Quantum Information

In unidirectional communication theory, two of the most prominent problems are those of compressing a source of information and of transmitting data noiselessly over a noisy channel. In 1948, Shannon introduced information theory as a tool to address both of these problems. Since then, information theory has flourished into an important field of its own. It has also been successfully extended to the quantum setting, where it has also served to address questions about quantum source compression and transmission of classical and quantum data over noisy quantum channels.

However, in interactive communication theory, more specifically communication complexity, it is much more recently that tools from information theory have been successfully applied. Indeed, the interactive nature of communication protocols in this setting imposes new constraints and tools specific to this setting need to be developed, both for the interactive analogue of source compression and that of coding for noisy channels. The exciting field of classical interactive information theory has been very active in recent years.

We discuss recent works for its quantum counterpart. In particular, we discuss joint work showing that a constant factor overhead is sufficient for robustly implementing interactive quantum communication over noisy channels [1]. We also discuss work introducing a new notion of quantum information complexity that exactly captures the amortized cost per copy for implementing many copies of a communication task in parallel, such that compressing to this information complexity leads to a bounded-round direct sum theorem [2].

For both of these, we further discuss many interesting potential research directions that follow.

[1] joint work with Gilles Brassard, Ashwin Nayak, Alain Tapp, Falk Unger, QIP’14, FOCS’14 [2] Merge of arXiv:1404.3733 and arXiv:1409.4391, to appear at QIP’15

Quantum many-body systems ranging from a many-electron atom to a solid material are described by effective Hamiltonians which are obtained from more accurate Hamiltonians by neglecting or treating weak interactions perturbatively. Quantum complexity theory asks about the quantum computational power of such quantum many-body models for both practical as well as fundamental purposes. Three distinct computational classes have emerged within this framework: namely (1) classical Hamiltonians such as the Ising model, (2) sign-free or stoquastic Hamiltonians such as the transverse field Ising model, and (3) fully quantum Hamiltonians such as the Heisenberg model. Each class can be characterized by certain prototype universal Hamiltonians which can encode the physics of any other Hamiltonian in that class. We will show how this encoding is established through the use of perturbation theory via perturbative gadgets. We will discuss the technical expression of this classification in terms of the complexity classes NP, Stoquastic MA and QMA and the power of these Hamiltonians for performing quantum adiabatic computation.

Quantum information theory has taught us that quantum theory is just one possible probabilistic theory among many others. In the talk, I will argue that this "bird's-eye" perspective does not only allow us to derive the quantum formalism from simple physical principles, but also reveals surprising connections between the structures of spacetime and probability which can be phrased as mathematical theorems about information-theoretic scenarios.