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Information and Randomness in Quantum Theory

Final Report Summary - IRQUAT (Information and Randomness in Quantum Theory)

Quantum mechanics is unique among the fundamental physical theories of nature in that it is inherently probabilistic. Indeed, the famous intellectual battles over its foundations, right from the very beginning in the 1920s, were ultimately all about the inherent indeterminism of quantum mechanics. Or rather, the fact that while a deterministic wave equation, the Schrödinger equation, governs the evolution of a closed system, the theory makes only probabilistic predictions regarding observations. Heisenberg’s, and later, uncertainty relations epitomise this trait in the form of a basic limitation on the statistical precision with which observable quantities can be measured. Whereas Bohr took this to mean that measurement results have no objective ontological status as long as no observation of a quantum system is made, the groundbreaking paper by Einstein, Podolsky and Rosen pointed to another non-classical feature of quantum theory, inextricably linked to its non-determinism: that a pure state can nevertheless necessitate correlations between separate, even distant systems. Today, after the work of Bell, and with the advent of quantum information and computation, we understand much better that quantum theory is in fact a theory of fundamentally irreducible correlations, thus showing randomness (of individual classical events) as a necessary feature of a description of nature which ultimately is about information (between subsystems, such as the degrees of freedom in elementary particles or measurement devices).
Indeed, we have come a long way from the days when entanglement appeared first, then as a bizarre feature of the new quantum mechanics, to today’s understanding of it as the prime resource in a future information technology, enabling secure communication, quantum teleportation, and quantum computational speedups. Nevertheless quantum theory remains deeply mysterious, conceptually disturbing and counterintuitive. A century after the initiation of quantum mechanics by Planck we continue to come across unexpected new features of it, now often couched in the language and perspective of quantum information.
This project is directly about this unique interplay between information and randomness, motivated by questions at the forefront of quantum communication (Shannon) theory, the foundations of quantum mechanics and statistical mechanics.

In particular, our understanding of the very notion of channel capacity to transmit classical or quantum information was advanced by the proof of so-called "strong converses" for various capacities, meaning that the error of any code at rates above the capacity necessarily goes to 1; this part also involved the development of techniques to do with generalized information measures (beyond the von Neumann entropy). In this context also several results on the inequalities of generalized entropies were found.
To assess quantum correlations and other, more abstract stronger-than-quantum correlations, new techniques were developed and applied. One of the most exciting bits of work here was an in-depth analysis of the set of operations one can perform on a composite system when restricted by locality (LOCC). Furthermore, in various collaborations, quantum zero-error communication was developed, as an arena for the study of non-local correlations, especially entanglement. Our work has shown that quantum channels give rise to generalised, “non-commutative” confusability graphs, permitting the generalization of various graph parameters, including independence number and Lovász number.
In the second half of the project, the emphasis shifted to resource theories inspired by physical constraints, such as thermodynamics and coherence. In the quantum thermodynamics of systems with several conserved quantities, a derivation from first principles of the form of the equilibrium state was given, which works even when the quantities correspond to non-commuting quantum observables. In the resource theory of coherence, groundbreaking results on the distillation and dilution of coherence were obtained, and an operational link to visibility on interferometric experiments exposed.
The core themes of IRQUAT continued to be active concerns, for example in the exploration of the monogamy of quantum entanglement, which was shown to be incompatible with other desiderata on entanglement measures. One of the highlights in entanglement theory from the project was the discovery that squashed entanglement measures the closeness of a state to being highly extendible. In another direction, we showed that quantum data hiding is actually a feature of very general non-classical state spaces and the way they compose to form bipartite systems.