next up previous
Next: Bibliography

My research has been developing into two main branches:

The entropy of Shannon measures the uncertainty of states of classical systems, while the dynamical entropy of Kolmogorov measures the unpredictability of the dynamics [1]. The role of the latter is threefold: the theorem of Shannon, Mc Millan and Breiman [2] 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 [3] in chaotic systems and Brudno's theorem [4] links it to the algorithmic complexity of their trajectories [5]. Quantum chaos [6] and quantum information theory [7] 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 [8]-[11], two of them look especially promising: the Connes-Narnhofer-Thirring (CNT) entropy [8], [12] and the Alicki-Lindblad-Fannes (ALF) entropy [9]. 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 [13], [14]. In particular, the CNT-entropy is based upon the notion of entropy of a subalgebra which was proved [15]- [17] 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 [18].

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 [7], 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 [19], [20]. Those local positive transformations that preserve positivity also of global entangled states are called completely positive and have a characteristic Stinespring-Kraus form [21]. 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 [22], [23]. 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 [24], which is embodied in the Kossakowski-Lindblad form [19],[25]. 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 [26]. 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 [27],[28]. In this case complete positivity can be proved to be necessary to avoid physical inconsistencies like the appearance of negative probabilities [29].


The standard setting is a finite dimensional quantum system with chaotic classical limit characterized by positive Lyapounov exponents. When the Hilbert space dimension goes to infinity, a semiclassical behaviour sets and the signature of chaos is a time-scale, during which classical and quantum behaviours are indistinguishable, which increases logarithmically in the Hilbert space dimension rather than as a power as would emerge from time-energy Heisenberg relations [6]. Despite being zero for finite quantum systems, both the CNT and ALF entropies show non-trivial semiclassical behaviours where the presence and role of the logarithmic time-scale will be studied. Concretely, the classical limit of quantum dynamical entropies will be studied for quantized hyperbolic automorphisms of the torus, the so-called quantum Arnold cat maps and for the perturbations which diminish the arithmetic rigidity of these systems.

For Bernoulli quantum sources, the Shannon-Mc Millan-Breiman theorem has a quantum counterpart given by Schumacher's theorem [2],[30],[31] which prescribes definite protocols how to reliably and maximally compress information by means of encodings via quantum states, the so-called qbits. In Schumacher's theorem, the von Neumann entropy plays the fundamental role, exactly as the Shannon entropy in the classical case. For ergodic quantum sources the results available so far indicate the von Neumann entropy per site, or mean entropy density, as the candidate for that role. For quantum spin chains with good decay of correlation functions and the shift on them as dynamics, both the CNT [8] and the ALF entropy [9] coincide with the mean entropy density. Thus, it will be investigated whether and how such models provide optimal protocols for such sources. Further, it will be examined the possibility of further applying the optimization analysis elaborated for the entropy of a subalgebra [15]-[17],[32] to other fundamental aspects of quantum information theory like the properties of the so-called entanglement of formation [33].

Following some recent applications of algorithmic complexity theory to the study of chaos [34],[35] and investigate the possible connections with quantum dynamical entropies. In particular, the ALF entropy offers a quantum way to compute the Kolmogorov entropy, by reducing the calculation to the evaluation of the von Neumann entropy of density matrices which model the evolving classical system and whose rank increases with time [36]. Such an approach looks particularly useful for discrete classical systems, with a finite number of possible configurations, where the measure theoretic approach of Kolmogorov entropy is not directly applicable [35]. In a sense, finitely discretizing continuous classical systems appears as a sort of quantization which leads to the suppression of chaos, which emerges again when the discretization goes back to the continuum. The presence of logarithmic time-scales in the behaviour of the ALF entropy will be studied for such classical discrete systems and its connections with Brudno's theorem [4] which examines the possibility to encode and to algorithmically compress the trajectories of classical ergodic systems.

The standard argument why it is necessary that physically acceptable linear transformations on the states of a quantum system S be completely positive is as follows. Considering the compound system S+S, the transformation acting as the given one on one of the two constituent systems and as the identity transformation on the other one, does not in general preserve the positivity of entangled states of S+S. Condition for the positivity of all states to be preserved is that the given transformation be completely positive in which case a theorem of Stinespring [21] completely characterizes its structure. The simplest positive transformation which is not completely positive is the transposition on the density matrices of a two-level system S: it does not alter the spectrum but if tensorized with the identity operation on another two-level system S, it generates negative eigenvalues when acting on the antisymmetric state of the compound system S+S. The identification of the role of the transposition as entanglement witness for bipartite two level systems and of positive, but not completely positive, linear maps in higher dimension, has been a major breakthrough in quantum information and communication theory [20]. However, if the considered transformation is the time-evolution of the system S, the tensor product with the identity operation on another system S looks somewhat abstract and the argument in favour of the necessity of complete positivity of the time-evolution loses much of its strength. More physically, the question arises whether such necessity may result by looking at the time evolution of the system S+S when both parties, not just one of them, evolve independently according to the same time-evolution . Such a situation is typical of many experimental contexts [37] where pairs of systems are in entangled states, do not interact and evolve according to time-evolutions that, in line of principle, as it happens for open quantum systems, need only be positivity preserving on states of the component systems. Notice that the problem is not trivial since the tensor product of the transposition with itself preserves the positivity of the antisymmetric state of two two-level systems [38].

In physical chemistry, when dealing with dissipative time-evolutions of molecules in heat baths, complete positivity is not accepted as necessary [26]. However, the markovian approximations that lead to reduced dynamics that are not completely positive, fail to preserve positivity of states as well [24]. However, even if not to semigroups which are positive on all possible states of single molecules, the markovian approximations may lead to positivity preserving time-evolutions on only those one-molecule states that emerge from the interaction with the environment after a transient [39] during which, due to memory effects, no Markov approximation is physically acceptable. It will be studied whether this escape route is still practicable when entangled states of two molecules evolve within a heat bath during a transient and according to a tensor product of two semigroups afterwards, which are acceptable in the single molecule case sketched above.

next up previous
Next: Bibliography
Fabio Benatti 2002-10-30