# Lecture 3 | Quantum Entanglements, Part 1 (Stanford) | Summary and Q&A

## Summary

In this video, the lecturer explains the concept of complex numbers and their relation to exponential functions. He introduces the concept of a complex number having a real part and an imaginary part and how it can be represented on a two-dimensional plane. Additionally, he discusses the notion of a unitary number and its significance in complex number calculations. The lecturer also covers the basics of quantum mechanics, including states, observables, and the importance of hermitian matrices. Finally, he explains the concept of eigenvalues and eigenvectors and their relevance to quantum mechanics.

## Questions & Answers

### Q: What is a complex number and how is it represented on a two-dimensional plane?

A complex number is a number that has a real part and an imaginary part. It can be represented on a two-dimensional plane, where the real part is represented on the horizontal axis and the imaginary part is represented on the vertical axis. Each point on the plane represents a complex number.

### Q: What is the significance of a unitary number in complex number calculations?

A unitary number is a complex number with a modulus of 1, meaning its distance from the origin of the complex plane is equal to 1. These numbers have the property that when they are multiplied by their complex conjugate (the same number with the imaginary part changing sign), the result is 1. Unitary numbers play a role in representing certain types of complex numbers and have significance in quantum mechanics.

### Q: What are some properties of the exponential function e to the I theta?

The combination of the exponential function e to the I theta, where theta is an angle, satisfies certain rules of exponentials. For example, when two angles theta and phi are multiplied as e to the I theta times e to the I phi, the result is a complex number with a real and an imaginary part. The real part is given by cosine theta times cosine phi minus sine theta times sine phi, and the imaginary part is given by cosine theta times sine phi plus cosine phi times sine theta. This combination of cosine and sine can simplify trigonometric calculations.

### Q: What is the relationship between the exponential function and complex conjugation?

The exponential function e to the I theta and its complex conjugate (e to the -I theta) have a special relationship. When these two numbers are multiplied together, the result is 1. This relationship is due to the fact that multiplying e to the I theta and e to the -I theta is equivalent to rotating a complex number counterclockwise and then clockwise, resulting in a net rotation of zero. This property is useful in various calculations involving complex numbers.

### Q: What is the definition of a hermitian matrix?

A hermitian matrix is a square matrix that is equal to its own hermitian conjugate. The hermitian conjugate of a matrix is obtained by transposing the matrix and then taking the complex conjugate of each element. For a hermitian matrix, the diagonal elements are real numbers, and the off-diagonal elements are complex conjugates of each other. This property has significance in quantum mechanics, where hermitian matrices represent observables.

### Q: What are eigenvalues and eigenvectors of a matrix?

Eigenvalues and eigenvectors are associated with matrices. An eigenvalue is a scalar number that is associated with a certain eigenvector. An eigenvector is a vector that, when multiplied by a matrix, yields the original vector multiplied by the eigenvalue. In other words, the matrix acts on the eigenvector as a scalar multiple. In the context of quantum mechanics, eigenvalues represent the possible measurable values of an observable, and eigenvectors represent the states in which those measurements yield a specific eigenvalue.

### Q: How can we find eigenvectors and eigenvalues of a matrix?

To determine the eigenvectors and eigenvalues of a matrix, one must solve the equation M times V equals lambda times V, where M is the matrix, V is the eigenvector, and lambda is the eigenvalue. This equation can be solved by finding the values of lambda for which the determinant of the matrix minus lambda times the identity matrix is equal to zero. The corresponding eigenvectors can then be found by plugging the eigenvalues back into the original equation and solving for V.

### Q: What are some properties of hermitian matrices?

A hermitian matrix has several important properties. One property is that its eigenvalues are always real numbers. Additionally, the matrix squared is equal to the identity matrix. This means that the eigenvalues, when squared, are equal to 1. Another property is that the eigenvectors corresponding to different eigenvalues are orthogonal to each other. These properties have significance in quantum mechanics, where hermitian matrices represent observables and their eigenvalues represent measurable quantities.

### Q: What are the square matrices Sigma 1 and Sigma 3, and what are their eigenvalues?

Sigma 1 and Sigma 3 are special matrices known as Pauli matrices, which are commonly used in quantum mechanics. Sigma 1 is equal to [1, 1; 0, 0], and Sigma 3 is equal to [1, 0; 0, -1]. The eigenvalues of Sigma 1 are +1 and -1, while the eigenvalues of Sigma 3 are also +1 and -1. Additionally, the square of both Sigma 1 and Sigma 3 is equal to the identity matrix [1, 0; 0, 1]. These matrices have significance in representing observable quantities and measurements in quantum mechanics.

### Q: What is the relationship between the eigenvectors of Sigma 1 and Sigma 3?

The eigenvectors of Sigma 1 and Sigma 3 are related to each other. The eigenvectors of Sigma 1 are [1, 0] (corresponding to eigenvalue +1) and [0, 1] (corresponding to eigenvalue -1). The eigenvectors of Sigma 3 are [1, 0] (corresponding to eigenvalue +1) and [0, -1] (corresponding to eigenvalue -1). Notice that the second eigenvector of Sigma 3 is equal to the negative of the second eigenvector of Sigma 1. This relationship between eigenvectors can have implications in quantum mechanics and the measurements of observable quantities.

## Takeaways

Complex numbers play a crucial role in quantum mechanics, particularly in representing and calculating probabilities of measurements. Hermitian matrices are of special importance in quantum mechanics as they represent observable quantities, with their eigenvalues corresponding to the possible measurable values. Eigenvectors associated with eigenvalues represent states where the measured value of the observable is certain. Additionally, the square of hermitian matrices and their eigenvalues often have notable properties, such as equaling the identity matrix or being equal to +1 or -1. Understanding concepts such as complex numbers, hermitian matrices, and eigenvalues is fundamental to gaining insight into the workings of quantum mechanics.