Temporal and spatio-temporal correlations in quantum physics

The correlations that can be realized within quantum mechanics differ from those of classical phy- sics. For example, quantum physics allows two parties that are situated at faraway places to ob- serve the following spatial correlations. When one party performs an arbitrary local measurement on the joint system, the other party instantaneously observes the opposite result when implemen- ting the same measurement, but for certain other measurement settings the outcomes are comple- tely random. This has been called spooky action at a distance by Albert Einstein, and lead to the development of Bell inequalities which can distinguish among quantum mechanics and classical theories. Correlations form a pillar of every physical theory as the only access we have to our surrounding world is by performing measurements and interpreting the results. It is then natural to investigate the correlations among successive measurements or local measurements by spatially separated parties. The study of correlations has allowed to obtain profound insight into physical theories, like quantum mechanics, and is also important for applications to understand, for example, why by using quantum mechanical systems one can outperform classical systems and to use the gained knowledge to find new applications. Spatial correlations such as Bell non-locality and entanglement have been shown to be useful for applications, like quantum communication and computation. Bell non-locality can also be used to detect entanglement or to certify that a joint system has all the desired properties, which is called self-testing. In this project entitled Temporal and spatio-temporal correlations in quantum physics the correla- tions arising from sequences of measurements on a single quantum system (temporal correlations) and sequences of local measurements performed by spatially separated parties (spatio-temporal correlations) are studied with the aim to investigate their usefulness and to gain a better under- standing. In particular, their usability for tasks, such as the estimation of the purity of a quantum system, entanglement detection and self-testing will be explored. Moreover, systems that show maximal spatio-temporal correlations and their entanglement properties will be characterized. By this, as well as the insight gained through the investigation of their role for entanglement detection, there will be shed light on the relation among entanglement and spatio-temporal correlations. In order to show the potential of temporal and spatio-temporal correlations for the proposed tasks, inequalities based on certain correlations will be devised which allow to certify entanglement or attain their maximum if the system has certain properties. In summary, the goal of this project is to show new applications of temporal and spatio-temporal correlations and to obtain insight into the relation among spatial and spatio-temporal correlations.

The study of correlations in the spatial realm has significantly advanced our understanding of quantum physics and has also lead to applications such as quantum communication protocols and randomness certification. In the temporal scenario -when sequences of measurements are considered- relevant questions concerning the structure of quantum correlations were lacking an answer and (spatio)-temporal correlations held promise to enable tasks beyond answering foundational questions and dimension witnessing. To find answers to these questions and applications was precisely the main goal of my research project "Temporal and spatio-temporal correlations in quantum physics". As an example for such applications I derived a witness for the purity of a quantum state based on temporal correlations. Combining this witness with known results from entanglement theory further allows one to deduce an upper bound on an entanglement measure from the temporal correlations observed on a subsystem. We also investigated the structure of temporal correlations. In particular, we showed that for systems with small dimension the set of temporal correlations is in general non-convex. Employing this non-convexity we could provide non-linear criteria which can certify a lower bound on the dimension and we studied the dimension required to realize extremal correlations. These results deepen our understanding on the structure of temporal correlations and their dimension-dependence. The correlations arising from sequences of local measurements on bipartite systems may reveal non-local behavior of states which appear local if subjected to a single measurement round. However, as I could show, if previous outcomes and settings can influence the measurements of any party in the next round no advantage can be gained beyond considering the non-locality of the state obtained after a single round (known as hidden non-locality). Quantum combs provide a way of storing the information about temporal quantum processes in multipartite quantum states. We could relate their bipartite entanglement properties to properties of the process and exemplify that well known entanglement classes and phenomena also exist for combs. Experimental advances allow nowadays for the distribution of bipartite entangled state with high fidelity. Also in view of the "Quantum internet" this poses the question what states or correlations can be generated if the sources distribute their states according to some network structure to the parties. This topic has attracted recently much interest. In particular, a semidefinite test which can detect that certain correlations cannot originate from a given network structure has been developed. This test can be evaluated numerically, which however becomes infeasible for large networks. In our work we used results from coherence theory to solve this test analytically for important cases.