Mathematical Physics

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.