Shemesh Theorem and Its Relation With the Zero-Error Quantum Information Theory

In this paper, our objective is to establish results that connect Shemesh theorem with quantum zero-error information theory. To achieve this, we will first introduce definitions and properties related to the zero-error capacity of a quantum channel, with particular emphasis on demonstrating that the eigenstates common to the Kraus operators representing the quantum channel are fixed points. This result has previously been employed to develop a test for determining whether a quantum channel possesses positive zero-error capacity. In this work, we demonstrate that the eigenstates common to the Kraus operators representing the channel reside within the Shemesh subspace. Consequently, the fixed points of the quantum channel act as a bridge between Shemesh theorem and quantum zero-error information theory, thereby enabling the derivation of new results on the zero-error capacity of quantum channels.