Before delving into quantum computation, it is crucial to understand the concept of orthogonal projection within a complex inner product space. To illustrate this concept, we shall begin with the real vector space ℝ³.

Why use ℝ³? Lets talk about it.

• The space ℝ³ allows us to represent a point or a vector in three dimensions, making mathematical concepts more tangible and comprehensible. In a three-dimensional space, it is easier to visualize how vectors are projected and how the inner product is computed.
• Concepts in ℝ³ can be generalized to more complex and abstract spaces, including complex inner product spaces.
• The physical world is often represented in three dimensions. Thus, using ℝ³ to mathematically model physical phenomena enhances realism and comprehension.

Consider two vectors defined as follows:

– (x, y, z) and (x^*, y^*, z^*)

The inner product of these vectors, which measures their relation, is defined by:

– <(x, y, z), (x^*, y^*, z^*)> = xx^* + yy^* + zz^*

Standard orthonormal basis vectors are:

– i =(1,0,0), j =(0,1,0), k = (0,0,1)

Orthonormal basis vectors are vectors in a given vector space that each have a length of 1 and are orthogonal (mutually perpendicular) to each other. These basis vectors allow any vector in the space to be expressed as a combination of them.

Here are the standard orthonormal basis vectors for three dimensional ℝ³ space:

1. i=(1,0,0)i=(1,0,0): The unit vector in the direction of the X-axis. This vector has a magnitude of 1 along the X direction and 0 in the other two components.
2. j=(0,1,0)j=(0,1,0): The unit vector in the direction of the Y-axis. This vector has a magnitude of 1 along the Y direction and 0 in the other two components.
3. k=(0,0,1)k=(0,0,1): The unit vector in the direction of the Z-axis. This vector has a magnitude of 1 along the Z direction and 0 in the other two components.

For example, the orthogonal projection of a point (x,y,z) onto the xy-plane can be expressed as (x,y,0). This means that to project this point onto the xy-plane, we draw a perpendicular line along the z-axis.

In this case, if we define the projection vector as :

• Pxy ( x , y , z )
  • ⟨i,(x,y,z)⟩i+⟨j,(x,y,z)⟩j=(x,y,0).

One point to consider here is that if we apply the projection operator twice, it still verifies the property, and the result remains unchanged. Mathematically, this can be expressed as:

– P_xy² = P_xy

A complex product space can be defined as a field where vectors are combined with an inner product. In this space, there is a basis composed of vectors that are orthogonal to each other (determined by taking averages). We have previously discussed that this basis is called an orthonormal basis. Now, let’s move a step further and discuss examples.

For instance, if we have the following basis:

– 𝜃₀, 𝜃₁, 𝜃₂

Each vector in this basis is orthogonal to the others and has unit length. If we select a few vectors from this basis, the selected vectors will form a subspace. Let’s assume we choose the vectors 𝝌₁ and 𝝌₂. These vectors form a subspace called 𝐻_x.

If we have a vector ψ, we may want to project this vector onto the chosen subspace. Projection is the process of mapping a vector onto a specific plane or space. Formulating this process results in the following expression:

𝑃_X = <𝝌₁, ψ> 𝝌₁ + <𝝌₂, ψ> 𝝌₂

Here:

• <𝝌₁, ψ> represents the inner product between 𝝌₁ and ψ, indicating how similar these two vectors are.
• The vectors 𝝌₁ and 𝝌₂ represent the projection of the vector ψ within the subspace 𝐻_x.

This projection process is significant in quantum computing and quantum mechanics because such mappings are used to measure or modify the state of a system. When a measurement is made, the state of the system is often projected onto a specific subspace, enabling the system to transition to a particular state.

Let us illustrate what we have discussed so far with an example using real numbers. Consider a point:

Let ψ = (3, 4). This point is located in a two-dimensional plane. If we wish to project this point onto an orthonormal basis composed of the vectors (1, 0) and (0, 1) (the x and y axes), the projection would proceed as follows:

1. The point is reflected onto (3, 0) along the x-axis and (0, 4) along the y-axis.
2. The resulting projection represents the point (3, 4) in the plane.

This serves as an example for understanding how vectors are represented in a complex inner product space and how they are projected onto subspaces.