Open Systems & Information Dynamics

We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators $ℒ$ are sums of other generators ${ℒ}_{\omega }$ associated with positive Bohr frequencies $\omega$ of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each ${ℒ}_{\omega }$ do so with respect to the same invariant state.

We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl $C*$-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.

An important kind of interaction between two moving two-level atoms and a field mode, where the coupling parameter is taken to be time-dependent, is presented in this paper. Nonlocal correlations between the atoms and the field have been investigated by means of concurrence and von Neumann entropy in terms of the involved parameters of the system. The results show that the atomic motion plays an essential role in the evolution of system dynamics, its nonlocal correlations and geometric phase. Moreover, an interesting correlation between the entanglement and the geometric phase during the evolution was observed. The presented system can be very useful for generating and maintaining high amount of entanglement by means of controlling the parameters of atomic motion.

We provide a straightforward generalization of a positive map in $M_3\left(ℂ\right)$ considered recently by Miller and Olkiewicz [5]. It is proved that these maps are optimal and indecomposable. As a byproduct we provide a class of PPT entangled states in $d\otimes d$.

The main result of this paper is a construction for complex Hadamard matrices: for $q>2$ any prime power and $m$ the size of a real Hadamard matrix, this construction yields a family of complex Hadamard matrices of order $m(q+1)$ with $q+1$ parameters, including Butson-type matrices of even type $d$ a divisor of $\text{lcm(}q-1,2\text{)}$. Only a few lowdimensional examples and the real Hadamard matrices obtained by this construction are already known. Also a small extension of Diţa’s construction (cf. Lemma 1) is given.

In this paper, we propose a modified construction of the quantum adiabatic algorithm for Boolean functions studied by M. Andrecut et al. [13, 14]. Our algorithm has the time complexity $O\left(1\right)$ for the evaluation of Boolean functions, without additional computational cost of implementing the driving Hamiltonian, which is required by the adiabatic evolution described in [13, 14].

A new quantum algorithm for the search problem and its computational complexity are discussed. Its essential part is the use of the so-called chaos amplifier, [8, 9, 10, 13]. It is shown that for the search problem containing $2^n$ objects time complexity of the method is polynomial in $n$.

In this paper, a method to solve functionally commutative time-dependent linear homogeneous differential equation is discussed. We apply this technique to solve some dynamical quantum problems.