combination. Orthonormal bases are crucial in quantum mechanics, especially in the definition of qubits; for example, the vectors |0⟩ and |1⟩ are the orthonormal bases of a qubit. Up to this point, we have discussed that vectors are represented in the vector space ℂ². The inner product in ℂ² can be defined as:
- (z₁, z₂) ⋅ (w₁, w₂) = z₁w₁ + z₂w₂
- The inner product is an operation that measures the similarity or the relationship between two vectors. In quantum mechanics, this measurement allows us to determine the probabilities of quantum states. Additionally, the inner product provides information about the magnitude and direction of vectors. For instance, if two vectors are orthogonal (i.e., perpendicular), their inner product is zero. This represents two quantum states that do not interfere with each other.
Now, let’s discuss the concept of Orthonormal Basis a bit further. We previously defined the Orthonormal Basis as “basis” vectors in a vector space that are both perpendicular (i.e., orthogonal) and unit length. This means that vectors representing quantum states must be independent and have a unit norm (length of 1). In quantum mechanics, basis vectors are used to represent the possible states of qubits.
The simplest orthonormal basis in ℂ² consists of the following vectors:
- (1, 0), which is also expressed as the quantum state |↑⟩ (spin up).
- (0, 1), which is expressed as |↓⟩ (spin down).
These two vectors are written using ket notation, known as Dirac notation, as follows:
– |0⟩ = (1, 0)
– |1⟩ = (0, 1)
Here, |0⟩ and |1⟩ represent the two fundamental states of a qubit. These two states are the building blocks for forming superpositions of qubits.
So far, we have discussed the fundamental information related to the Orthonormal Basis. Now, let us ask: Why do we use an Orthonormal Basis? In short, to accurately express quantum states. Qubits can exist in any quantum state, and these states are expressed as combinations of the basis vectors (e.g., |0⟩ and |1⟩) in various proportions.
A qubit state can be written as:
- ψ = α|0⟩ + β|1⟩
- Here, the coefficients α and β determine the probability of the qubit being in the 0 or 1 state.
The representation of qubits defined over the complex number field ℂ² forms a fundamental structure in quantum computing. From the Stern-Gerlach experiment, it becomes apparent that qubits must be defined in a vector space where amplitudes, determining measurement outcomes, are based on inner products. The Frobenius theorem and Hamilton’s work reveal that two-dimensional spaces can only be defined over certain fields, and in the development of quantum theory, the complex number field (ℂ) was chosen. Thus, quantum bits are represented as unit vectors expressed with complex number components in superposition states, forming the foundational structure of quantum computing.