QUANTUM COMPUTING FOUNDATIONS [Part 2]: Properties of the Hilbert Space (Inner and outer product)

In this article, the properties, fundamental to quantum computation only, of the Hilbert space will be illustrated and articulated.

Before disserting about inner and outer product, let's make some further consideration on the Hilbert space by making an example to help us reconvene with the content of the previous article, namely QUANTUM COMPUTING FOUNDATION [ Part 1]: Hilbert Space Characterisation.

We have spoken of the Hilbert space like a linear and Normalised space (we, let's say orthonormal) and we closed the article by noting that a quantum system is characterised as follows:

Ψ = aΨ1 + bΨ2

We also introduced a normalisation constant to enforce the normalisation of any quantum system so that it can be defined in the Hilbert space:

N = 1 / sqrt(|a|**2 + |b|**2).

Let's now make an example by considering an electron with its bases states (spin bases):

the electron spin is 1/2 (half circle) and ms takes values in the interval [-s, +s].

Let's characterise ms:

Whenever we consider the electron motion in the Hydrogen atom (or any atom, generally speaking), 2 types of motion can be identified: one which is orbital and another which is spin motion. When considering the spin, we will refer to the SPIN ANGULAR MOMENTUM of the electron. Let's represent it:

The angular momentum characterises the precession movement of the electron around the reference axis. This movement represents one of the states the electron can be found following a measurement.

Without going into great details, the precession happens in 2 ways which correspond to + 1/2 and -1/2 when referenced to the vertical axis (z for the sake of convention, even if FIGURE 2 illustrates an unlabelled vertical axis). What we are saying then is that the electron will be in precession according to two different states.

We will define, then, the +1/2 state as upspin and the -1/2 state as downspin and we weill use the following notation to refer to them:

Let's quantify these 2 states (we will not provide a formal demonstration for simplicity; these values are the results produced according to spectroscopy)

This is the same concept we can derive from an Hydrogen atom:

From FIGURE 6 we can see that the electron moves according to a spherical shape (given by the combination of the 3 orbital angular momenta). There are 3 different precession and definetely, further to a measurement the electron can be available in the following states:

| 1, 1 > , | 1, 0 >, | 1, -1 > which can also be referred as Px, Pz, Py.

This is called Vector atomic model (Not the Rutherford Model)

INNER PRODUCT AND OUTER PRODUCT

We have defined above and in QUANTUM COMPUTING FOUNDATION [ Part 1]: Hilbert Space Characterisation, what is a Hilbert space and we made two examples

Let's now ask ourselves this question:

How can we operate in the Hilbert Space?

We can do it by introducing operators. Like a scalar product and a vector product is defined for other spaces, we can define an INNER PRODUCT and an OUTER PRODUCT to operate in the Hilbert space.

Given | Ψi >, | Ψf > belonging to the Hilbert space, Let' s define the inner product.

The inner product is a scalar product which in this case returns a number in the complex field.

For example let's consider the representation below:

According to FIGURE 7, the particle is in the state where the point is: that is our initial state. Let's use a "torch light" (funny way to say "...let's make a measurement..." to find where the particle is. What the particle will do is to absorb that light and move to another state or level.

[This is simple theory: in the case of an Hydrogen atom whose ground state energy is -13.6 eV, if we send some energy (light) to detect the electron than what the electron will do is to absorb that energy and move to a higher energy state].

Therefore Initially we are on Ψi state and after the measurement we end up in Ψf state.

Question: What is the probability of finding that particle in Ψf if I make a measurement in Ψi?

The response is the inner product: the inner product is telling us the probability amplitude of finding the system in the final state once we make a measurement on the initial state.

Let's formalise in symbols the above:

Let's make an example:

Let's Define the Outer product:

The outer product is very important and its result returns a complex Matrix.

What is the meaning and significance of the projection operator? In other words what it represents?

Let's explain it by making a similarity with a space in the Real field:

Let's see how this result can be extended into the quantum field via tensor notation:

Intuitively, if in the real space the scalar product of a vector with a unitary vector gives the projection of that vector along the unitary vector axis, the same must apply to tensors as natural extension of the result. In this case, however, we are given a vector that will project any vector in its own bases (in its own space) [Note that tensors do not need a reference system. Tensors are independent from any reference system, that is why they are convenient to describe the behaviour of the universe as Universe itself cannot be described accurately with a "traditional" reference system]