**Remarks on hypercontractivity**** for quantum channels**

*Chris King*

Hypercontractivity for matrix algebras has been used to obtain new results in quantum information theory, including spectral bounds for local Hamiltonians, improved estimates for convergence rates of quantum channel semigroups, and constraints on local state transformations. After a review of the background and recent applications, some open problems and their connections to multiplicativity questions will be discussed.

**Quantum Information in Statistical Mechanics and Thermodynamics**

*Fernando Brandao*

I will give an overview of applications of ideas and techniques from quantum information science in quantum statistical mechanics and thermodynamics. In particular I will discuss the use of quantum information theoretic ideas in understanding equilibration and thermalization of quantum many-body systems, as well as in developing a framework for thermodynamics in the nanoscale.

**Quantum information
as asymptotic geometric analysis**

*Patrick **Hayden*

Quantum states are
represented as vectors in an inner product space. Because the dimension of
that state space grows exponentially with the number of its constituents,
quantum information theory is in large part the asymptotic theory of finite
dimensional inner product spaces, a field with its own long history. I’ll
highlight some examples of how abstract mathematical results from that area,
such Dvoretzky’s theorem, manifest themselves in quantum information theory as
improvements in quantum teleportation and as the raw material for
counterexamples to the field’s famous additivity conjecture. More recently,
this perspective has led to methods for encrypting arbitrarily long messages
using constant-sized secret keys.

**Do we need random states and random channels? **

*Stanislaw Szarek*

In recent years, some
important results in quantum information involved random constructions,
arguably the most noteworthy being the Hastings-Hayden-Winter
counterexamples to additivity conjectures. Such arguments bring up two
questions. First, can we come up with explicit objects solving the
problems at hand? Second, are the random models that lead to such constructions
physically realistic? The first question is primarily mathematical, and it
resembles similar problems studied in computer science and geometric functional
analysis that are relevant - for example - to compressed sensing.

**Hamiltonian simulation with nearly optimal dependence on all parameters**

*Andrew Childs*

We present an
algorithm for sparse Hamiltonian simulation that has optimal dependence on all parameters
of interest (up to log factors). Previous algorithms had optimal or
near-optimal scaling in some parameters at the cost of poor scaling in others.
Hamiltonian simulation via a quantum walk has optimal dependence on the
sparsity d at the expense of poor scaling in the allowed error epsilon. In contrast,
an approach based on fractional-query simulation provides optimal scaling in epsilon at the expense of poor scaling in d. Here we combine the two approaches,
achieving the best features of both. By implementing a linear combination of
quantum walk steps with coefficients given by Besselfunctions, our algorithm
achieves near-linear scaling in tau := d ||H||_{max} t and sublogarithmic scaling
in 1/epsilon. Our dependence on epsilon is optimal, and we prove a new lower bound showing
that no algorithm can have sublinear dependence on tau. [Joint work with Dominic
Berry and Robin Kothari.]

**Quantum LDPC
codes and homology theory**

*Sergey Bravyi*

In this talk I will
review some of the recent developments and open problems concerning quantum
LDPC codes that are of relevance for quantum fault-tolerance, Hamiltonian
complexity and the theory of topological quantum order. I will argue that
several interesting open problems concerning quantum LDPC codes are connected
to deep mathematical questions arising in the Z_2 homology theory of simplicial
complexes and Riemannian manifolds. I will discuss two examples of such
problems: (1) characterizing tradeoffs between the encoding rate and the
minimum distance of quantum LDPC codes, and (2) constructing quantum locally
testable codes. The latter are closely related to the so-called high dimensional
expanders that have been actively studied recently.

**Semidefinite hierarchies in quantum information**

*Pablo Parrilo*

In the last 15 years, semidefinite programming (SDP) has proved to be a very powerful technique for many problems in quantum information, including among others entanglement detection, quantum coin flipping, query complexity, and nonlocal games. In particular, methods based on sum of squares decompositions have provided hierarchies of SDP relaxations for optimization over separable states and the quantum correlation/moment problem. In this talk we survey these methods, highlight some of the associated mathematical and computational challenges, and outline directions for possible future research.

**Quantum hypothesis testing
and full counting statistics in open quantum systems**

*Vojkan Jaksic*

The mathematical
theory of non-equilibrium quantum statistical mechanics has developed rapidly in recent years.
The current research efforts are centered around the theory of entropic fluctuations.

Since Shannon's
rediscovery of Gibbs-Boltzmann entropy there has been a close interplay between information theory and
statistical mechanics. One of the deepest links is provided by the theory of large deviations.
In this context, it is natural to try to connect recent results in non-equilibrium statistical
mechanics with recent developments in quantum information theory.

In this
talk I will comment on the following link. Consider the large deviation
principle for the full
counting statistics for the repeated quantum measurement of the energy/entropy flow over
the time interval [0; t] in an open quantum system. Let I(theta) be the rate
function and e(s)
its Legendre transform. Let e'(s) be the Chernoff error exponent in the quantum hypothesis
testing of the arrow of time. Then e(s) = e'(s).

The principal goal
of the presentation is point to a surprising (and I believe fundamental) link
between two directions of research which are largely unaware of each other.