- QUANTUM ENTROPIES, CHAOS AND INFORMATION
The entropy of Shannon measures the uncertainty of states of classical systems, while the dynamical entropy of Kolmogorov measures the unpredictability of the dynamics. The role of the latter is threefold: the theorem of Shannon, Mc Millan and Breiman links it to the maximal reliable compression rate of the information emitted by classical stationary ergodic sources; Ruelle and Pesin theorems link it to the positive Lyapounov exponents in chaotic systems and Brudno's theorem links it to the algorithmic complexity of their trajectories. Quantum chaos and quantum information theory ask for an extension of Kolmogorov entropy. While the quantum partner of Shannon entropy is the entropy of von Neumann, when the dynamics is concerned, the extension is not unique. Among the various proposals, two of them look especially promising: the Connes-Narnhofer-Thirring (CNT) entropy, and the Alicki-Lindblad-Fannes (ALF) entropy. Such entropies are useful notions also for classical systems in which case they coincide with Kolmogorov entropy. Instead, they are zero for finite quantum systems, non zero and in some cases different for infinite quantum systems. In particular, the CNT-entropy is based upon the notion of entropy of a subalgebra which was proved to be related to the maximal accessible information of a quantum transmission channel, and to the so-called entanglement of formation which measures the entanglement content of a mixed state of a bipartite system and is of primary importance in quantum communication theory.
- COMPLETE POSITIVITY AND OPEN QUANTUM SYSTEMS
Quantum states are density matrices whose positive eigenvalues are probabilities; physically consistent state transformations must preserve the positivity of the spectrum. In the typical protocols of quantum cryptography, quantum communication and quantum teleportation, two partners share two particles in a globally entangled state and are allowed local quantum operations on their own party and classical communication between them. Positive local operations on one party can fail to preserve the positivity of global entangled states of the two parties. Those local positive transformations that preserve positivity also of global entangled states are called completely positive and have a characteristic Stinespring-Kraus form. The time-evolution is the most natural state transformation; when generated by a Hamiltonian operator the transformations of the corresponding dynamical group are authomatically completely positive. This is no longer true for the time-evolutions of open quantum systems weakly interacting with suitable environments. These systems exhibit typical dissipative and damped behaviours described by semigroups of linear maps which are obtained by eliminating the environment degrees of freedom and by operating suitable Markov approximations. Remarkably, Markov approximations yielding positivity preserving dynamical maps, ensure complete positivity, too, which is embodied in the Kossakowski-Lindblad form. Consequently, the characteristic decay times of the open quantum systems obey certain order relations which are sometimes disputed as technical artifacts rather than physical necessities. In the context of elementary particle physics, the role of complete positivity in relation to quantum entanglement has been studied in the case of neutral mesons produced in entangled states and evolving independently, each of them as an open quantum system subjected to the fluctuations of a background of gravitational origin. In this case complete positivity can be proved to be necessary to avoid physical inconsistencies like the appearance of negative probabilities.
Some of my publications are listed here