### Abstract

This chapter provides a review of the mathematical theory of linear quantum systems, which is based on the Hudson–Parthasarathy quantum stochastic calculus as a mathematical tool for describing Markov open quantum systems interacting with external propagating quantum fields. A precise definition of linear quantum systems is given as well as quantum stochastic differential equations representing their linear equation of motion in the Heisenberg picture. The important notion of physical realizability for linear quantum stochastic differential equations is introduced, and necessary and sufficient conditions for physical realizability reviewed. Complete parameterizations for linear quantum systems are given, and transfer functions defined. Also, the special class of completely passive linear quantum systems is introduced and the notion of stability for linear quantum systems is developed.

Original language | English |
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Title of host publication | Communications and Control Engineering |

Publisher | Springer International Publishing |

Pages | 35-71 |

Number of pages | 37 |

Edition | 9783319551999 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

### Publication series

Name | Communications and Control Engineering |
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Number | 9783319551999 |

ISSN (Print) | 0178-5354 |

ISSN (Electronic) | 2197-7119 |

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### Keywords

- Covariance
- Shale

### ASJC Scopus subject areas

- Computer Networks and Communications
- Control and Systems Engineering
- Control and Optimization

### Cite this

*Communications and Control Engineering*(9783319551999 ed., pp. 35-71). (Communications and Control Engineering; No. 9783319551999). Springer International Publishing. https://doi.org/10.1007/978-3-319-55201-9_2