Quantum Physics

Chris Isham in pre-recorded video, with Andreas Doring fielding questions and clarifications. Like watching commentators Scott Hamilton and Katarina Witt analyze Kristi Yamaguchi\'s performance at the World Figure Skating Championships for CBS News, join us for something different in quantum foundations. Chris Isham parries the intricacies of topos theory; Andreas Doring shows us how to see the moves in slow motion. Bring your own popcorn and plenty of questions.

Quantum measure theory describes quantum theory as a generalization of a classical stochastic process, which may be fruitful for quantum gravity. I will describe the approach, and show that, in the context of an EPRB setup with two distant experimenters, two alternative experiments, and two outcomes per experiment, any set of no signaling probabilities can be realized, albeit by violating a `strong positivity\' condition.

The talk concerns a generalization of the concept of a minimum uncertainty state to the finite dimensional case. Instead of considering the product of the variances of two complementary observables we consider an uncertainty relation involving the quadratic Renyi entropies summed over a full set of mutually unbiased bases (MUBs). States which achieve the lower bound set by this inequality were introduced by Wootters and Sussman, who proved existence for every prime power dimension, and by Appleby, Dang and Fuchs who showed that in prime dimension the fiducial vector for a for a symmetric informationally complete positive operator valued measure (SIC-POVM) covariant under the Weyl-Heisenberg group is a state of this kind. Subsequently Sussman proved existence for a class of odd prime power dimensions. The purpose of this talk is to complete the existence proof by showing that minimum uncertainty states exist in every prime power dimension, without exception. Along the way we establish a number of properties of the Clifford group, and Galois extension fields, which might be of some independent interest.

It is usually expected that nonrelativistic many-body Schroedinger equations emerge from some QFT models in the limit of infinite masses. For instance, from Yukawa\'s QFT, if the initial state contains 2 fermions, we expect to recover a 2-fermion nonrelativistic Schroedinger equation with 2-body Yukawa potential (in the limit of infinite fermion mass). I will give an easy (but still heuristic) derivation of this, based on the analysis of the corresponding Feynman diagrams and on the behaviour of the complete propagators for large spacetime distances. Then, I may outline another possible derivation based on the Schroedinger picture and dressed particles.

Coin flipping by telephone (Blum \'81) is one of the most basic cryptographic tasks of two-party secure computation. In a quantum setting, it is possible to realize (weak) coin flipping with information theoretic security. Quantum coin flipping has been a longstanding open problem, and its solution uses an innovative formalism developed by Alexei Kitaev for mapping quantum games into convex optimization problems. The optimizations are carried out over duals to the cone of operator monotone functions, though the mapped problem can also be described in a very simple language that involves moving points in the plane. Time permitting, I will discuss both Kitaev\'s formalism, and the solution that leads to quantum weak coin flipping with arbitrarily small bias.

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.

Decoherence attempts to explain the emergent classical behaviour of a quantum system interacting with its quantum environment. In order to formalize this mechanism we introduce the idea that the information preserved in an open quantum evolution (or channel) can be characterized in terms of observables of the initial system. We use this approach to show that information which is broadcast into many parts of the environment can be encoded in a single observable. This supports a model of decoherence where the pointer observable can be an arbitrary positive operator-valued measure (POVM). This generalization makes it possible to characterize the emergence of a realistic classical phase-space. In addition, this model clarifies the relations among the information preserved in the system, the information flowing from the system to the environment (measurement), and the establishment of correlations between the system and the environment.

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.

We prove that all non-conspiratorial/retro-causal hidden variable theories has to be measurement ordering contextual, i.e. there exists *commuting* operator pair (A,B) and a hidden state \\\\lambda such that the outcome of A depends on whether we measure B before or after. Interestingly this rules out a recent proposal for a psi-epistemic due to Barrett, Hardy, and Spekkens. We also show that the model was in fact partly discovered already by vanFraassen 1973; the only thing missing was giving a probability distribution on the space of ontic states (the hidden variables).

The purpose of this talk is to describe bosonic fields and their Lagrangians in the causal set context. Spin-0 fields are defined to be real-valued functions on a causal set. Gauge fields are viewed as SU(n)-valued functions on the set of pairs of elements of a causal set, and gravity is viewed as the causal relation itself. The purpose of this talk is to come up with expressions for the Lagrangian densities of these fields in such a way that they approximate the Lagrangian densities expected from regular Quantum Field Theory on a differentiable manifold in the special case where the causal set is a random sprinkling of points in the manifold. I will then conjecture that that same expression is appropriate for an arbitrary causal set.

I should like to show how particular mathematical properties can limit our metaphysical choices, by discussing old and new theorems within the statistical-model framework of Mielnik, Foulis & Randall, and Holevo, and what these theorems have to say about possible metaphysical models of quantum mechanics. Time permitting, I should also like to show how metaphysical assumptions lead to particular mathematical choices, by discussing how the assumption of space as a relational concept leads to a not widely known frame-invariant formulation of classical point-particle mechanics by Föppl and Zanstra, and related research topics in continuum mechanics and general relativity.

Mutually unbiased bases (MUBs) have attracted a lot of attention the last years. These bases are interesting for their potential use within quantum information processing and when trying to understand quantum state space. A central question is if there exists complete sets of N+1 MUBs in N-dimensional Hilbert space, as these are desired for quantum state tomography. Despite a lot of effort they are only known in prime power dimensions. I will describe in geometrical terms how a complete set of MUBs would sit in the set of density matrices and present a distance between bases–a measure of unbiasedness. Then I will explain the relation between MUBs and Hadamard matrices, and report on a search for MUB-sets in dimension N=6. In this case no sets of more than three MUBs are found, but there are several inequivalent triplets.