How To Find If Given Kets Form An Orthonormal Basis?

Learn about a vital calculation that will help you design quantum gates and algorithms.

Photograph of Max Born (1882–1970), a genius physicist who introduced the Born Rule, which interprets the wave function’s square as the probability of finding a particle in a specific state, forming the foundation of quantum mechanics. (Source)

Now that you know most of the mathematics required for Quantum Computing, here’s an important calculation you must familiarize yourself with.

This one will help you design Quantum gates and algorithms.

Let’s begin!

In case you missed the previous lessons on the mathematics required for quantum mechanics and quantum computing, here they are:

• Lesson 1: Imaginary and Complex Numbers
• Lesson 2: Trigonometry
• Lesson 3: Differential Calculus
• Lesson 4: Probability
• Lesson 5: Linear Algebra (1)
• Lesson 6: Linear Algebra (2)
• Lesson 7: Linear Algebra (3)
• Lesson 8: Linear Algebra (4)
• Lesson 9: Linear Algebra (5)
• Lesson 10: Linear Algebra (6)

We know that the state of a quantum system is represented by a ket vector in a complex Hilbert space.

We also know that this state is represented using a set of vectors called Orthonormal basis.

To revise:

The word “Orthonormal” means the following:

• “Ortho” means that the vectors are orthogonal or perpendicular to one another
• “Normal” means that the vectors are of unit length or that they have a norm of 1

For representing a quantum system in n-dimensions, the orthonormal basis will consist of a set of n unit length kets that are orthogonal to each other.

An important calculation is to determine whether a given set of kets forms an orthogonal basis.

Let’s learn how to do this.

Check If Two Ket Vectors Form An Orthonormal Basis

The problem goes like this:

Given two ket vectors |a> and |b>, we need to find out if they form an Orthonormal basis.