The Kählerian geometry of quantum model order reduction with applications in the simulation of open quantum systems

This article presents numerical techniques for simulating high-temperature and non-equilibrium quantum spin systems that are continuously measured and controlled. The notion of a “spin system” is broadly conceived, to encompass test masses as the limiting case of large-j spins, and in general the systems simulated are spatially inhomogeneous. The simulation technique has three stages: first the deliberate introduction of noise into the simulation, then the conversion of that noise into a informatically equivalent continuous measurement and control processes, and finally, projection of the trajectory onto a Kählerian state-space manifold having reduced dimensionality and possessing a Kähler potential of multilinear (i.e., product-sum) functional form; these state-spaces can be regarded as ruled algebraic varieties. To provide a unifying geometric context for this technique, the sectional curvature of ruled state-spaces is analyzed, and proved to be non-positive upon all sections that contain a rule. It is further shown that ruled state-spaces include the Slater determinant wave-functions of quantum chemistry as a special case and that these Slater determinant manifolds possess a Kähler-Einstein metric. It is suggested that the Riemannian curvature properties of ruled state-spaces generically account for the fidelity, efficiency, and robustness of projective trajectory simulation on these state-spaces. The resulting formalism is used to construct a positive P-representation for the thermal density matrix, and then to prove a formal equivalence between Caves’ amplifier noise limit and the standard quantum limit in test-mass monitoring. Single-spin detection by magnetic resonance force microscopy (MRFM) is then simulated, and the data statistics are shown to be those of a random telegraph signal with additive white noise, to all orders, in excellent agreement with experimental results. Then a larger-scale spin-dust model is simulated, having no spatial symmetry and no spatial ordering; the high-fidelity projection of numerically computed quantum trajectories onto low-dimensionality Kähler state-space manifolds is demonstrated. Finally, a method for accomplishing public key exchange via spin-dust simulations is demonstrated; this method establishes the geometric equivalence of a set of problems in physics, engineering, and information theory.