Statistical mechanics and quantum entanglement

Hi everyone!

Jonathon Riddell here. Today we take a closer look at how entanglement entropy plays a role in the emergence of statistical mechanics. We find that the entanglement entropy between a part of a system obeying eigenstate thermalization, and the rest of the system, scales like the size of the subsystem. This is called a "volume law" and is a key indicator that an energy eigenstate may obey the Eigenstate Thermalization Hypothesis.

First video on ETH:

Density matrices:

Recommended textbooks:
Quantum mechanics:
Statistical mechanics:
Quantum information:

00:00 Intro and announcements
02:30 Constructing our Hilbert space
08:24 Density matrix essentials
13:17 Reviewing ETH prediction
15:30 Entanglement for pure states
18:23 Von Neumann entropy and scaling
21:51 Entanglement entropy volume law

[ahsanhayat]

Thanks dude, You literally explain almost all the topics that appeared to me as a next question while listening to you

[frankconley]

Great video. So glad you worked so hard and now you can enlighten others. Unfortunately for me you used more terms i have never heard in my life in 24 minutes than any 24 minutes of my life.

[kshitisneh]

Hi, thanks for the awesome video! I have a question regarding the entropy part: the entropy of the complete microcanonical ensemble is following the volume law, but on the other hand, the entropy of the pure system on the left hand side is zero right since it's pure? How do the two situations then relate? Maybe I am missing something fundamental here..

[nathanborak]

At the end you said volume law entanglement entropy is a way of seeing if ETH is satisfied. Question: if you have enough information to check volume law entanglement entropy, then surely you have enough info to just check ETH directly by computing matrix elements, right? I don't see the point of using volume law entanglement entropy as a way of seeing if ETH holds in a system, when it seems no easier to show than ETH directly.

Also, isn't the point of all this to explain thermalization? Why are we trying to show ETH specifically when the volume law entanglement already seems sufficient to explain thermalization.