Zero, pole and fragility of quantum systems

New fundamental limits on design and control of linear quantum systems are presented. A main difference between quantum and classical systems comes from the fact that quantum signals are represented by operator-valued vectors. The transfer function representation is useful to describe a system in the quantum setting for the same reason as it is useful classically. It is shown that the noncommutative property of quantum signals can be characterized by a specific relation between zeros and poles of a quantum transfer function. Then, two quantum mechanical tradeoffs follows from the transfer function constraint. One is a well known tradeoff, Heisenberg´s uncertainty relation, and the other is a tradeoff between the ability of noise reduction and sensitivity to modeling error. While these tradeoffs hold at each frequency, quantum systems have another constraint between different frequencies based on Bode integral as classical systems usually do. In the quantum case, however, this constraint is also dependent on quantum mechanical parameters.