Symmetric Spaces of Qubits and Gaussian Modes

The understanding of the properties of multipartite systems is a long-standing challenge in quantum theory that signals the need for new ideas and alternative frameworks that can shed light on the intricacies of quantum behavior. In this work, we argue that symmetric spaces provide a common language to describe two-qubit and two-mode Gaussian systems. Our approach relies on the use of equivalence classes that are defined by a subgroup of the maximal symmetry group of the system and involves an involution which enables the Cartan decomposition of the group elements. We work out the symmetric spaces of two qubits and two modes to identify classes which include an equal degree of mixing states, product states, and X states, among others. For three qubits and three modes, we point out how the framework can be generalized and report partial results about the physical interpretations of the symmetric spaces.