标题： 度量和模糊空间上的测地线 显示英文标题

标题： Metrics and geodesics on fuzzy spaces

摘要： 我们研究模糊空间（作为非对易流形的特殊例子），并研究其准相干态，以找到它们相关的度量。 我们证明它们自然地具有两个自然的“量子度量”，这些度量与“路径”的量子涨落有关。 第一个度量提供平均路径的长度，而第二个度量提供涨落路径的平均长度。 对于与准相干态相关的经典流形（即在最小化其量子不确定性的状态中坐标可观测量的平均值的流形），这两个度量提供了两个最小化测地线方程。 此外，由于模糊空间不是无挠的，我们在将探测器移动到模糊空间时，还存在两个不同的与不同绝热过程相关的测地线方程。 我们将这些数学结果应用于BFSS矩阵模型中的量子引力，以及受大量子环境噪声影响的受控量子比特的量子信息理论。 显示英文摘要

摘要： We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural "quantum metrics" which are associated with quantum fluctuations of "paths". The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment.