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Proceedings of the Steklov Institute of Mathematics

Generating Quantum Channels

Renat Nelsonovich Gumerov, Ruslan Linarovich Khazhin

Published Mar 1, 2024• 2 citations
Publisher

Abstract

Abstract For composite quantum systems, we consider quantum channels that uniquely determine the channels of the subsystems. Such channels of composite systems are called generating channels. Examples of generating channels are given by tensor products of two quantum channels of subsystems and their convex combinations. The paper deals with the properties of generating channels. In particular, we show that these channels form a convex compact set in the norm topology. We prove a criterion for a quantum channel to be generating. For composite systems consisting of two qubits, we construct generating phase-damping channels. For subsystems, these channels generate both phase-damping channels and depolarizing channels. Examples of nongenerating phase-damping channels are also presented.

DOI
10.1134/s0081543824010085

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Abstract For composite quantum systems, we consider quantum channels that uniquely determine the channels of the subsystems. Such channels of composite systems are called generating channels. Examples of generating channels are given by tensor products of two quantum channels of subsystems and their convex combinations. The paper deals with the properties of generating channels. In particular, we show that these channels form a convex compact set in the norm topology. We prove a criterion for a quantum channel to be generating. For composite systems consisting of two qubits, we construct generating phase-damping channels. For subsystems, these channels generate both phase-damping channels and depolarizing channels. Examples of nongenerating phase-damping channels are also presented.
We consider one-parameter families of generating quantum channels. Such families are called the generating quantum dynamical mappings or the generating quantum processes. By the generating channels of composite quantum systems, we understand the channels that allow the channels of constituent subsystems, called the induced channels, to be uniquely defined. Using the criterion for generating and induced linear mappings, we study the properties of bijective quantum channels and the properties of quantum processes consisting of such channels. Using a generating quantum dynamical mapping, we naturally construct the induced dynamical mapping. We show that the properties of continuity and completely positive divisibility of generating quantum dynamical mappings are hereditary for induced dynamical mappings. As an application of the obtained results, we construct continuous completely positive evolutions. For generating quantum dynamical mappings taking values in the set of phase-damping channels, we obtain a criterion for the completely positive divisibility.
It is commonly stated that decoherence in open quantum systems is due to growing entanglement with an environment. In practice, however, surprisingly often decoherence may equally well be described by random unitary dynamics without invoking a quantum environment at all. For a single qubit, for instance, pure decoherence (or phase damping) is always of random unitary type. Here, we construct a simple example of true quantum decoherence of two qubits: we present a feasible phase damping channel of which we show that it cannot be understood in terms of random unitary dynamics. We give a very intuitive geometrical measure for the positive distance of our channel to the convex set of random unitary channels and find remarkable agreement with the so-called Birkhoff defect based on the norm of complete boundedness.
This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings is analyzed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assign preconditioners to matrices in numerical linear algebra. Thus, the known pre-conditioner maps are viewed as quantum channels in finite dimensions. In addition, the infinite-dimensional analogue of preconditioners is introduced and certain limit theorems are discussed; this is with an aim to analyze asymptotic methods in quantum channels analogous to problems in asymptotic linear algebra.
Special sets of states, called optimal, which are related to the Holevo capacity and to the minimal output entropy of a quantum channel, are considered. By methods of convex analysis and operator theory, structural properties of optimal sets and conditions of their coincidence are explored for an arbitrary channel. It is shown that strong additivity of the Holevo capacity for two given channels provides projective relations between optimal sets for the tensor product of these channels and optimal sets for the individual channels.
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.