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The Impact of State Representation and Superposition in Hilbert Spaces

By Luis Gerardo Ayala Bertel posted Sat October 07, 2023 01:05 AM


Cheers esteemed IBM community,

Along this fascinating journey of quantum learning, concepts like state representation and superposition lie at the heart of understanding the behavior of quantum systems. These concepts are deeply rooted in the mathematical framework of Hilbert spaces, which provide a rigorous mathematical foundation for quantum physics. Let's check together the mathematical aspects of state representation and superposition within Hilbert spaces, using matrices and operators to explain these principles.

To start this topic it is necessary to formally define what is Hilbert space,

Definition: A Hilbert space denoted as H, is a complete inner product space over the complex numbers C equipped with an inner product ⟨·, ·⟩ that satisfies the following properties:

These properties are linearity in the first argument, conjugate symmetry and positive-definiteness.

In quantum mechanics, a state of a quantum system is represented as a normalized vector in a Hilbert space H. Let ψ be such a vector, then it is a valid state representation if ⟨ψ, ψ⟩ = 1.

Superposition is a fundamental principle in quantum mechanics, where a quantum system can exist in a linear combination of multiple states. Mathematically, if ψ1, ψ2, . . . , ψn are valid state representations, then any linear combination

is also a valid state representation, where ci are complex coefficients. For valid state representations ψ1 and ψ2, and complex coefficients c1 and c2, the linear combination

is also a valid state representation.

Example: State Representation

Let’s consider a one-dimensional Hilbert space H with a basis vector ψ1. The state representation ψ can be any normalized linear combination of ψ1, such as:

The top row represents the properties of the state ψ. The arrow labeled "N" for normalization indicates that the state ψ is normalized, which means that the sum of the squared magnitudes of its components is equal to 1. It’s saying that the inner product of ψ with itself, denoted as ⟨ψ, ψ⟩, equals 1. This is a fundamental requirement for valid quantum states. 

The bottom row represents the basis vector ψ1. This vector is an example of a possible linear combination of ψ1, where the coefficient is chosen such that the resulting satisfy the normalized state. The arrow connecting the two rows, represents the fact that the state ψ is equal to the linear combination. In other words, it shows how a specific quantum state (ψ) is constructed as a combination of basis states.

Example: Superposition

Let's consider a two-dimensional Hilbert space H with basis vectors ψ1 and ψ2. We can create a superposition state ψ as follows:

The top row represents the properties of the basis vectors ψ1 and ψ2. The arrow labeled ψ1 + ψ2 indicates that any linear combination of these basis vectors is a valid state representation. In this context, c1 and c2 are coefficients representing how much of each basis vector is used to create a state. So, the graph is highlighting that states in this space can be constructed by linear combinations of the basis vectors. The bottom row represents the specific superposition state. The term "entanglement" is often used to describe the phenomenon of superposition in multiple particles or quantum systems. In quantum mechanics, when two or more particles are in a superposition of states, their individual states become correlated in such a way that the properties of one particle are dependent on the properties of the others, even when they are separated by large distances.

This is an example of a state created by taking a linear combination of the basis vectors ψ1 and ψ2. The coefficients chosen in a way that makes the state normalized, ensuring that the inner product of this state with itself equals 1. The arrow connecting the two rows, represents the fact that this specific superposition state ψ is constructed as a linear combination of the basis vectors, as shown in the equation.

State vectors, normalized to unity, serve as our mathematical descriptions of quantum states, ensuring they adhere to the principles of quantum physics. Superposition, on the other hand, showcases the unique quantum feature where states can exist in a blend of different possibilities, characterized by linear combinations of state vectors. These mathematical concepts are fundamental to understanding quantum systems and are at the core of the rapidly advancing field of quantum computing. I hope we can keep learning along this interesting theory, feel comfortable to share any insights or comment.

Kind regards and greetings from the distance.