All The Math That You Need To Start Doing Quantum Computing (Part 9)

Lesson 9: Linear Algebra (5) - Matrices With Bra & Kets

Photograph of Wolfgang Pauli, a theoretical physicist who made fundamental contributions to quantum mechanics, including the discovery of the Pauli exclusion principle that earned him the Nobel Prize in Physics in 1945. (Source)

In this lesson, we learn about matrices composed of bras and kets, and learn to perform operations on them.

You should read this chapter with Lesson 7: Linear Algebra (3) to get the most out of it.

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)

Matrices As Bras & Kets

To revise,

A ket ∣v⟩ represents a column vector.

A bra represents a row vector.

The bra <v| is the Hermitian conjugate (transpose and complex conjugate) of the ket |v>.

Matrices can be seen as:

• multiple bras stacked on top of each other

• multiple kets stacked side by side

Matrix A as shown below:

It can be visualised as a combination of its bras as follows:

And it can be written as:

Similarly, it can be visualised as a combination of its kets as follows:

And it can be written as:

---

Matrix Addition and Subtraction

For two matrices A and B:

They can be broken down into bras as follows: