**Copyright Information:**All rights reserved to Prof. Leonard Susskind, Stanford University.

### Lecture Description

This lecture takes a deeper look at entanglement. Professor Susskind begins by discussing the wave function, which is the inner product of the system's state vector with the set of basis vectors, and how it contains probability amplitudes for the various states. He relates these probability amplitudes to the expectation values of observables discussed in previous lectures. He then examines more deeply the difference between product and entangled states. For product states, the wave function factorises which allows the two (or more) sub-systems to be treated as independent systems. He also describes the properties of a maximally entangled two spin system, and introduces the concept of density matrices, which express everything we can know about one part of an entangled system. Professor Susskind then moves on to discuss measurement versus entanglement. There are two views of measurement: one in which the measuring apparatus becomes entangled with the system under measurement, and the other in which the wave function of the system under measurement collapses when measured. He then discusses locality beginning with Einstein's famously skeptical phrase "spooky actions at a distance." He distinguishes between actual instantaneous action at a distance - which is impossible - and simple correlation. What is strange about quantum mechanics is not correlation in entangled states, but rather that we can know everything about this system as a whole, without knowing anything about the individual states of the entangled elements. Professor Susskind concludes the lecture by revisiting the example of the computer simulation from the last lecture, which is an example of Bell's theorem that local hidden variables are not sufficient to explain quantum mechanics. Topics: - Quantum wave function - Product vs. entangled states - Singlet state - Maximum entanglement - Density matrices - Measurement - Locality - Spooky action at a distance - Computer simulation of product and entangled states - Bell's theorem - Hidden variables Recorded on February 20, 2012.

### Course Index

- Introduction to Quantum Mechanics
- The Basic Logic of Quantum Mechanics
- Vector Spaces and Operators
- Time Evolution of a Quantum System
- Heisenberg Uncertainty Principle & The Schrödinger Equation
- Entanglement: Entangled, Singlet, & Triplet States
- Entanglement and the Nature of Reality
- Particles Moving in One Dimension and their Operators
- Fourier Analysis applied to Quantum Mechanics
- The Uncertainty Principle and Classical Analogs

### Course Description

Quantum theory governs the universe at its most basic level. In the first half of the 20th century physics was turned on its head by the radical discoveries of Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, and Erwin Schroedinger. An entire new logical and mathematical foundation—quantum mechanics—eventually replaced classical physics. We will explore the quantum world, including the particle theory of light, the Heisenberg Uncertainty Principle, and the Schrödinger Equation. This course is second-part of a six course sequence given by Prof. Leonard Susskind that explores the theoretical foundations of modern physics - the Theoretical Minimum. Topics in the series include classical mechanics, quantum mechanics, theories of relativity, electromagnetism, cosmology, and black holes.