Mathematical Physics

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).

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)