Abstract:
Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.

Abstract:
We show that the Fano representation leads to a particularly simple and appealing form of the quantum process tomography matrix $\chi_{_F}$, in that the matrix $\chi_{_F}$ is real, the number of matrix elements is exactly equal to the number of free parameters required for the complete characterization of a quantum operation, and these matrix elements are directly related to evolution of the expectation values of the system's polarization measurements. These facts are illustrated in the examples of one- and two-qubit quantum noise channels.

Abstract:
The diamond norm measures the distance between two quantum channels. From an operational vewpoint, this norm measures how well we can distinguish between two channels by applying them to input states of arbitrarily large dimensions. In this paper, we show that the diamond norm can be conveniently and in a physically transparent way computed by means of a Monte-Carlo algorithm based on the Fano representation of quantum states and quantum operations. The effectiveness of this algorithm is illustrated for several single-qubit quantum channels.

Abstract:
We propose a quantum algorithm for the purification of a generic mixed state $\rho$ of a $n$-qudit system by using an ancillary $n$-qudit system. The algorithm is optimal in that (i) the number of ancillary qudits cannot be reduced, (ii) the number of parameters which determine the purification state $|\Psi>$ exactly equals the number of degrees of freedom of $\rho$, and (iii) $|\Psi>$ is easily determined from the density matrix $\rho$. Moreover, we introduce a quantum circuit in which the quantum gates are unitary transformations acting on a $2n$-qudit system. These transformations are determined by parameters that can be tuned to generate, once the ancillary qudits are disregarded, any given mixed $n$-qudit state.

Abstract:
Any pure quantum state can be equivalently represented by means of its wave function psi(q) or of the Fermi function g_F(q,p), with q and p coordinates and conjugate momenta of the system under investigation.We show that a Gaussian wave packet can be conveniently visualized in phase space by means of the curve g_F(q,p)=0. The evolution in time of the g_F=0 curve is then computed for a Gaussian packet evolving freely or under a constant or a harmonic force. As a result, the spreading or shrinking of the packet is easily interpreted in phase space. Finally, we discuss a gedanken prism microscope experiment for measuring the position-momentum correlation. This gedanken experiment, together with the well-known Heisenberg microscope and von Neumann velocimeter, is sufficient to fully determine the state of a Gaussian packet.

Abstract:
The working of a quantum computer is described in the concrete example of a quantum simulator of the single-particle Schrodinger equation. We show that a register of 6-10 qubits is sufficient to realize a useful quantum simulator capable of solving in an efficient way standard quantum mechanical problems.

Abstract:
Quantum chaotic maps can efficiently generate pseudo-random states carrying almost maximal multipartite entanglement, as characterized by the probability distribution of bipartite entanglement between all possible bipartitions of the system. We show that such multipartite entanglement is robust, in the sense that, when realistic noise is considered, distillable entanglement of bipartitions remains almost maximal up to a noise strength that drops only polynomially with the number of qubits.

Abstract:
We study the time dependence of the ionization probability of Rydberg atoms driven by a microwave field. The quantum survival probability follows the classical one up to the Heisenberg time and then decays inversely proportional to time, due to tunneling and localization effects. We provide parameter values which should allow one to observe such decay in laboratory experiments. Relations to the $1/f$ noise are also discussed.

Abstract:
In classical mechanics the complexity of a dynamical system is characterized by the rate of local exponential instability which effaces the memory of initial conditions and leads to practical irreversibility. In striking contrast, quantum mechanics appears to exhibit strong memory of the initial state. Here we introduce a notion of complexity for a quantum system and relate it to its stability and reversibility properties.