Gaussian Open Quantum Dynamics and Isomorphism to Superconformal Symmetry

Understanding the Lie algebraic structure of a physical problem often makes it easier to find its solution. In this paper, we focus on the Lie algebra of Gaussian-conserving superoperators. We construct a Lie algebra of $n$-mode states, $\mathfrak{go}(n)$, composed of all superoperators conserving Gaussianity, and we find it isomorphic to $\mathbb{R}^{2n^2+3n}\oplus_{\mathrm{S}}\mathfrak{gl}(2n,\mathbb{R})$. This allows us to solve the quadratic-order Redfield equation for any, even non-Gaussian, state. We find that the algebraic structure of Gaussian operations is the same as that of super-Poincar\'e algebra in three-dimensional spacetime, where the CPTP condition corresponds to the combination of causality and directionality of time flow. Additionally, we find that a bosonic density matrix satisfies both the Klein-Gordon and the Dirac equations. Finally, we expand the algebra of Gaussian superoperators even further by relaxing the CPTP condition. We find that it is isomorphic to a superconformal algebra, which represents the maximal symmetry of the field theory. This suggests a deeper connection between two seemingly unrelated fields, with the potential to transform problems from one domain into another where they may be more easily solved.