### Fall 2019: PHYS 57000(A) “Quantum Information and Geometry”

Tue-Thu 4:45-6:00 pm

Room 338

**Prerequisites**: Quantum Mechanics and/or Operator Algebra, Statistical Mechanics, Classical Mechanics. Knowledge of Quantum Field Theory is not required; however, familiarity with the basics of General Relativity and Riemannian geometry will be assumed.

**Course description: **This course is an introduction to information-theoretic methods in fundamental physics. It is aimed at graduate students and advanced undergraduate students in Physics and Mathematics. I strongly encourage interested Mathematics students with knowledge of operator algebras to participate.

Topics that we will cover:

**1) Entanglement Theory and General Quantum Systems**

- Lecture 1:
**A review of the axioms of quantum mechanics** - Lecture 2:
**A review of the density matrix, Purification, Entanglement**(pre-class reading John Preskill’s lecture notes: Foundations: States and Ensembles Sections 2 to 2.5) - Lecture 3:
**Classical information, compression, typicality, Shannon entropy, quantum information and von Neumann entropy**(pre-class reading Mark Wilde’s “From Classical to Quantum Shannon Theory” Sections 2 to 2.1.3, 10 to 10.3 and 11 to 11.2) - Lecture 4:
**Properties of entropy, Conditional entropy, Mutual information, and Jaynes maximum entropy principle**(pre-class reading Mark Wilde’s “From Classical to Quantum Shannon Theory” 10.4 to 10.9 and 11.3 to 11.9) - Lecture 5:
**Distance measures, Trace distance, Fidelity**(pre-class reading Section 9 of Mark Wilde’s “From Classical to Quantum Shannon Theory”) - Lecture 6:
**Distinguishability, Relative entropy and free energy, Quantum operations and monotonicity**(pre-class reading Vedral’s “The role of relative entropy in quantum information theory” Sections I and II) - Lecture 7:
**Separable Hilbert space, Equivalence of p****rojections, Factors of type I and General quantum system**(pre-class reading Daniel Harlow’s “The Ryu-Takayangi formula from Quantum Error Correction” Appendix A) - Lecture 8:
**Infinite tensor products, trace,****Factors of type II and III**(pre-class reading Edward Witten’s “Notes On Some Entanglement Properties of Quantum Field Theory” Sections 6 to 6.5)

**2) Tensor Networks, Many-body Quantum Systems and Geometry**

- Lecture 1:
**Tensor diagrams**(pre-class reading “Tensor networks and graphical calculus for open quantum systems” Section II) - Lecture 2:
**Thermodynamic limit, Mean-field theory, and Quantum Circuits**(pre-class reading John McGreevy’s Lectures on “Quantum Information is Physical” Section 1) - Lecture 3:
**Kadanoff spin blocking and Renormalization**(pre-class reading Wilson and Kogut’s review “The Renormalization Group and the epsilon expansion” Sections 1 to 3) - Lecture 4:
**Compressing quantum states, Wilson’s numerical renormalization and Matrix Product States (MPS)**(pre-class reading Verstraete, Murg and Cirac’s “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems” Sections 1 to 4.3; extra reading: John McGreevy’s Lectures on “Quantum Information is Physical” section 7; talk by Guiffre Vidal on Tensor Networks from It-from-Qubit summer School at PI 2019) - Lecture 5:
**Projected Entangled Pair States (PEPS) and coarse-graining of gapless 1d systems** - Lecture 6:
**Multi-scale Entanglement Renormalization Ansatz (MERA)****and Geometry (**Brian Swingle’s

**3) Quantum Fields and Entanglement**

- Lecture 1:
**General quantum system as non-commutative probability theory**(Redei and Summers “Quantum Probability Theory” up to Section 7) - Lecture 2:
**Thermodynamic limit, superselection sectors and Modular theory** - Lecture 3:
**Relativistic quantum mechanics, Non-relativistic quantum fields**( Mark Srednicki “The Quantum Field Theory” Sections 1 and 2) - Lecture 4:
**Relativistic quantum fields**(Edward Witten’s “Notes On Some Entanglement Properties of Quantum Field Theory” Section 5) - Lecture 5:
**Relative entropy and monotonicity**(Edward Witten’s “Notes On Some Entanglement Properties of Quantum Field Theory” Section 4.3 and Nielsen and Petz “Simple Proof of the Strong Subadditivity Inequality”) - Lecture 6:
**Modular operator in Rindler space**(Edward Witten’s “Notes On Some Entanglement Properties of Quantum Field Theory” Sections 5)

**4) Basics of Conformal Field Theory**

- Lecture 1:
**Quantum field theory and Euclidean path-integrals**( Mark Srednicki “The Quantum Field Theory” Sections 6, 7, 8, 9) - Lecture 1:
**Scale-invariance, conformal transformations, operator product expansion**(Paul Ginsparg’s “Applied conformal field theory” up to section 2.3) - Lecture 3:
**Massless free fields as conformal field theory**(Paul Ginsparg’s “Applied conformal field theory” Sections 2.3 and 2.4, Robert Dijgraph’s talk on “Introduction to topological and conformal field theory”) - Lecture 4:
**Entanglement in conformal field theory**(Calabrese and Cardy “Entanglement entropy and conformal field theory” Up to section 4) - Lecture 5:
**Conformal field theories and Anti-de Sitter space**(“Large N field theories, string theory and gravity” Section 2)

**5) Towards the Emergence of Spacetime and Holography**

- Lecture 1:
**Blackhole thermodynamics** - Lecture 2:
**AdS/CFT basics 1** - Lecture 3:
**AdS/CFT basics 2** - Lecture 4:
**Eternal black holes, multiple boundaries** - Lecture 5:
**Holographic entanglement entropy** - Lecture 6:
**Emergence of space, ER=EPR**

If you have suggestions/comments regarding the course please do not hesitate to email me.

To keep the course interactive and the students can play a more active role in choosing the focus of the course, we will use the application Slack to post course announcements, reports, discuss ideas and share interesting papers we have come across. If you are planning to attend the class please send me an email and I will add you to the course channel on Slack.

Some books on the basics of quantum information theory:

- “Quantum Information and Quantum Statistics” Denes Petz
- “Quantum Computation and Quantum Information” Nielsen, Chuang
- “Alice and Bob Meet Banach” Auburn, Szarek
- “From Classical to Quantum Shannon Theory” Mark Wilde

Some references on quantum many-body quantum systems and tensor networks:

- “Quantum Information Meets Quantum Matter” Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen

Some references on von Neumann algebras and Modular Theory:

- “von Neumann algebras” Vaughn Jones
- “Operator Algebras and Quantum Statistical Mechanics 1” Bratelli and Robinson
- “Theory of Operator Algebras I, II, III” Takesaki

Some references on local algebras of quantum fields:

- “Local Quantum Physics” Haag
- “Mathematical Theory of Quantum Fields” Araki
- “On revolutionizing quantum field theory with Tomita’s modular theory” Borchers

Some references on conformal field theory:

- “TASI Lectures on conformal boostrap” David Simmons-Duffin
- “Conformal field theory” Di Francesco, Matheiu, Senechal
- “Applied conformal field theory” Paul Ginsparg
- “EPFL Lectures on conformal field theory in D>=3” Salva Rychkov