Two Brandeis professors have been awarded highly prestigious and competitive Simons Fellowships for 2017. Daniel Ruberman received a 2017 Simons Fellowship in Mathematics. Matthew Headrick was awarded a 2017 Simons Fellowship in Theoretical Physics. This is the first of two articles where each recipient’s award-winning research is described.

Matthew Headrick’s research studies the phenomenon of entanglement in certain quantum systems and its connection to the geometry of spacetime in general relativity. This very active area of research is the culmination of three developments in theoretical physics over the past 20 years.

First, in 1997, string theorists discovered that certain quantum systems involving a large number of very strongly interacting constituents — whose analysis would normally be intractable — are secretly equivalent to general relativity — a classical theory describing gravity in terms of curved spacetime — in a space with an extra dimension. For example, if the quantum system has two dimensions of space, then the general relativity has three; the phenomenon is thus naturally dubbed “holography”.

This equivalence between two very different-looking theories is incredibly powerful, and has led to much progress in understanding both strongly-interacting quantum systems and general relativity. However, it is still not fully understood how or precisely under what conditions such an equivalence holds.

Meanwhile, around 15 years ago, theorists began studying entanglement as a way to understand the behavior of quantum systems. “Entanglement” refers to correlations between two parts of a quantum system that occurs at the level of the wave function (unlike more familiar classical, or statistical, correlations). Entanglement is responsible for much of the apparent weirdness of quantum mechanics, as well as the power of such potential technologies as quantum cryptography and quantum computers. Although the concept of entanglement has been recognized almost since the birth of quantum mechanics, only recently has it been understood how quantifying entanglement (using certain kinds of entropies) provides powerful insights into the behavior of quantum systems ranging from many-body condensed-matter systems to theories of particle physics.

These two developments came together in 2006, when two physicists, Ryu and Takayanagi — one a condensed-matter theorist and the other a string theorist — made a dramatic conjecture connecting them. They posited that, in a holographic system, the entanglement is directly related to the geometry of space on the general relativity side; more specifically, given a decomposition of the quantum system into two parts, the amount of entanglement is given by the area of a certain minimal surface on the general relativity side. This development has led to many advances in our understanding of entanglement in strongly-interacting systems, and has provided a new framework for thinking about the emergence of space out of quantum constituents.

Since the Ryu-Takayanagi conjecture was originally made, Headrick has been a leader in testing, generalizing, and applying it. Working with Michael Freedman (Microsoft Research), he recently discovered a new way to understand entanglement in holographic systems in terms of microscopic “bit threads” running through space; using the so-called max flow-min cut theorem from network theory, they showed that this picture precisly reproduces the predictions of the Ryu-Takayanagi formula. With his students and other collaborators, Headrick is currently building on this picture. A major focus, for example, is to understand the subtle role that time plays in entanglement and holography.

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