Positivity structures in quantum many-body systems

In order to describe very small physical systems, such as atoms and electrons, in physics we use quantum theory. This theory works well if we have very few systems, for example less than twenty, but it very quickly becomes too complicated after that. At that point, there are so many variables that one cannot compute or predict anything. Fortunately, atoms, electrons and other quantum systems are usually in very special states. This is because physical interactions are local, that is, an atom or an electron usually interacts only with its neighbors. As a consequence, these states can be described with less variables. This insight has been very useful to describe idealised quantum states (called pure states) for example, systems at zero temperature, or when we have perfect knowledge of the state of the system. For non-idealised states (called mixed states), it is more difficult to use the idea of locality. The main problem is the following. On the one hand, the state is positive, as this ensures that probabilities of measurement outcomes are positive. On the other hand, locality implies that local descriptions of the state require few variables. And these two conditions (positivity and locality) cannot be satisfied simultaneously: if we force the local description to be positive, then we have to deal with many variables. But if we allow the local description to be negative, then we may lose track of the positivity of the state. Fortunately, this problem is not exclusive to quantum theory, but appears in several areas of mathematics and data analysis. In fact, it appears whenever positivity interacts with locality. In this project, we will use these connections to other fields to develop the local descriptions of mixed states. Specifically, we will focus on the following three aspects. First, this problem may be milder in the approximate case. That is, if we are happy with a state which is close the exact one, there may be locally positive descriptions which do not use too many variables. Second, we will study the computational complexity of this problem. Computational complexity studies how difficult it is to solve problems by a machine some can be solved rapidly because there is a very good algorithm, others take a very long time, and others cannot be solved (they are undecidable). By studying the positivity problem from this perspective, we will gain insight into what is doable and what is difficult. Finally, we will study the positivity problem in the presence of symmetries. Symmetries are very important in physics, as they are linked to conservation laws. Symmetric systems are usually easier to describe, as they have less variables. But to locally represent the positivity and the symmetry gives rise to many interesting problems, which we will also explore.

Quantum theory predicts the outcomes of observables on quantum states. Observables can be thought of as questions, such as "Is the position of the quantum system here?" Quantum states are something that can give us these answers, albeit only probabilistically. It is unclear whether quantum states represent reality, our state of knowledge or both. However, observables have a complicated mathematical relation with quantum states, essentially because one can add or subtract observables, but only add quantum states. When the system has many parts, the relation on the whole can hardly be expressed as relations on the parts. Moreover, the relation depends on the internal dimension of the quantum system, which is related to the number of independent answers the system can give. If the internal dimension is one, we recover classical physics; if greater than one, it is quantum physics. Adventuring into quantum physics is like opening the door to a new universe, which is non-commutative, as observables and states are represented by mathematical objects that do not commute - A times B is different than B times A. This project sheds light on this relation from various perspectives. First, imagine that the quantum state has a symmetry -- for example, it is arranged on a circle so that shifting each system to the right results in the same state. While it is hard to witness this symmetry in the individual systems, we find that it is much easier to do it approximately. Second, any quantum state can be given in terms of simple `ideal' quantum states. This is very useful. We consider quantum states arranged on a grid, so that they define a quantum magic square. We find that they cannot be given in terms of simple or ideal quantum magic squares, in contrast to classical magic squares. Third, another mysterious fact about quantum theory is its reliance on complex numbers. We consider quantum theory over more extravagant numbers, the hypercomplex, and find that we can solve a long-standing problem in quantum information theory. It is unclear what this means for usual quantum theory. Fourth, we borrow techniques from computer science to study the hardness of some problems. We find that asking questions for a certain number of systems is often difficult (NP-hard), whereas doing so for any number is unsolvable (undecidable). Finally, we consider the relation for all internal dimensions at the same time, as well as other notions of positivity, i.e. of addition of states. We generalise many results to other notions of positivity. This highlights in what way the mathematical structure of quantum states is special. Overall, this project sheds light on the mathematical richness of the quantum world compared to the classical world.