Mathematical Physics

The modern understanding of quantum field theory underlines its effective nature: it describes only those properties of a system relevant above a certain scale. A detailed understanding of the nature of the neglected information is essential for a full application of quantum information-theoretic tools to continuum theories. I will present an operationally motivated method for deriving an effective field theory from any microscopic description of a state. The approach is based on dimensional reduction relative to a quantum distinguishability metric. It relies on a microscopic description of experimental limitations, such as a finite spatial resolution. In this picture, the emergent field observables represent cotangent vectors on the manifold of states, and are not necessarily endowed with the full semantic of standard quantum observables.

Our subject is Entropic Dynamics, a framework that emphasizes the deep connections between the laws of physics and information. In attempting to understand quantum theory it is quite natural to assume that it reflects laws of physics that operate at some deeper level and the goal is to discover what these underlying laws might be. In contrast, in the entropic view no fundamental underlying dynamics is invoked. Quantum theory is an application of entropic methods of inference and the goal is to make the best possible predictions on the basis of some limited information represented by appropriate constraints. It is through the choice of microstates and of these constraints that the “physics” is introduced. In Entropic Dynamics a relational notion of entropic time is introduced as a book-keeping device to keep track of changes. We show that a non-dissipative entropic dynamics naturally leads to generic forms of Hamiltonian dynamics, and notions of information geometry naturally lead to those specific Hamiltonians (that is, those that include the correct quantum potential) that describes quantum mechanics.

In the last decade there were proposed several new information theoretic frameworks (in particular, symmetric monoidal categories and "operational" convex sets), allowing for an axiomatic derivation of finite dimensional quantum mechanics as a specific case of a larger universe of information processing theories. Parallel to this, there was an influential development of quantum versions of bayesianism and causality, and relationships between quantum information and space-time structure. In the face of structural problems encountered when moving beyond finite dimensional quantum mechanics, as well as the lack of a mathematically and predictively sound nonperturbative framework for quantum field theories, a question appears: which of the existing structural assumptions of quantum information theory should be relaxed, and how? In this talk I will present a new approach to the information theoretic foundations of a "general" quantum theory (i.e., beyond quantum mechanics), that is a specific answer to the above question, with a hope to reconstruct both emergent space-times and emergent QFTs. Its mathematical setting is based on using quantum information geometry and integration over noncomutative algebras as structural and conceptual replacements of spectral theory and probability theory, respectively. This corresponds to a paradigmatic change: considering expectation values as more fundamental than eigenvalues. We construct a nonlinear generalisation of quantum kinematics using quantum relative entropies and spaces of states over W*-algebras. Unitary evolution is generalised to nonlinear hamiltonian flows, while Bayes' and Lueders' rules are generalised to constrained relative entropy maximisations. Combined together, they provide a framework for nonlinear causal inference (information dynamics), that is a generalisation and replacement of completely positive maps. As a result, we construct a large class of information processing theories, containing Hilbert space based QM and probability theory as two special cases. On the conceptual level, we propose a new approach to quantum bayesianism, that is ontically agnostic, intersubjective, and concerned with the relationships between experimental design, model construction, and their mutual predictive verifiability. Finally, we propose a procedure for the emergence of space-times from the geometry of quantum correlations and quantum causality structure, and discuss (briefly) the possibility of reconstructing emergent QFTs.

In science, we often see new advances and insights emerging from the intersection of different ideas coming from what appeared to be disconnected research areas. The theme of my seminar will be an ongoing collision between the three topics listed in my title which has been generating interesting new insights into a variety of fields, eg, condensed matter physics, quantum field theory and quantum gravity.

The second law of thermodynamics appears to be a universal law of physics. This universality suggests that entropy and with it information theory is part of the foundations of physics. In this talk I take the opposite (probably more conservative) approach: I assume that the dynamics of relational degrees of freedom form the foundation of physics. Physical information is an emergent phenomenon. This approach has an interesting consequence: Typical (initial) data for a gravitationally dominated universe leads to the spontaneous emergence of a gravitational arrow of time for the universe as a whole. This primary gravitational arrow of time generates secondary thermodynamic arrows of time in sufficiently isolated subsystems of the universe, which coincide with the gravitational arrow of time. This coincidence explains the universality of the second law of thermodynamics. I conclude the talk with a speculation about the emergence of quantum information: I assume (1) that the purpose of a physical law is the prediction of future properties of the universe based on the knowledge of records and (2) that gravity generates physical records (as in the classical part of the talk). This suggests a scenario in which stable quantum information is spontaneously generated. Collaborators for the classical part of the talk where Julian Barbour and Flavio Mercati

Classical and quantum theories are very different, but the gap between them may look narrow particularly if the notion of classicality is broadened. For example, if we do not impose all the classical assumptions at the same time, hidden variable theories reproduce the results of quantum mechanics. If a quantum system is restricted to Gaussian states, evolution and measurements, then classical phase space mechanics with a finite resolution fully reproduces its behavior. We discuss two examples of such extensions. In a version of the delayed-choice experiment we allow an otherwise classical system to exhibit two types of behavior ("P" or "W"), requiring, however, objectivity: the system is at any moment either "P" or "W", but not both. It turns out that the three conditions of objectivity, determinism, and independence of hidden variables are incompatible with any theory, not only with quantum mechanics. We then consider two harmonic oscillators with a Gaussian interaction between them. If one is treated as quantum and one is described by a classical theory with a finite phase space resolution, no consistent description of this interaction is possible. The lesson is that it is hard to be a little bit quantum: it is either pointless or quantumness takes over altogether.

Quantum mechanics is derived from the principle that the universe contain as much variety as possible, in the sense of maximizing the distinctiveness of each subsystem. This is an expression of Leibniz's principles of sufficient reason and the identity of the indiscernible. The quantum state of a microscopic system is defined to correspond to an ensemble of subsystems of the universe with identical constituents and similar preparations and environments. A new kind of interaction is posited amongst such similar subsystems which acts to increase their distinctiveness, by extremizing the variety. In the limit of large numbers of similar subsystems this interaction is shown to give rise to Bohm's quantum potential. As a result the probability distribution for the ensemble is governed by the Schroedinger equation. The measurement problem is naturally and simply solved. Microscopic systems appear statistical because they are members of large ensembles of similar systems which interact non-locally. Macroscopic systems are unique, and are not members of any ensembles of similar systems. Consequently their collective coordinates may evolve deterministically. This proposal could be tested by constructing quantum devices from entangled states of a modest number of quits which, by its combinatorial complexity, can be expected to have no natural copies.

To make precise the sense in which nature fails to respect classical physics, one requires a formal notion of "classicality". Ideally, such a notion should be defined operationally, so that it can be subjected to a direct experimental test, and it should be applicable in a wide variety of experimental scenarios, so that it can cover the breadth of phenomena that are thought to defy classical understanding. Bell's notion of local causality fulfills the first criterion but not the second, because it is restricted to scenarios with two or more systems that are space-like separated. The notion of noncontextuality fufills the second criterion, because it is applicable to any experiment (even those on a single system), but it is a long-standing question whether it can be made to fulfill the first. Previous attempts to experimentally test noncontextuality have all presumed certain idealizations that do not hold in real experiments, namely, noiseless measurements and exact operational equivalences. In this talk, I will describe how one can devise experimental tests that are free of these idealizations using an operational notion of noncontextuality that applies to both preparations and measurements. These new theoretical insights raise the bar significantly for any claim of an experimental demonstration of nonclassicality. They also provide the means of determining, for any phenomenon that is typically thought to defy classical explanation, which experimentally-testable features of that phenomenon, if any, conflict with the assumption of a noncontextual model.

I will discuss my work (in progress) to formulate General Relativity as an operational theory which includes probabilities and also agency (knob settings). The first step is to find a way to discuss operational elements of GR. For this I adapt an approach due to Westman and Sonego. I assert that all directly observable quantities correspond to coincidences in the values of scalar fields. Next we need to include agency. Usually GR is regarded as a theory in which a solution is simply stated for all space and time (the Block Universe view). Here, instead, we find a way to treat agents as making choices. Finally, we need to incorporate probabilities. For this purpose we take a compositional point of view. We associate a generalized state with regions of the space consisting of coincidences in the values of scalars. We then show how to combine these generalized states to make probabilistic predictions using the duotensor machinery developed previously.

It has become conventional wisdom to say that quantum theory and gravitational physics are conceptually so different, if not incompatible, that it is very hard to unify them. However, in the talk I will argue that the operational view of (quantum) information theory adds a very different twist to this picture: quite on the contrary, quantum theory and space-time are highly fine-tuned to fit to each other. After a recap of ideas by von Weizsacker, Wootters, and Popescu and Rohrlich, I will show how uncertainty relations, the number of degrees of freedom of the Bloch ball, and the existence of entangled states and possibly the Tsirelson bound can be understood from space-time geometry alone. Conversely, I will show how the 3+1 Lorentz group of space-time can be derived from a purely informational communication scenario of two observers that describe local quantum physics in different Hilbert space bases (joint work with Philipp Hoehn).