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.
Sections 2.3, 2.4, and 2.7 contain reprinted excerpt with permission from [18]. Copyright (2010) by the American Physical Society.
Section 2.5.3 contains some materials reprinted, with permission, from [25] \(\copyright \) 2014 IEEE.
Section 2.7.2 contains some materials reprinted from [19] \(\copyright \) 2014 with permission of Springer.
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Notes
- 1.
Technically, the existence of a unique solution of the Hudson–Parthasarathy QSDE that is unitary is guaranteed whenever the coefficients S, L, H are all bounded operators. When they are unbounded then additional technical assumptions need to be assumed, see, e.g., [8].
- 2.
We take the common convention that units are taken such that \(\hbar =1\). Also, we will take (2.3) as the default CCR for the position and momentum operators of multiple distinct oscillators. However, it is easy to adapt the results of this chapter for a different definition of these operators that satisfy a different set of commutation relations; see Remark 2.2.
- 3.
To see why this is the case, consider a complex \(k\times k\) matrix \(E\ge 0\) with \(E_{11}=0\). Taking \(u\left( t\right) =\left( tx_{1},x_{2},\ldots ,x_{k}\right) ^{\top }\) we have that \( 0\le u\left( t\right) ^{*}E u\left( t\right) =2t\)Re{\( \sum _{j>1}x_{1}^{*}E_{1j}x_{j}+\sum _{j,k>1}x_{j}^{*}E_{jk}x_{k}\}\). However, for this inequality to hold for all real t it must be that \(\mathfrak {R}\{\sum _{j>1}x_{1}^{*}E_{1j}x_{j}\}=0\). Now, replacing t with \(\imath t\) shows that \(\mathfrak {I}\{ \sum _{j>1}x_{1}^{*}E_{1j}x_{j}\}\) also vanishes. As this is true for any \(x_{j}\), it must be the case that \(E_{1j}=E_{j1}^{*}=0\) for all \(j>1\).
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Nurdin, H.I., Yamamoto, N. (2017). Mathematical Modeling of Linear Dynamical Quantum Systems. In: Linear Dynamical Quantum Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55201-9_2
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