Talks

Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. SU(2)), the bipartite entanglement of formation for the MMIS scales logarithmically ∼logN with the number of qudits N.

The description of a quantum many-body system lacking quasiparticles is a longstanding challenge at the forefront of physics. A solvable example is provided by the Sachdve-Ye-Kitaev (SYK) model, which has attracted much attention due to its intriguing connections to black holes and intertwined questions on non-Fermi liquid metals, thermalization, and many-body quantum chaos. After a brief review of the phenomenology of Fermi (FL) and non-Fermi (NFL) liquids, I will give a pedagogical introduction to the large-N saddle-point method, spectral and thermodynamics properties, and the computation of the out-of-time-ordered correlation and Lyapunov exponent in the SYK model. I will then discuss a few generalizations/extensions of the SYK model with NFL-FL and other dynamical transitions. I will conclude with a discussion of an experimental realization of the SYK model in condensed matter systems. 

We will discuss connections between quantum relative entropy and other distance measures between probability distributions and quantum states. We will discuss the connection to hypothesis testing. If there is time, we will present applications of classical and quantum information theory to problems in combinatorics and computer science.

We will present the basic inequalities concerning the classical entropic quantities introduced in the first lecture. We will then proceed to the quantum analogue of Shannon entropy. We will first briefly review pure and mixed states, and their evolution. We will then present Von Neumann entropy, and other related quantum mechanical entropic quantities, study their properties, and present the basic inequalities.

We will see that Shannon entropy arises naturally from the problem of efficiently representing the outcome of a probabilistic experiment as a string of bits. We will review entropic quantities such as relative entropy, conditional entropy and mutual information, and relate them to notions such as the rate of growth of the number of typical sequences and asymptotic limits on the rate of information transmission across a noisy channel.