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Exploring the Basics in Mathematics: Vectors, Inner Products, and Hilbert Spaces

By Luis Gerardo Ayala Bertel posted Wed October 04, 2023 10:28 PM


Greetings esteemed IBM Community,

This post will explore three fundamental concepts that are essential for understanding quantum systems: vectors, inner products, and Hilbert spaces.

Understanding these mathematical constructs is important for several reasons. First and foremost, they serve as the language through which we describe and manipulate quantum states. Quantum computing, with its promise potential to revolutionize fields ranging from cryptography to materials science, relies on the precise representation and manipulation of quantum states. 

Vectors and Vector Spaces

In quantum computing, quantum states are represented as vectors in a complex vector space. These vectors obey the
properties of linearity, making them elements of a vector space. Typically, a quantum state is represented as a column
vector, like |ψ, where ψ is the state.

- Vector Addition

Vector addition in quantum computing is straightforward. Given two quantum states |ψ⟩ and |φ⟩, their sum is represented
as follows:


Here, ψ and ϕ are complex numbers representing the components of the vectors.

- Scalar Multiplication

Scalar multiplication is another fundamental operation. If c is a complex number and |ψ⟩ is a quantum state, then the
result of scalar multiplication is:

- Inner Product and Hilbert Spaces

The Inner product (or scalar product) of two quantum states |ψ⟩ and |ϕ⟩ is a crucial operation used to calculate
probabilities and measure quantum states. It’s often denoted as ⟨ψ|ϕ⟩ and is calculated as the sum of the products of the
complex conjugate of components:

Here, ∗ denotes the complex conjugate of a complex number.

Hilbert Space in quantum states reside in a complex vector space known as a Hilbert space. Has several important

  • It is a complete inner product space, meaning that limits of convergent sequences of vectors are also within the
  • It is a separable space, implying that it contains a countable dense subset (a subset with elements arbitrarily
    close to any given vector).
  • It is a vector space over the complex numbers.

In quantum mechanics, Hilbert spaces are typically infinite-dimensional, as quantum systems can have infinitely many
possible states.


Let's consider two vector states |ψ⟩ and |ϕ⟩ with the following components:

  • Vector Addition:
  • Scalar Multiplication: Let’s multiply |ψ⟩ by the scalar 2:
  •  Inner Product:

These mathematical foundations are indispensable on this great journey into the quantum frontier; Since Hilbert spaces provide a mathematical stage where quantum phenomena unfold and are rigorously described, the comprehension of these mathematical concepts is not merely an academic pursuit; it is the gateway to understand State Representation, Superposition, Entanglement, Unitary Evolution and Measurement.

Feel comfortable to share your insights and be part of this learning discussion, any comment will be appreciated.

Kind regards.