Linear Dynamical Quantum Systems by Hendra I. Nurdin & Naoki Yamamoto

Author:Hendra I. Nurdin & Naoki Yamamoto
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

4.1 Quantum Conditional Expectations

4.1.1 Quantum Probability Space

In the framework of quantum filtering, we will only consider a restricted notion of quantum conditional expectations, as linear projective mappings from the space of observables to an algebra of commutative observables. Hence, we first need to specify such an algebra of observables and the notion of a state on this algebra. For the purpose of illustration and to emphasize the basic underlying idea, in the following we discuss only the case of finite-dimensional systems, wherein the dimension of the Hilbert space is finite-dimensional and an observable can be represented by a complex matrix. Quantum harmonic oscillators are thus, strictly speaking, excluded because they are defined on an infinite-dimensional Hilbert space. Nonetheless, the notions and results below have an infinite-dimensional analogue, which readers can find in, e.g., [1] and the references cited therein.

The content we describe here partly follows [2]. First recall that a quantum mechanical random variable is represented by a linear self-adjoint operator on a Hilbert space. This axiom means that quantum random variables in general do not commute with one another, and thus, the conventional classical probability space needs to be replaced by a quantum probability space, which is composed of a -algebra and a state defined as follows.

Definition 4.1

( -algebra) Let be a finite-dimensional complex Hilbert space. A -algebra is a set of linear operators such that for any and . is called commutative if for any .

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