Geometry and Reconstruction of Bipartite Quantum Correlations

Abstract

The first part of this thesis provides a mathematical description for bipartite quantum correlations, aiming to analyze the geometry of several sets of correlations. We explain why quantum entanglement can be used to simulate shared randomness: Cloc(Γ) ⊆ Cqd(Γ) for a sufficiently large d. The known bound for this dimension d in the literature is d ≥ dim(Cloc(Γ))+1, but we improve this by showing that the inclusion is always true for d ≥ dim(Cloc(Γ)). For the proof of this bound, we show that the set Cprivate(Γ) of correlations using private randomness is connected, which allows the use of an improved version of Carathéodory’s Theorem. In the second part of this thesis, we define and analyze a see-saw method to determine the state and measurement operators that reconstruct both the correlation itself as its entanglement dimension, by solving consecutive semidefinite programs. One of the strengths of the algorithm is its generality: it applies to different dimensions, question sets, and answer sets. Some numerical experiments demonstrated that the method can indeed reconstruct quantum correlations, although some highly entangled correlations failed to be reconstructed due to the computationallimitations. The numerical experiments motivated several new theorems, for example the fact that every correlation with |A|= 1 or |B|= 1 has entanglement dimension 1, which means that it can be written as a private randomness correlation. The proof of this result is based on the earlier described improvement for the dimension d.