Open Systems & Information Dynamics

We present the reduced operator approximation: a simple, physically transparent and computationally efficient method of modelling open quantum systems. It employs the Heisenberg picture of the quantum dynamics, which allows us to focus on the system degrees of freedom in a natural and easy way. We describe different variants of the method, low- and high-order in the system–bath interaction operators, defining them for either general quantum harmonic oscillator baths or specialising them for independent baths with Lorentzian spectral densities. Its wide applicability is demonstrated on the examples of systems coupled to different baths (with varying system–bath interaction strength and bath memory length), and compared with the exact pseudomode and the popular quantum state diffusion approach. The method captures the decoherence of the system interacting with the bath, while conserving the total energy. Our results suggest that quantum coherence effects persist in open quantum systems for much longer times than previously thought.

The aim of this paper is to derive the global Hamiltonian form for a quantum system and bath, or more generally a quantum network with multiple quantum input field connections, based on the local descriptions. We give a new simple argument which shows that the global Hamiltonian for a single Markov component arises as the singular perturbation of the free translation operator. We show that the Fermi analogue takes an equivalent form provided the parity of the coefficients is correctly specified. This allows us to immediately extend the theory of quantum feedback networks to Fermi systems.

In quantum compound systems, joint (or compound) states usually do not exist [26]. The so-called Ohya compound state was defined by using the von Neumann–Schatten decomposition [23] of an input state and a quantum communication channel. Quantum mutual entropy [10] was introduced by using quantum relative entropy [24] between the Ohya compound state and the trivial (product) compound state on the compound (input and output) system. One of the remarkable results related to state entanglement is the Jamiolkowski isomorphism [5]. In this paper, we use the Jamiolkowski isomorphism for the construction of Ohya compound state and we discuss the existence of a completely positive map between compound entangled states and the Ohya compound states. The efficiency of information transmission is investigated by using the mean entropy and the mean mutual entropy [12, 6] of a connected channel.

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3×3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

A probabilistic approach to decision-making is developed in which the states of the underlying stochastic process, assumed to be of the Markov type, represent the competing options. The principal parameters determining the dominance of a particular option versus the others are identified and the transduction of information associated to the transitions between states is quantified using a set of entropy-like quantities.