Correlation constraints and the Bloch geometry of two qubits

Abstract

We present a novel inequality on the purity of a bipartite state depending solely on the difference of the local Bloch vector lengths. For two qubits this inequality is tight for all marginal states and so extends the previously known solution for the 2-qubit marginal problem and opens a new research avenue. We further use this inequality to construct a 3-dimensional Bloch model of the 2-qubit quantum state space in terms of Bloch lengths, thus providing a geometrically pleasing visualization of this difficult to access high-dimensional state space. This allows to characterize quantum states relying on a strongly reduced set of parameters alone and to investigate the interplay between local properties of the marginal systems and global properties encoded in the correlations.