Dynamical
A and
B maps have been employed extensively by Sudarshan and co-workers to investigate open-system evolution of quantum systems. A canonical structure of the
A map is introduced here. It is shown that this canonical
A map enables us to investigate whether the dynamics is completely positive (CP) or not completely positive (NCP) in an elegant way and, hence, it subsumes the basic results on open-system dynamics. Identifying memory effects in open-system evolution is gaining increasing importance recently and, here, a criterion of non-Markovianity, based on the relative entropy of the dynamical state is proposed. The relative entropy difference of the dynamical system serves as a complementary characterization---though not related directly---to the fidelity difference criterion proposed recently. Three typical examples of open-system evolution of a qubit, prepared initially in a correlated state with another qubit (environment), and evolving jointly under a specific unitary dynamics---which corresponds to a NCP dynamical map---are investigated by employing both the relative entropy difference and fidelity difference tests of non-Markovianity. The two-qubit initial states are chosen to be (i) a pure entangled state, (ii) the Werner state, which exemplifies both entangled and separable states of qubits, depending on a real parameter, and (iii) a separable mixed state. Both the relative entropy and fidelity criteria offer a nice display of how non-Markovianity manifests itself in all three examples.
The role of completely positive--indivisibility (CP-indivisibility) and incompatibility as valuable resources for various information-theoretic tasks is widely acknowledged. This study delves into the intricate relationship between CP-divisibility and channel compatibility. Our investigation focuses on the behavior of incompatibility robustness of quantum channels for a pair of generic dynamical maps. We show that the incompatibility robustness of channels is monotonically nonincreasing for a pair of generic CP-divisible dynamical maps. Further, our explicit study of the behavior of incompatibility robustness with time for some specific dynamical maps reveals nonmonotonic behavior in the CP-indivisible regime. Additionally, we propose a measure of CP-indivisibility based on the incompatibility robustness of quantum channels. Our investigation provides valuable insights into the nature of quantum dynamical maps and their relevance in information-theoretic applications.
The pure quantum entanglement is generalized to the case of mixed compound states to include the classical and quantum encodings as particular cases. The true quantum entanglements are characterized as transpose-CP but not CP maps. The entangled information is introduced as the relative entropy of the mutual and the input state and total information of the entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the supremum of the information over all c-entanglements, and the true quantum entropy, which is achieved at the standard entanglement. The q-capacity, defined as the supremum over all entanglements, doubles the c-capacity in the case of the simple algebra. The conditional q-entropy is positive, and q-information of a quantum channel is additive.