Aplicações do teorema de Shemesh e dos Conceitos de subespaço invariante e canal quântico não-ergódico na teoria da informação quântica erro-zero.

Abstract

Quantum Information Theory is a science that utilizes the paradigms of Quan tum Mechanics to study the ultimate limits of processing and transmitting informa tion through a quantum channel. One of its sub-area of research is the capacity of quantum channels, which is understood as the supremum of rates at which the prob ability of error asymptotically tends to zero as the code length approaches infinity when information is transmitted through quantum channels. In certain contexts, there is interest in studying the capacity of quantum channels to transmit infor mation with an error probability exactly equal to zero. In this case, the channel is said to have positive or non-trivial zero-error capacity. For a quantum chan nel to transmit information with an error probability exactly equal to zero, certain conditions must be satisfied. With the proposal of the definition of zero-error capac ity of a quantum channel in the first decade of this century, a necessary condition for the zero-error capacity of a quantum channel was demonstrated, based on the orthogonality of quantum states at the channel output. More recently, in 2019, an other condition for the zero-error capacity of quantum channels was proven, based on the orthogonality of quantum states with the subspace spanned by all pairwise products of Kraus operators representing the quantum channel. Following the line of proposing conditions for the zero-error capacity of quantum channels, this the sis focuses on the study of mathematical conditions for quantum channels to have zero-error capacity. In this regard, a capacity condition is presented based on the common eigenstates of the Kraus operators representing the quantum channel. It is also proven that quantum channels with common invariant subspaces are capable of transmitting information with an error probability exactly equal to zero. Con tinuing the emphasis on the concept of zero-error capacity of quantum channels, a class of quantum channels with positive zero-error capacity, called non-ergodic quan tum channels, is presented. Additionally, some connections between the concept of zero-error capacity of a quantum channel and the Shemesh Theorem are discussed.