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arXiv:1907.11942 · Entropy

A New Mechanism of Open System Evolution and Its Entropy Using Unitary Transformations in Noncomposite Qudit Systems

Julio A. López-Saldívar, Octavio Castaños, Margarita A. Man’ko, Vladimir I. Man’ko

Published Jul 27, 2019• 9 citations
arXivHTMLPDF

Abstract

The evolution of an open system is usually associated with the interaction of the system with an environment. A new method to study the open-type system evolution of a qubit (two-level atom) state is established. This evolution is determined by a unitary transformation applied to the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels employing unitary transformations into the qutrit system. In particular, we study the phase damping and spontaneous-emission quantum channels. In addition, we mention a proposal for quasiunitary transforms of qubits, in view of the unitary transform of the total qutrit system. The experimental realization is also addressed. The probability representation of the evolution and its information-entropic characteristics are considered.

DOI
10.3390/e21080736

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The evolution of an open system is usually associated with the interaction of the system with an environment. A new method to study the open-type system evolution of a qubit (two-level atom) state is established. This evolution is determined by a unitary transformation applied to the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels employing unitary transformations into the qutrit system. In particular, we study the phase damping and spontaneous-emission quantum channels. In addition, we mention a proposal for quasiunitary transforms of qubits, in view of the unitary transform of the total qutrit system. The experimental realization is also addressed. The probability representation of the evolution and its information-entropic characteristics are considered.
Abstract We present a review of the general approach to the problem of correlations in classical statistics and quantum statistics of systems with different numbers of subsystems and demonstrate the information-entropic relations for systems without subsystems recently obtained for Shannon entropies. We present the example of a single-qudit state corresponding to the N-level atom, consider explicitly the qutrit state, and show that qutrit can be interpreted as a set of several qubits. For each of these qubits, there exist corresponding von Neumann entropies, and constraints for these entropies determine the hidden correlations between the qubits in spite of the fact that the qutrit does not contain any subsystem. These constraints are expressed in terms of nonnegativity of the mutual information introduced, which usually exists only for the states of systems with subsystems. The value of information parameterizes the hidden correlations of artificial qubits in the system. We discuss examples of some qudits.
Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established. The entropy of a channel, also called the map entropy, is defined as the entropy of the state corresponding to the channel by the Jamiolkowski isomorphism. In Part III of the thesis, the additivity of the entropy of a channel is proved. The minimal output entropy, which is difficult to compute, is estimated by an entropy of a channel which is much easier to obtain. A class of quantum channels is specified, for which additivity of channel capacity is conjectured. The last part of the thesis contains characterization of Davies channels, which correspond to an interaction of a state with a thermal reservoir in the week coupling limit, under the condition of quantum detailed balance and independence of rotational and dissipative evolutions. The Davies channels are characterized for one-qubit and one-qutrit systems.
An important problem in quantum technologies is the generation of a target evolution of an open quantum system. A core component of this task is the ability to determine whether the actual evolution of the system coincides with the target evolution. For the generation of unitary quantum channels in open quantum systems, it was shown by Goerz, Reich and Koch [New J. Phys. {\bf 16} 055012 (2014)] that for determining whether the actual evolution coincides with a desired unitary it is sufficient to compare their action on three special density matrices. In this work, we consider controlled generation and unique determination of non-unitary quantum channels in open quantum systems with particular emphasis on rank two single-qubit quantum channels. We prove that to uniquely determine such quantum channels it is sufficient to consider their action on only three suitably chosen density matrices. Based on this theoretical result, we numerically investigate generation of various non-unitary target single-qubit quantum channels in open quantum systems using coherent and incoherent controls.
We study a hybrid quantum open system consisting of two interacting subsystems formed by one two-level atom (qubit) and one three-level atom (qutrit). The quantum open system is coupled to an external environment (cavity) via the qubit-cavity interaction. It is found that the feedback control on different parts of the system (qubit or qutrit) gives dramatically different asymptotical behaviors of the open system dynamics. We show that the local feedback control mechanism acting on the qutrit subsystem is superior than that on the qubit in the sense of improving the entanglement. Particularly, the qutrit-control scheme may result in an entangled steady state, depending on the initial state.
The evolution of complex composite quantum systems which can be reduced to systems with a small dimensionality of the Hilbert space (of the qutrit, ququart type, etc.) is considered. In the case of the interaction of an ensemble of two-level atoms with light, the conditions are found under which a qutrit is produced from the light and atomic states. The properties and possible applications of a qutrit based of the Fock states of light in which two photons are distributed among three modes are discussed. It is shown that this state has the nonclassical photon statistics, is entangled and can be used as a quantum channel for the teleportation, dense coding, and key distribution.