Colloquium

There are two notions that play a central role in the mathematical theory of computation. One is that of a computable problem, i.e., of a problem that can, in principle, be solved by an (idealized) computer. It is known that there exist problems that \'have answers\', but for which those answers are not computable. The other is that of the difficulty of a computation, i.e. of the number of (idealized) steps required actually to carry out that computation. It is known that, given any appropriate \'degree of difficulty\', there exists a problem that, while computable, is at least that difficult. These two notions, while purely mathematical, are designed to reflect, in some broad sense, the physics of the computation process. But there are indications that physics may have something further to say about them. Indeed, it has been suggested that, by using general relativity, some problems that are (mathematically) non-computable may become computable; and that, by using quantum mechanics, some problems that are (mathematically) difficult may become less so. Are there, in principle, any limitations on what physics can do for us in this area?

At a very basic level, physics is about what we can say about propositions like \'A has a value in S\' (or \'A is in S\' for short), where A is some physical quantity like energy, position, momentum etc. of a physical system, and S is some subset of the real line. In classical physics, given a state of the system, every proposition of the form \'A is in S\' is either true or false, and thus classical physics is realist in the sense that there is a \'way things are\'. In contrast to that, quantum theory only delivers a probability of \'A is in S\' being true. The usual instrumentalist interpretation of the formalism leading to these probabilities involves an external observer, measurements etc.In a future theory of quantum gravity/cosmology, we will have to treat the whole universe as a quantum system, which renders instrumentalism meaningless, since there is no external observer. Moreover, space-time presumably does not have a smooth continuum structure at small scales, and possibly physical quantities will take their values in some other mathematical structure than the real numbers, which are the \'mathematical continuum\'. In my talk, I will show how the use of topos theory, which is a branch of category theory, may help to formulate physical theories in a way that (a) is neo-realist in the sense that all propositions \'A is in S\' do have truth values and (b) does not depend fundamentally on the continuum in the form of the real numbers. After introducing topoi and their internal logic, I will identify suitable topoi for classical and quantum physics and show which structures within these topoi are of physical significance. This is still very far from a theory of quantum gravity, but it can already shed some light on ordinary quantum theory, since we avoid the usual instrumentalism. Moreover, the formalism is general enough to allow for major generalisations. I will conclude with some more general remarks on related developments.

It is common to assert that the discovery of quantum theory overthrew our classical conception of nature. But what, precisely, was overthrown? Providing a rigorous answer to this question is of practical concern, as it helps to identify quantum technologies that outperform their classical counterparts, and of significance for modern physics, where progress may be slowed by poor physical intuitions and where the ability to apply quantum theory in a new realm or to move beyond quantum theory necessitates a deep understanding of the principles upon which it is based. In this talk, I demonstrate that a large part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over classical states that can be prepared. This restriction implies a fundamental limit on the amount of knowledge that any observer can have about the classical state. I will also discuss the quantum phenomena that are not captured by this principle, and I will end with a few speculations on what conceptual innovations might underlie the latter set and what might be the origin of the statistical restriction.

The history of human knowledge is often highlighted by our efforts to explore beyond our apparent horizon. In this talk, I will describe how this challenge has now evolved into our quest to understand the physics at/beyond the cosmological horizon, some twenty orders of magnitude above Columbus\' original goal. I then recount how the study of physics on the horizon scale has led to the successful development of inflationary cosmology, and how we can use the Integrated Sachs-Wolfe effect in the Cosmic Microwave Background to probe cosmological physics, such as late-time inflation, the nature of gravity, and primordial non-gaussianity on the horizon scale.

A convergence of climate, resource, technological, and economic stresses gravely threaten the future of humankind. Scientists have a special role in humankind\\\'s response, because only rigorous science can help us understand the complexities and potential consequences of these stresses. Diminishing the threat they pose will require profound social, institutional, and technological changes -- changes that will be opposed by powerful status-quo special interests. Do scientists have a responsibility to articulate the dangers of inaction to a broader event beyond simply publishing their findings in scholarly journals? Should they become more actively involved in the politics of global change?

The Achilles\\\' heel of quantum information processors is the fragility of quantum states and processes. Without a method to control imperfection and imprecision of quantum devices, the probability that a quantum computation succeed will decrease exponentially in the number of gates it requires. In the last ten years, building on the discovery of quantum error correction, accuracy threshold theorems were proved showing that error can be controlled using a reasonable amount of resources as long as the error rate is smaller than a certain threshold. We thus have a scalable theory describing how to control quantum systems. I will briefly review some of the assumptions of the accuracy threshold theorems and comment on experiments that have been done and should be done to turn quantum error correction into an experimental reality.

The basic problem of much of condensed matter and high energy physics, as well as quantum chemistry, is to find the ground state properties of some Hamiltonian. Many algorithms have been invented to deal with this problem, each with different strengths and limitations. Ideas such as entanglement entropy from quantum information theory and quantum computing enable us to understand the difficulty of various problems. I will discuss recent results on area laws and use these to prove that we can use matrix product states to efficiently represent ground states for one-dimensional systems with a spectral gap, while certain other one-dimensional problems, without the gap assumption, almost certainly have no efficient way for us to even represent the ground state on a classical computer. I will also discuss recent results on higher-dimensional matrix product states, in an attempt to extend the remarkable success of matrix product algorithms beyond one dimension.

As physicists, we have become accustomed to the idea that a theory\\\'s content is always most transparent when written in coordinate-free language. But sometimes the choice of a good coordinate system is very useful for settling deep conceptual issues. Think of how Eddington-Finkelstein coordinates settled the longstanding question of whether the event horizon of a Schwarzschild black hole corresponds to a real spacetime singularity or not. Similarly we believe for an information-oriented or Bayesian approach to quantum foundations: That one good coordinate system may (eventually!) be worth more than a hundred blue-in-the-face arguments. This talk will motivate and chronicle the search for one such candidate coordinate system---the so-called Symmetric Informationally Complete Measurement---which has caught the attention of a handful of us here at PI and a handful of our visitors.

After almost a century of observations, the ultra-high energy sky has finally displayed an anisotropic distribution. A significant correlation between the arrival directions of ultra-high cosmic rays measured by the Pierre Auger Observatory and the distribution of nearby active galactic nuclei signals the dawn of particle astronomy. These historic results have important implications to both astrophysics and particle physics.

The Universe offers environments with extreme physical conditions that cannot be realized in laboratories on Earth. These environments provide unprecedented tests for extensions of the Standard Model. I will describe three such \"astrophysical laboratories\", which are likely to represent new frontiers in cosmology and astrophysics over the next decade. One provides a novel probe of the initial conditions from inflation and the nature of the dark matter, based on 3D mapping of the distribution of cosmic hydrogen through its resonant 21cm line. The second allows to constrain the metric around supermassive black holes based on direct imaging or the detection of gravitational waves. The third involves the acceleration of high-energy particles in cosmological shock waves. I will describe past and future observations of these environments and some related theoretical work.

This will be an introductory talk about Topological Quantum Computation. TQC is attractive because it is intrinsicaly decoherence free. We introduce the basic notions, such as non abelian anyons, quantum symmetries and topological order. A topologically ordered phase is a gapped phase in which the basic degrees of freedom are of a topological nature (denoted as anyons), charactetized by their fusion and braiding properties. If time permits possible implementations based on Quantum Hall systems will be discussed as well.

Laser cooling and precision spectroscopy provide powerful tools for exploring quantum measurement and metrology using atoms as sensors. In this talk I will discuss our ongoing work to bring together abstract ideas of quantum parameter estimation and concrete physical details of atom-photon interactions in the specific context of magnetometry. I will also present some new ideas on how laser probing of cold atoms could provide a basis for developing entanglement-enhanced spin gyroscopes.