This page is devoted to the previous Solstice of Foundations 2019, which took place on June 17-21, 2019, in Zürich, Switzerland, followed by a workshop on the interface between mathematical physics and quantum information theory, hosted by SwissMap.

**Organizers**

**Foundations summer school**

Lídia del Rio, ETH Zurich

Nuriya Nurgalieva, ETH Zurich

**SwissMap workshop**

Anton Alexeev, U. Geneva

Renato Renner, ETH Zurich

Lídia del Rio, ETH Zurich

**Sponsors**

**Poster session**

### Lectures and programme

#### Thomas Galley, Perimeter Institute, Canada: **Deriving quantum postulates from principles [slides]**

There are many ways of understanding quantum theory: as a non-classical logic, a non-classical probability theory and more generally as some form of operational theory. In this lecture I will show that different ways of viewing quantum theory allow us to have more efficient axiomatisations than in the standard approach. I will cover operational quantum logic and use Gleason’s theorem to show how we can recover quantum theory from a minimal number of postulates about the structure of measurements. I will then proceed to describe quantum theory as a generalised probabilistic theory (GPT) and show that many quantum properties are in fact generically non-classical. Following this I will show how to make modifications to the measurements postulates of quantum theory, and how to derive state spaces for systems in theories with the same pure states and dynamics as quantum theory, but different measurement postulates. In a final part I will show how composition of systems imposes strict constraints on how we can modify the measurement postulates of quantum theory, and specifically how the property of associativity of systems rules out all modifications. This allows us to derive the measurement postulates of quantum theory from basic features of the operational framework. References:

- A. Wilce, “Quantum logic and probability theory,” in The Stanford Encyclopedia of Philosophy (E. N. Zalta, ed.), Metaphysics Research Lab, Stanford University, spring 2017 ed., 2017.
- P. Janotta and H. Hinrichsen, “Generalized probability theories: what determines the structure of quantum theory?,” Journal of Physics A Mathematical General, vol. 47, p. 323001, Aug 2014.
- L. Masanes, T. D. Galley, and M. P. Müller, “The measurement postulates of quantum mechanics are operationally redundant,” arXiv:1811.11060, Nov 2018

#### Valerio Scarani, National University of Singapore:** Bell non-locality and applications [book chapter]**

- * Definition, loopholes, interpretations
- * Formalisation: Fine theorem, local polytope
- * Application: device-independent certification.
- * Example: Self-testing.

Main reference: my lecture notes on Bell’s theorem.

#### Tony Short, University of Bristol, UK: **Time in quantum mechanics**

These lectures will explore our understanding of time in quantum theory. We will begin with a general overview, considering in particular the differences between space and time in quantum theory. Next we will consider quantum clocks, beginning with a model of an `ideal’ clock (with *H=vp*) and then considering finite dimensional and thermodynamic clocks. Key questions are how clocks should be defined, how to assess their accuracy, and the ultimate limitations on them. Finally, we will consider a possible way to understand the flow of time in terms of correlations between a system and a clock within a single static entangled state.

Interesting general perspectives on time:

- “Time in quantum mechanics – vol 1”, editors J. Gonzalo Muga, Rafael Sala Mayato, Iñigo L. Egusquiza, Lecture Notes in Physics 734, Springer (2008)
- “Time in Physics”, editors Renato Renner and Sandra Stupar, Tutorials, Schools, and Workshops in the Mathematical Sciences, Springer (2017)

Quantum clocks:

- “Autonomous Quantum Machines and Finite-Sized Clocks”, Mischa P. Woods, Ralph Silva, and Jonathan Oppenheim. Annales Henri Poincaré, 20, 125-218 (2019)
- “Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?”, Paul Erker, Mark T. Mitchison, Ralph Silva, Mischa P. Woods, Nicolas Brunner, and Marcus Huber. Phys. Rev. X 7, 031022 (2017)
- “Quantum clocks are more accurate than classical ones, Mischa P. Woods, Ralph Silva, Gilles Pütz, Sandra Stupar, and Renato Renner. quant-ph arXiv:1806.00491 (2018)
- “Clock-driven quantum thermal engines”, Artur S.L. Malabarba, Anthony J. Short, and Philipp Kammerlander. New J. Phys. 17, 045027 (2015)

Time as correlations in a static entangled state:

- “Evolution without evolution: Dynamics described by stationary observables”, Don N. Page and William K. Wootters. Phys. Rev. D 27, 2885 (1983)
- “Quantum time”, Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Phys. Rev. D 92, 045033 (2015)

#### Sandu Popescu, University of Bristol, UK: **Dynamic **non-locality

#### Ralph Silva, ETH Zurich, Switzerland: **Pre- and post-selection**

#### Lev Vaidman, Tel Aviv University, Israel: **Past of a quantum particle [slides]**

Bohr preached not to talk about it, Wheeler said we can, but only when we have a definite retrodiction. According to Bohm the past might be surrealistic and according to Aharonov not only the past, but even the present and future are fixed. I will argue that not only we can talk about the past in our world, but we can say much more than about the past of a classical system.

#### Mirjam Weilenmann, University of Vienna, Austria: **Causality I: classical, quantum and post-quantum causality**

Understanding cause-effect relationships is crucial for our scientific reasoning. In this lecture we will speak about causality and introduce mathematical tools for analysing cause-effect relationships. We will first introduce classical causality and the notion of interventions. We will further discuss the causal inference problem, including recent techniques involving entropy measures and graph inflation. In the second part, we will discuss the inadequacy of these classical tools in the quantum realm and introduce avenues to extend them. With this we set the scene for the lecture Causality II, where some tools for analysing quantum causality and its applications are discussed in-depth.

References and related literature:

- For an introduction to classical causality: Causality by Judea Pearl, https://dl.acm.org/citation.cfm?id=331969.
- For recent techniques for causal inference: graph inflation, https://arxiv.org/abs/1609.00672, and a review of entropic techniques, https://arxiv.org/abs/1709.08988.
- For the inadequacy of classical models for quantum causality: https://arxiv.org/abs/1208.4119.
- For quantum causal models: https://arxiv.org/abs/1609.09487.

#### V.Vilasini, University of York, UK: **Causality II: different methods and applications [**lecture notes part 1, part 2**]**

In these lectures, I will start by contrasting classical and quantum causality. To this effect, I will highlight some of the peculiar features of non-classical notions of causality and describe some theoretical possibilities beyond quantum theory that remain compatible with relativistic causality. While there are several theoretical frameworks for analysing these general notions of causality, I will focus on two of them, namely the Causal Box and the Process Matrix frameworks and describe how they handle superpositions of temporal/causal orders. The quantum switch is claimed to be an example of a physically implementable superposition of causal orders. By comparing the description of the quantum switch in these frameworks, I hope to fuel a discussion on whether it implements a superposition of causal orders or a superposition of temporal orders. Lastly, I will touch upon some of the applications that follow from asking these fundamental questions: such as that of superpositions of orders in communication/query complexity problems and the Causal Box framework in relativistic quantum cryptography.

References:

These lectures will be heavily based on my master thesis. Most of the relevant references for this talk can be found in the thesis. Following is a list of some of the main references:

- For classical causality: https://dl.acm.org/citation.cfm?id=331969.
- Why classical causal models cannot satisfactorily explain quantum causality: https://iopscience.iop.org/article/10.1088/1367-2630/17/3/033002.
- The quantum switch https://journals.aps.org/pra/abstract/10.1103/PhysRevA.88.022318, and a physical implementation of it https://advances.sciencemag.org/content/3/3/e1602589.
- Some frameworks for studying non-classical notions of causality (there are many others but we will focus on these): Causal Boxes https://ieeexplore.ieee.org/document/7867830, Process Matrices https://www.nature.com/articles/ncomms2076.
- Applications: computational advantage, complexity https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.250402,https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.100502, relativistic cryptography https://iopscience.iop.org/article/10.1088/1367-2630/ab0e3b.

#### Matthew Leifer, Chapman University, US – **Interpretations of quantum theory [lecture notes]**

Although quantum theory is about a century old, there is still little consensus about what it all means. In these lectures, I will start by explaining what I think should be the goals of an interpretation of quantum theory and why textbook accounts of quantum theory are inadequate. I will briefly review decoherence theory, which plays a role in accounting for the appearance of classical reality in most interpretations of quantum theory. Dozens of interpretations of quantum theory have been developed over the years, but four of them currently attract the most interest. These are modern variants of the Copenhagen interpretation (which I call Copenhagenish interpretations) as well as the “big three” realist interpretations: de Broglie-Bohm theory, Everett/many-worlds, and spontaneous collapse theories. I will give an overview of the modern perspective on de Broglie-Bohm theory, Everett/many-worlds, and Copenhagenish interpretations. Spontaneous collapse theories will not be covered in detail (mainly due to lack of time, but also because I view them as the least plausible of the “big three”).

References:

These are more relevant for supplementary reading after the summer school for those who want to know more details.

- Lectures 12-15 in my Perimeter Institute course on Quantum Foundations http://pirsa.org/C19002/2 cover similar material to my summer school lectures.* Maximilian Schlosshauer, Decoherence and the Quantum to Classical Transition (Springer, 2007). A detailed overview of the technical details of decoherence theory.
- Travis Norsen, Foundations of Quantum Mechanics (Springer, 2017). Chapters 7 and 9 are good on de Broglie Bohm and Spontaneous Collapse Theories.
- David Wallace, The Emergent Multiverse, (Oxford University Press, 2012) is a much more detailed account of modern Everett/Many-Worlds than I could ever hope to achieve.

#### Matthew Pusey, University of Oxford, UK – **Contextuality [lecture notes]**

I will present the Kochen-Specker theorem, which rules out pre-existing results for quantum measurements unless those results are ‘contextual’. I will then discuss the challenges of turning contextuality from a mathematical result about quantum theory into an experimental result about nature itself. I will focus on the approach to this due to Spekkens but will also attempt to summarise and compare other approaches.

I would encourage students to prepare by reading Spekkens’ original paper. It may also be of interest to examine a simple proof of the Kochen-Specker theorem presented by Mermin.

#### Gilles Brassard, Université de Montréal, Canada – **Could Einstein have been right? Quantum theory ***can* be local and realistic! [slides on pdf, keynote]

*can*be local and realistic!

One of the most surprising aspects of quantum theory is that it tells us that we live in a nonlocal universe. This idea was completely abhorrent to Einstein, who dismissed it as “spooky action at a distance”. Recent so-called loophole-free Bell experiments have confirmed nonlocality beyond any reasonable doubt. But have they really? I shall demonstrate that the mere experimental violation of a Bell inequality cannot be used as evidence for nonlocality since local-realistic universes that violate such inequalities are easy to imagine. Furthermore, no experiment whose purpose is to confirm the predictions of quantum theory can possibly be used as an argument in favour of nonlocality because any theory of physics that does not allow instantaneous signalling to occur and has reversible dynamics (such as unitary quantum theory) can be explained in a purely local and realistic universe. And if Einstein was right after all… once again?

Joint work with Paul Raymond-Robichaud (here and here).

#### Markus Müller, University of Vienna, Austria – **From observers to physics via information theory [notes on AIT and Solomonoff Induction]**

In these lectures, I will describe several puzzles and paradoxes from physics and philosophy, and show that they hint towards a view on the physical world that differs substantially from the usual intuition. These hints and ideas can be made rigorous with the tools of algorithmic information theory, which I will briefly introduce in the second part.

In the third part, I will discuss the metaphysically counterintuitive but consistent view that arises from such an approach. In particular, such views have the potential to resolve the paradoxes mentioned in the first part, they are consistent with everything that we observe, and they lead to surprising predictions like “probabilistic zombies”.

See also the abstract of my paper: https://arxiv.org/abs/1712.01826

More detailed list of topics and references:

- Algorithmic information theory / algorithmic probability

A good description can be found in the book “An Introduction to Kolmogorov Complexity and Its Applications” by Li and Vitanyi (Springer Verlag). - Solomonoff Induction

The following “Less Wrong” blog post gives a nice introduction:

https://www.lesswrong.com/posts/Kyc5dFDzBg4WccrbK/an-intuitive-explanation-of-solomonoff-induction - Paradoxes

To see that there is a lot that we do not yet understand, and to bring yourself into some serious (but necessary) metaphysical vertigo, read for example about Parfit’s “teletransportation paradox”, or have a look at the book “The Mind’s I” by Dennett and Hofstadter

#### Paolo Perinotti, University of Pavia, Italy –** From cellular automata to relativity**

In this lecture we will illustrate a recent approach to the foundations of relativistic quantum fields based on informational accounts of the theory of elementary systems. The framework of Operational Probabilistic Theories (OPTs) encompasses a wide variety of alternate theories abut elementary systems, tests and processes they can undergo, and it consists in universal rules about composition of systems and tests, and probabilities of events. Specific theories are then determined by the particular choice of mathematical objects representing tests and events. Relevant examples are quantum theory—which can be derived from informational axioms—, quantum theory on real Hilbert spaces, classical information theory, and Fermionic theory. The nature of elementary systems of an OPT is that of elementary information carriers, and there is no natural way of directly introducing mechanical concepts—such as space-time, mass, energy, and so on—in an OPT without referring to some additional structure outside the OPT itself.

In the first part, we will provide a fast review of the framework of OPTs we will then introduce the notion of a physical law in a purely informational scenario, and conclude that it is well described by a cellular automaton. We will show how a geometric picture of Space-time can be derived once a cellular automaton is provided. We will then restrict attention to Fermionic cellular automata, and furthermore we will take the simplest case of cellular automata on Cayley graphs of Abelian groups. There are surprisingly few of them: essentially two, which we call Weyl automata because once we embed them in their space-time, they are very well approximated by Weyl’s equations.

In the second part we will discuss the notion of change of inertial frame in a scenario where space-time is not fundamental, recurring to the original formulation of the principle of relativity. We then show how the changes of reference frame for the Weyl automaton correspond to a group isomorphic to the semi-direct product of the Poincaré group by a group of dilations, thus recovering the basic symmetry of Minkowski space-time.