4.1.5. Dirac Bra-Ket Notation and Vectors
In quantum computing, a special notation called Dirac bra-ket notation is used. A ket is represented by the symbol ∣⟩ and can be considered a column vector. For example, we define a vector d below and represent it using the ket symbol:

∣𝑑⟩ =

5 − 𝑖
−9 − 2𝑖

This is the representation of a ket vector that represents the state of a quantum system. Similarly, a bra is represented by the symbol ⟨∣ and is considered a row vector. For instance, let’s define a vector c below and express it using the bra notation:

⟨𝑐∣ = (2 + i, 5 + 3i)

In this case, a bra is the row vector obtained by taking the Hermitian conjugate of a ket.

4.1.6. Bra-Ket Hermitian Conjugate

As discussed earlier, the terms Bra and Ket were introduced. Now, we will discuss the concept of the Hermitian conjugate based on these definitions. The transformation of a Ket to a Bra is done by converting the ket into a row vector composed of the same elements, but with each element being the complex conjugate of the corresponding element.

For example:

∣𝑑⟩ =

5 − 𝑖
−9 − 2𝑖

The Hermitian conjugate of this expression is:

⟨𝑑∣ = (5 + i, −9 + 2i)

When taking the Hermitian conjugate, the complex conjugate of each complex number is taken, meaning that the real part remains the same while the imaginary part is reversed in sign. Here, a 2-element vector is formed, not a 4-element one.

4.1.7. Basic Vectors Used in Quantum Computing

In quantum computing, particularly in ℂ² (i.e., with complex numbers), the following basic vectors are commonly used:

These vectors represent the basic quantum states and play a significant role in quantum computing. These states are often used to represent the two states (0 and 1) of a computer.