18th International Conference on
Quantum Physics and Logic

Self-testing is a method for certifying the production of quantum states with making minimal assumptions about the devices. In recent years a plethora of quantum states have been shown to be amenable to self-testing, but the question of whether an arbitrary quantum state can be self-tested, and up to which transformations, is an open question. I will show a method for self-testing an arbitrary (pure) quantum state, up to local transformations and global complex conjugation. The method is compositional in nature, utilising simple quantum networks with EPR pairs distributed throughout. I will also indicate how the global complex conjugation symmetry can be removed if the sources of EPR pairs are assumed to be independent. This is from joint work with Ivan Šupić, Joe Bowles, Marc-Olivier Renou and Antonio Acín.

Incompatibility of quantum measurements is one of the fundamental non-classical features of quantum theory. As a crucial ingredient in many quantum information protocols, incompatibility has become an important resource for quantum information theory, similar to entanglement. It is therefore a natural question how much of this resource is available in a given situation, characterized by the dimension of the quantum system, number of measurements and their outcomes.

It is known that incompatibility is not restricted to quantum mechanics but is present in any non-classical theory, in the framework of general probabilistic theories (GPT). This broader setting allows us to study and characterize incompatibility of measurements from different perspectives and using different mathematical tools. In this talk, we first concentrate on two-outcome measurements (or effects) and characterize their incompatibility in terms of tensor norms on Banach spaces. For measurements with more outcomes this does not seem possible, so we use a characterization by a GPT generalization of free spectrahedra and their inclusion constants, and by extensibility and separability properties of certain positive maps. As an application we explore compatibility regions and degrees of several GPT's of interest, in particular, we find a tight lower bound on incompatibility of any number of qubit effects.

The talk is based on a joint work with Andreas Bluhm and Ion Nechita, arxiv:2011.06497.

Recently, table-top experiments involving massive quantum systems have been proposed to test the interface of quantum theory and gravity. In particular, the crucial point of the debate is whether it is possible to conclude anything on the quantum nature of the gravitational field, provided that two quantum systems become entangled due to solely the gravitational interaction. Typically, this question has been addressed by assuming an underlying physical theory to describe the gravitational interaction, but no systematic approach to characterise the set of possible gravitational theories which are compatible with the observation of entanglement has been proposed. Here, we introduce the framework of Generalised Probabilistic Theories (GPTs) to the study of the nature of the gravitational field. This framework has the advantage that it only relies on the set of operationally accessible states, transformations, and measurements, without presupposing an underlying theory. Hence, it provides a framework to systematically study all theories compatible with the detection of entanglement generated via the gravitational interaction between two non-classical systems. Assuming that such gravitationally mediated entanglement is observed we prove a no-go theorem stating that gravity cannot simultaneously satisfy the following conditions i) it is a theory with no faster-than-light signalling; ii) it mediates the gravitational interaction via a physical degree of freedom; iii) it is classical. We further show what the violation of each condition implies, and in particular, when iii) is violated, we provide examples of non-classical and non-quantum theories which are logically consistent with the other conditions.

Traditionally, quantum Shannon theory has focused on scenarios where the internal state of the information carriers is quantum, while their trajectory is classical. However, as illustrated by the iconic double slit experiment, quantum particles can also propagate in a quantum superposition along multiple trajectories. In this talk, I shall discuss the recent extension of quantum Shannon theory to a second level of quantisation, where both the information and its propagation in spacetime is treated quantum mechanically.

First, I shall discuss our theoretical framework for formalising these scenarios, showing that when a single particle propagates through a superposition of multiple paths, the joint action of the independent noisy processes on each path is uniquely determined by their individual action on the vacuum state [G Chiribella & HK, Proc R Soc A 475.2225, 2019]. Secondly, I shall show how the same formalism can be extended further to describe the transmission of a quantum particle at a superposition of alternative moments in time [HK, W Mao, G Chiribella, arXiv:2004.06090, 2020]. When successive uses of a transmission line are correlated, we find that contrary to classical intuition, these correlations can be probed by a single quantum particle propagating at a superposition of times, and exploited to carry a larger amount of information per channel use.

Finally, I shall show that the mathematical structures arising in the physical scenarios of the second-quantised Shannon theory cannot be adequately described within the standard framework of quantum circuits. Consequently, I shall provide a brief introduction to our extended framework of routed quantum circuits [A Vanrietvelde, HK, J Barrett, arXiv:2011.08120, 2020], the details of which are left for another talk.

In this tutorial I will review some tools from theoretical computer science and descriptive complexity and show how can they be used to provide new insights into quantum foundations. In particular, I will show how Kolmogorov complexity and algorithmic randomness can be used in the context of quantum non-locality to analyse syntactic properties of sequences of experiments and provide a different notion of non-locality [1], new loopholes [2], new insight into the nature of quantum correlations [3] and open the doors into new informational principles for correlations.

1. [1] Wolf, S. (2015). Nonlocality without counterfactual reasoning. Physical Review A, 92(5), 052102.
2. [2] Bendersky, A., De La Torre, G., Senno, G., Figueira, S., & Acín, A. (2016). Algorithmic pseudorandomness in quantum setups. Physical review letters, 116(23), 230402.
3. [3] Bendersky, A., Senno, G., De La Torre, G., Figueira, S., & Acin, A. (2017). Nonsignaling deterministic models for nonlocal correlations have to be uncomputable. Physical review letters, 118(13), 130401.

Natural language processing (NLP) is a field of artificial intelligence that looks at representing natural language in a way that computers can take in, process, and interpret. One branch of NLP has had astonishing success over the last decade or so by representing words as vectors in vector spaces. A natural question then arises: can the tools and techniques of quantum theory be usefully applied in the area of NLP? In this tutorial I will give an overview of the use of vector semantics in NLP and an outline of applications of quantum techniques within NLP. These will include a principled approach to word and phrase composition in NLP, use of quantum structures such as density matrices for text representation, and an overview of how to move towards implementing these techniques on quantum computers.