Mathematical Physics

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).

I will argue that, apart from their ever growing number of applications to physics, information theoretic concepts also offer a novel perspective on the physical content and architecture of quantum theory and spacetime. As a concrete example, I will discuss how one can derive and understand the formalism of qubit quantum theory by focusing only on what an observer can say about a system and imposing a few simple rules on the observer’s acquisition of information.

How should we describe the thermodynamics of extreme quantum regimes, where features such as coherence and entanglement dominate? I will discuss possible limitations of a traditional statistical mechanics approach, and then describe work that applies modern techniques from the theory of quantum information to the foundations of thermodynamics. In particular I discuss recent progress in quantum resource theories and argue that to properly encapsulate the thermodynamic structure of quantum coherence and entanglement we must make use of concepts beyond free energies.

Much progress has recently been made on the fine-grained thermodynamics and statistical mechanics of microscopic physical systems, by conceiving of thermodynamics as a resource theory: one which governs which transitions between states are possible using specified "thermodynamic" (e.g. adiabatic or isothermal) means. In this talk we lay some groundwork for investigating thermodynamics in generalized probabilistic theories. We describe simple, but fairly strong, postulates: unique spectrality, projectivity, and symmetry of transition probabilities, that imply that a system has a well-behaved convex analogue of the spectrum, and show that the spectrum of a state majorizes the outcome probabilities of any fine-grained measurement, allowing the operationally defined measurement entropy (and Schur-concave analogues) to be calculated from the spectrum. These are implied by, but probably weaker than, Axioms 1 (weak spectrality) and 2 (strong symmetry) of a recent characterization of Jordan-algebraic and quantum systems by Barnum, Mueller, and Ududec. It is an open question whether theories beyond the Jordan-algebraic ones satisfy them. We describe how part of von Neumann's argument that spectral entropy is a good candidate for thermodynamic entropy generalizes to systems satisfying our postulates, and discuss whether its assumptions are reasonable there, suggesting that the extendibility of certain processes to reversible ones is crucial. We will discuss further postulates and results that might suffice to obtain, in this more general setting, a thermodynamical resource theory similar to the one that is emerging for quantum theory. (Joint work with Jon Barrett, Marius Krumm, Matt Leifer, Markus Mueller.)

In the classical world of Newton and Laplace, fundamental physics and thermodynamics do not blend well: the former puts forward a picture of nature where states are pure and processes are fundamentally reversible, while the latter deals with scenarios where states are mixed and processes are irreversible. Many attempts have been made at reconciling the two paradigms, but ultimately the source of all troubles remains: if every particle possesses a definite position and a definite velocity, why should experimental data depend on the expectations of agents who have only partial information? Quantum theory offers a way out. Thanks to entanglement, a system and its environment can be jointly in a pure state, while the system alone is in a mixed state. Similarly, the evolution of system and environment can be jointly reversible, but locally irreversible. At the level of axioms, these facts are captured by the Purification Principle [1], from which the whole of quantum theory was derived in Ref. [2]. In this talk I will report on a new research programme, aimed at establishing the Purification Principle as the conceptual foundation for thermodynamics. I will start from the idea that thermodynamical transformations can be mapped into transformations that degrade entanglement. This will lead to a duality between entanglement and thermodynamics, which will be used to define measures of entanglement/entropy and to explore operational tasks like information erasure. At this level, thermodynamical requirements lead directly to requirements on the particular type of Purifications allowed by the theory: for example, forbidding the existence of Entropy Sinks (systems that can absorb entropy from the environment without increasing their internal entropy) is equivalent to imposing the existence of Symmetric Purifications (purifications in which the environment is a mirror image of the system). I will conclude by giving a glimpse on quantitative measures of entanglement/entropy. By adding a requirement about the existence of sharp measurements, I will show that Purification leads directly the an operational version of the spectral theorem, allowing one to define a zoo of well-behaved entropies. [1] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Probabilistic Theories with Purification, Phys. Rev. A 81, 062348 (2010) [2] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Informational Derivation of Quantum Theory, Phys. Rev. A 84, 012311 (2011)

In QBism, a quantum state represents an agent's personal degrees of belief regarding the consequences of her actions on any part of her external world. The quantum formalism provides consistency criteria that enable the agent to make better decisions. QBism thus gives a central role to the agent, or user of the theory, and explicitly rejects the ontological model framework introduced by Harrigan and Spekkens. This talk addresses the status of agents and the notion of locality in QBism. Our definition of locality is independent of the assumption of an ontological model. Instead it depends on an appropriate formalization of the idea of a causal structure.

One of the most deeply rooted concepts in science is causality: the idea that events in the present are caused by events in the past and, in turn, act as causes for what happens in the future. If an event A is a cause of an effect B, then B cannot be a cause of A. Recently we proposed a framework that assumes that operations in local laboratories are described by quantum mechanics, but where no reference is made to any global causal relations between these operations. The central notion of the formalism is “process” which is a generalization of the causal notion of “quantum state”. The framework allows for processes in which two operations are neither causally ordered nor in a probabilistic mixture of definite causal orders, i.e. one cannot say that A is before or after B. However a physical interpretation of such correlations was lacking. I will show that the “superposition of quantum circuits” – in which the gate ordering is not fixed but controlled by a quantum system – is an example of a process with indefinite causal order. This process provides a reduction in query complexity of certain computational problems and can be realized by placing the laboratories in the gravitational field of a massive object in a spatial superposition.

It has become a platitute to say that black holes are fascinating objects—but they really are, in part because they challenge our understanding of the fundamental reversibility of physical processes.

In the first part of the talk, I will review some of the classical ways black holes behave as dissipative systems, such as the "hair loss” phenomenon and the monotonic growth of horizon area. In the second part, I will explain how quantum mechanics (more precisely, the coupling of black holes to the quantum vacuum) affects the classical picture at late times, notably through particle creation and evaporation. I will argue that techniques from two-dimensional field theory can help bring clarity to the associated “information loss”

problem, and perhaps also point to new, unexpected predictions. My approach will be as model-independent as possible; that is, rather than investigating a particular scenario for black hole evaporation, I will aim to derive generic consequences from basic assumptions regarding the reversibility of black hole evaporation.

Rideout and Sorkin proposed a classical dynamics for causal sets based upon a sequential growth model. Comparing it with models for sequential growth in other systems, and with the dual goals of generating manifold-like causal sets and finding a quantum dynamics for them, we propose some modifications to their model. The resulting, admittedly speculative, proposal is a type of quantum random walk. We explore its properties in some simple cases.

After the seminal work of Connes and Tretkoff on the Gauss-Bonnet theorem for the noncommutative 2-torus and its extension by Fathizadeh and myself, there have been significant developments in understanding the local differential geometry of these noncommutative spaces equipped with curved metrics. In this talk, I will review a series of joint works with Farzad Fathizadeh in which we compute the scalar curvature for curved noncommutative tori and prove the analogue of Weyl's law and Connes' trace theorem. Our final formula for the curvature matches precisely with the one computed independently by A. Connes and H. Moscovici. I will then report on our recent work on the computation of scalar curvature for noncommutative 4-tori (which involves intricacies due to violation of the Kähler condition). We show that metrics with constant curvature are extrema of the analogue of the Einstein–Hilbert action. A purely noncommutative feature in these works is the appearance of the modular automorphism from Tomita–Takesaki theory of KMS states in the final formulas for the curvature.