Quantum control theory and applications: A survey <s> Linear quantum stochastic differential equations <s> "Elegantly written, with obvious appreciation for fine points of higher mathematics...most notable is [the] author's effort to weave classical probability theory into [a] quantum framework." - The American Mathematical Monthly "This is an excellent volume which will be a valuable companion both for those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students." - Mathematical Reviews An Introduction to Quantum Stochastic Calculus aims to deepen our understanding of the dynamics of systems subject to the laws of chance both from the classical and the quantum points of view and stimulate further research in their unification. This is probably the first systematic attempt to weave classical probability theory into the quantum framework and provides a wealth of interesting features: The origin of Ito's correction formulae for Brownian motion and the Poisson process can be traced to communication relations or, equivalently, the uncertainty principle. Quantum stochastic interpretation enables the possibility of seeing new relationships between fermion and boson fields. Quantum dynamical semigroups as well as classical Markov semigroups are realized through unitary operator evolutions. The text is almost self-contained and requires only an elementary knowledge of operator theory and probability theory at the graduate level. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Linear quantum stochastic differential equations <s> A quantum stochastic model for the Markovian dynamics of an open system under the nondemolition unsharp observation which is continuous in time, is given. A stochastic equation for the posterior evolution of a quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the case of a quasi-free Hamiltonian in the initial CCR algebra with a linear output channel, and the posterior dynamics corresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of a quantum oscillator that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Linear quantum stochastic differential equations <s> The purpose of this paper is to formulate and solve a H∞ controller synthesis problem for a class of non-commutative linear stochastic systems which includes many examples of interest in quantum technology. The paper includes results on the class of such systems for which the quantum commutation relations are preserved (such a requirement must be satisfied in a physical quantum system). A quantum version of standard (classical) dissipativity results are presented and from this a quantum version of the Strict Bounded Real Lemma is derived. This enables a quantum version of the two Riccati solution to the H∞ control problem to be presented. This result leads to controllers which may be realized using purely quantum, purely classical or a mixture of quantum and classical elements. This issue of physical realizability of the controller is examined in detail, and necessary and sufficient conditions are given. Our results are constructive in the sense that we provide explicit formulas for the Hamiltonian function and coupling operator corresponding to the controller. <s> BIB003
In the BLM (5), MME (13) and SME (14), we use the Schrödinger picture of quantum mechanics where equations describing the time dependence of quantum states are given. In some cases, it is more convenient to adopt the Heisenberg picture where the time-dependent operators on H are used in describing the quantum dynamics. An interesting case is a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology, especially in linear quantum optics BIB003 . This class of systems can be described by the following linear quantum stochastic differential equation (LQSDE) BIB003 : where A, B, C and D are, respectively, R n×n , R n×nw , R ny×n and R ny×nw matrices (n, n w , n y are positive integers), and x(t) = [x 1 (t) . . . x n (t)] T is a vector of self-adjoint possibly noncommutative system variables. The initial system variables x(0) = x 0 consist of operators (on an appropriate Hilbert space) satisfying the commutation relations where Θ jk are the components of a real antisymmetric matrix Θ. For simplicity, we may take Θ to have one of the following forms: . . , J), where 0 < n ′ ≤ n, n ′ is the number of classical variables, and J = [ 0 1 −1 0 ]. The vector quantity w describes the input signals and is assumed to admit the decomposition wherew(t) is the noise part of w(t) and β w (t) is a self-adjoint, adapted process (see BIB001 , BIB002 ). The noisew(t) is a vector of self-adjoint quantum noises with Itô table where Fw is a positive semidefinite Hermitian matrix (see BIB001 , BIB002 for details). Equation (18) can describe quantum systems such as linear quantum optical systems. However, it does not necessarily represent the dynamics of a meaningful physical system. We may need to add some additional constraints to ensure that a system described by equations of the form (18) is physically realizable (for details, see Section 4 or BIB003 ). 3 Controllability and open-loop control of quantum systems