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We consider a controlled quantum system whose finite dimensional state is governed by a discretetime nonlinear Markov process. In open-loop, the * This work was supported in part by the "Agence Nationale de la Recherche" (ANR), Project QUSCO-INCA, Projet Jeunes Chercheurs EPOQ2 number ANR-09-JCJC-0070 and Projet Blanc CQUID number 06-3-13957, and measurements are assumed to be quantum nondemolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence of the controlled system to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system.

In this note, we show that the operator theoretic concept of Kolmogorov numbers and the number of degrees of freedom at level ǫ of a communication channel are closely related. Linear communication channels may be modeled using linear compact operators on Banach or Hilbert spaces and the number of degrees of freedom of such channels is defined to be the number of linearly independent signals that may be communicated over this channel, where the channel is restricted by a threshold noise level. Kolmogorov numbers are a particular example of s-numbers, which are defined over the class of bounded operators between Banach spaces. We demonstrate that these two concepts are closely related, namely that the Kolmogorov numbers correspond to the "jump points" in the function relating numbers of degrees of freedom with the noise level ǫ. We also establish a useful numerical computation result for evaluating Kolmogorov numbers of compact operators. I. INTRODUCTION The number of degrees of freedom of a communication channel plays an important role in evaluating the channel's Shannon capacity (see e.g. [1]). In a physical communication system, because of con-straints such as finite transmission power and noise at the receiver, only finitely many (linearly independent) signals may be exchanged between the transmitter and receiver. This physical intuition may be captured using the concept of number of degrees of freedom of the communication channel. The number of degrees of freedom of a communication channel has been used in evaluating the Shannon capacity for several physically realistic channels 1 (see e.g. [1], [2], [3], [4]). This concept has also been used in various other problem domains such as multi-antenna communication [5], [6], [7], optics [8], [9] and electromagnetic field sampling [10]. Kolmogorov numbers are a particular example of so called s-number sequences. In the theory of snumbers one associates with every bounded linear operator T , mapping between any two Banach spaces, a scalar sequence s 1 (T ) ≥ s 2 (T ) ≥ . . . ≥ 0 (see e.g. [11], [12]). A classical example in the more 1 The concept of number of degrees of freedom is used implicitly in evaluating the capacity of Shannon's classical bandwidth limited, additive Gaussian white noise channel. In particular it can be shown that as the bandwidth becomes large the number of degrees of freedom at level ǫ approaches the well know constant 2W T , for all noise levels ǫ (see Gallager [4, Ch. 8]). DRAFT restricted category of compact operators mapping between Hilbert spaces is the sequence of singular values. s-number sequences of various types have primarily been used to classify operators based on the behaviour of the sequence s n as n → ∞. In particular, various interesting operator ideals have been obtained based on the behaviour of these sequences. In this note we establish the connection between Kolmogorov numbers and degrees of freedom of a communication channel. The remainder of this note is organised as follows: in Section II, we recall the definitions of ...

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