Lecture 4 | Quantum Entanglements, Part 1 (Stanford)

The speaker begins by introducing the topic of quantum entanglement, which involves the peculiar correlations between particles like electrons. They mention that Bell's theorem and the concept of entanglement will be discussed later, emphasizing the complexity of understanding quantum systems with more than one part. The focus is on the simplest quantum system: the spin of a single electron, and the challenge of measuring its spin along an arbitrary axis.

The paragraph delves into the representation of physical quantities like magnetic moments or spin vectors as matrices. The speaker explains how these matrices operate on vectors, leading to measurable outcomes. The concept of hermitian matrices is introduced, which have real eigenvalues and is crucial for quantum observables. The speaker also discusses the properties of sigma matrices and their role in representing spin measurements.

The discussion shifts to the probabilities involved in measuring the spin of a particle along different axes. The speaker explores the question of finding the probability of measuring a certain spin component along one axis given that the particle was prepared along a different axis. The importance of rotational symmetry in quantum mechanics is highlighted, leading to the conclusion that the probability depends only on the angle between the two axes.

The speaker outlines the historical development of quantum mechanics, starting with Einstein's recognition of probabilities in quantum theory, through the contributions of Born, Heisenberg, and Dirac. The paragraph emphasizes that quantum mechanics is based on experimental facts and a calculus of probabilities. The process of calculating probability amplitudes and their squares to find probabilities is explained, which is central to quantum mechanics.

The focus is on the process of quantum measurement and how it affects the state of a system. The speaker explains that measuring a quantum system, such as an electron's spin, changes its state. The concept of eigenstates and eigenvalues is introduced, and the importance of the inner product in quantum mechanics is discussed. The speaker also touches on the normalization of quantum states.

The speaker explores the idea that quantum mechanics allows for probabilities rather than certainties. They discuss the case where two measurement axes are perpendicular and the expected probability outcomes. The concept of a quantum state being a filter is introduced, and the impossibility of simultaneously measuring two non-commuting observables, like Sigma 1 and Sigma 2, is explained.

The discussion moves to the commutator, a mathematical tool that determines whether two observables can be simultaneously measured. The speaker presents a theorem stating that if two matrices commute, they share the same set of eigenvectors, and thus their corresponding observables can be measured at the same time. The concept of entangled states is introduced, where the state of one particle cannot be described independently of the other.

The speaker discusses the geometric interpretation of quantum states, emphasizing that any quantum state can be expressed as an eigenvector of some sigma matrix. This is demonstrated through counting equations and variables, showing that for any given state, there is always a direction in space where the electron's spin is definitely up or down. The concept of degrees of freedom in quantum states is explored.

The speaker explores the concept of entanglement further by considering a system of two electrons. They explain how the state of two electrons can be described in a four-dimensional vector space and how the spin operators for each electron act on these states. The importance of distinguishing between product states, where each electron has its own independent state, and entangled states, where the electrons are linked in a way that the state of one cannot be described without the other, is highlighted.

The speaker describes how to create an entangled state of two electrons where the spin of one electron is always opposite to that of the other, regardless of the direction of measurement. This state is known as the singlet state. The process involves bringing two electrons close enough for their magnetic fields to interact, leading to a state where the electrons' spins are anti-aligned. The speaker also touches on the philosophical implications of entanglement and the difference between classical and quantum correlations.