LIMO-the limits of quantum models

To organize and explain experimental data, or just to formulate our beliefs about what we think is actually going on, we build theories, or models. By postulating relations between different physical entities, models impose limits on the space of feasible experimental behaviors. Determining the operational limits of a physical model is, however, a hard task. Think of neural networks: defining them is almost trivial, but determining the class of problems which they can solve is a topic of current research. In the last decades, there has been a profusion of quantum models, targeted to make sense of complex many-body quantum scenarios. Some of such scenarios appear in condensed matter physics, where the goal is to understand the collective behavior of a macroscopic sample containing 1023 atoms. Others appear naturally when we study communication tasks involving multiple agents, like quantum cryptography. The goal of LIMO is to develop a general theory of the correspondence between quantum models and their limitations. Efforts will be structured on two fronts: (i) Condensed matter theories: the aim will be to estimate the minimum complexity of the models required to produce a condensed matter state from realistic experimental data. More specifically, we will develop mathematical tools to identify the limits of the region of experimental data achievable with condensed matter models of low complexity and then propose experimental setups with the potential to violate those limits. Such violations must be understood as a benchmark for the experimental control of solid state systems. (ii)Quantum Causal networks: the goal is to determine when an assumption on the causal relations in a complex network of quantum systems is not compatible with the accessible data. This may have applications in quantum cryptography, since it can be used to bound the amount of private information that a spy may hold. It also has consequences for the foundation of quantum mechanics, as it identifies the limits enforced by the structure of quantum theory. Despite decades of intense research on solid state models, the problem of lower bounding the complexity of experimental condensed matter states has not been raised in the past. Consequently, there currently exist no general methods to attack this problem. We intend to devise these methods by exploiting a connection between condensed matter models and matrix theory. The identification of limits for quantum causal networks is a more conventional problem, which we will tackle via Non- Commutative Polynomial Optimization theory, a branch of convex optimization co-founded by the applicant in 2010.

Atoms and molecules, and, in general, any small physical system, follow very counter-intuitive physical laws, the laws of quantum mechanics. For instance, most of the time, the position of a given atom is not just unknown, but also undefined: regarding it as a vector, as we do with large, "classical" objects, such as cars, does not make any sense in quantum physics. The interaction of the quantum system with the environment, however, sometimes masks these quantum effects. In those situations, the presence of quantum components manifests in much more subtle ways. Project LIMO sought to understand how the laws of classical and quantum physics limit the experimental behavior of general physical systems, be they fundamental particles or tigers. In parallel, we also aimed to find new ways of tinkering with the way quantum systems change with time -we wished to control their dynamics. One of the first outcomes of the project was the mathematical characterization of non-classical behavior in experiments with macroscopic solid objects, such as a metal bar. This allowed us to provide experimental criteria to detect quantum phenomena in solids. Next, we identified some of the constraints on experimental behavior enforced by the structure of quantum theory in scenarios where different parts of a large system interact. An experimental violation of such constraints would signify that quantum physics does not have universal validity. In short, that our world is not quantum. Finally, we studied how to influence the time evolution of quantum systems. Our main discovery is the existence of physical processes that move quantum systems through time, in almost the same way that one ordinarily moves objects through space. More specifically, we show that one can devise a sort of "universal remote control" that, aimed to any physical system, will "rewind" it to the configuration that it had some time before the experiment started. We also found that there exist universal physical processes that, applied to two systems for a given amount of time (say, 5 seconds) would make one of them age twice as much (namely, 10 seconds), while keeping the other system in the same state that it had at the beginning of the experiment. The physical interpretation of this phenomenon is that the time that the second system should have aged during the experiment has been "transferred" to the first system. The above time translation results look a bit like science-fiction, but the simplest rewinding processes have already been implemented in the lab. In one such experiment, the state of a photon, the particle that light consists of, was rewound for a few nanoseconds. It is still too early to see where this line of research will lead. But the prospects are very exciting!