Abstract: A central problem in quantum information theory is determining how much information can be reliably transmitted through a noisy quantum channel, known as the "quantum capacity." Unlike classical channels, computing quantum capacity is notoriously difficult due to a phenomenon called non-additivity: using multiple copies of a channel together can sometimes achieve higher rates per channel copy than with a single copy of a channel. This leads to an optimization problem over an exponentially large space. In this talk, I'll introduce the quantum capacity problem and explain why additivity (or lack thereof) makes it computationally intractable. I'll then present joint work with Felix Leditzky (arXiv:2508.09978) where we exploit permutation symmetry to make progress. When the input to many channel copies is symmetric under permutations, the output can be efficiently described using the representation theory of the symmetric group. This reduces an optimization problem that scales exponentially in the number of qubits to something tractable, allowing us to evaluate the relevant figure of merit for up to 100 channel copies. Applying this to several physically motivated noise models, we obtain improved lower bounds on the quantum capacity and identify a surprisingly effective family of input states resembling repetition codes.

Abstract: In the study of dynamical systems, one comes across the Lyapunov Exponent, a quantity that characterizes the exponential rate of separation of infinitesimally close trajectories. One of the interesting applications of this is in the study of the discrete Schrodinger Equation with random potential function. In this case, the Lyapunov exponent holds information about the exponential decay of the fundamental solution of the Schrodinger Equation, with it either being positive or zero. Any zero of the Lyapunov Exponent is called critical energy.  In 1999, Germinet and De Bièvre proved that if the random potential has Bernoulli random variables, and the random variables repeat every two consecutive nodes, then there exist critical energies and they compute them. We discuss how to extend this result to any number of consecutive repetitions and compute the roots explicitly.