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Within the framework of the Accardi-Fagnola-Quaegebeur (AFQ) representation free calculus of cite{b}, we consider the problem of controlling the size of a quantum stochastic flow generated by a unitary stochastic evolution affected by quantum noise. In the case when the evolution is driven by first order white noise (which includes quantum Brownian motion) the control is shown to be given in terms of the solution of an algebraic Riccati equation. This is done by first solving the problem of controlling (by minimizing an associated quadratic performance criterion) a stochastic process whose evolution is described by a stochastic differential equation of the type considerd in cite{b}. The solution is given as a feedback control law in terms of the solution of a stochastic Riccati equation.
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We introduce a concept of a quantum wide sense stationary process taking values in a C*-algebra and expected in a sub-algebra. The power spectrum of such a process is defined, in analogy to classical theory, as a positive measure on frequency space taking values in the expected algebra. The notion of linear quantum filters is introduced as some simple examples mentioned.
We show that every finite-dimensional quantum system with Markovian time evolution has an autonomous unitary dilation which can be dynamically decoupled. Since there is also always an autonomous unitary dilation which cannot be dynamically decoupled, this highlights the role of dilations in the control of quantum noise. We construct our dilation via a time-dependent version of Stinespring in combination with Howlands clock Hamiltonian and certain point-localised states, which may be regarded as a C*-algebraic analogue of improper bra-ket position eigenstates and which are hence of independent mathematical and physical interest.
Based on the Chen--Moller--Sauvaget formula, we apply the theory of integrable systems to derive three equations for the generating series of the Masur--Veech volumes ${rm Vol} , mathcal{Q}_{g,n}$ associated with the principal strata of the moduli spaces of quadratic differentials, and propose refinements of the conjectural formulas given in [12,4] for the large genus asymptotics of ${rm Vol} , mathcal{Q}_{g,n}$ and of the associated area Siegel--Veech constants.
This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen-Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.
Quadratic matrix equations occur in a variety of applications. In this paper we introduce new permutationally invariant functions of two solvents of the n quadratic matrix equation X^2- L1X - L0 = 0, playing the role of the two elementary symmetric functions of the two roots of a quadratic scalar equation. Our results rely on the connection existing between the QME and the theory of linear second order difference equations with noncommutative coefficients. An application of our results to a simple physical problem is briefly discussed.
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