Krzysztof Wódkiewicz

Depolarizing channel as a completely positive map with memory

The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. This paper examines master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel. A physical model that possesses such an exponential memory kernel is presented. This model contains a classical, fluctuating environment based on random telegraph signal stochastic variables.

Quantum Markov channels for qubits

We examine stochastic maps in the context of quantum optics. Making use of the master equation, the damping basis, and the Bloch picture we calculate a nonunital, completely positive, trace-preserving map with unequal damping eigenvalues. This results in what we call the squeezed vacuum channel. A geometrical picture of the effect of stochastic noise on the set of pure state qubit density operators is provided. Finally, we study the capacity of the squeezed vacuum channel to transmit quantum information and to distribute Einstein-Podolsky-Rosen states.

Operational theory of homodyne detection

We discuss a balanced homodyne detection scheme with imperfect detectors in the framework of the operational approach to quantum measurement. We show that a realistic homodyne measurement is described by a family of operational observables that depends on the experimental setup, rather than a single field quadrature operator. We find an explicit form of this family, which fully characterizes the experimental device and is independent of a specific state of the measured system. We also derive operational homodyne observables for the setup with a random phase, which has been recently applied in an ultrafast measurement of the photon statistics of a pulsed diode laser. The operational formulation directly gives the relation between the detected noise and the intrinsic quantum fluctuations of the measured field. We demonstrate this with two examples: the operational uncertainty relation for the field quadratures, and the homodyne detection of suppressed fluctuations in photon statistics.

Hydrogen atom in phase space: the Wigner representation

The hydrogen atom is a fundamental exactly soluble system for which the Wigner function, being a quantum analogue of the joint probability distribution of position and momentum, is unknown. In this paper, we present an effective method of calculating the Wigner function, for all bound states of the nonrelativistic hydrogen atom. The formal similarity between the eigenfunctions of the nonrelativistic hydrogen atom in the momentum representation and the Klein–Gordon propagator has allowed the calculation of the Wigner function for an arbitrary bound state of the hydrogen atom, using a simple atomic integral as a generator. These Wigner functions for some low-lying states are depicted and discussed.

Wigner function evolution of quantum states in the presence of self-Kerr interaction

A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium (chi(3) nonlinearity) is presented. We numerically solved this equation taking a coherent state as an initial condition. The dissipation effects are discussed. We provide examples of quantum interference, sub-Planck phase space structures, and Gaussian vs non-Gaussian dynamical evolution of the state. The results also apply to the description of a nanomechanical resonator with an intrinsic Duffing nonlinearity.

Separability of entangled qutrits in noisy channels

We present an analysis of noisy atomic channels involving qutrits. We choose a three-level atom with V-configuration to be the qutrit state. Gell-Mann matrices and a generalized Bloch vector (8-dimensional) are used to describe the qutrit density operator. We introduce quantum quasi-distributions for qutrits that provide a simple description of entanglement. Studying the time-evolution for the atomic variables we find the Kraus representation of spontaneous emission quantum channel (SE channel). Furthermore, we consider a generalized Werner state of two qutrits and investigate the separability condition in the presence of spontaneous emission noise. The influence of spontaneous emission on the separability of Werner states for qutrit and qubit states is compared.

Separability of two-party Gaussian states

We investigate the separability properties of quantum two-party Gaussian states in the framework of the operator formalism for the density operator. Such states arise as natural generalizations of the entangled state originally introduced by Einstein, Podolsky, and Rosen. We present explicit forms of separable and nonseparable Gaussian states.

Spin quasi-distribution functions

Two-classes of phase-space spin quasi-distribution functions are introduced and discussed. The first class of these distributions is based on the delta function construction. It is shown that such a construction can be carried out for an arbitrary spin s and an arbitrary ordering of the spin operators. The second class of the spin distributions is constructed with the help of the spin coherent states. The connection of the spin coherent states to the Stratonovich formalism is established and discussed. It is shown that the c-number phase-space description of quantum fluctuations provides a simple statistical picture of quantum fluctuations of spinoperators in terms of random directions on a unit sphere. For quantum states of the spin system the statistics of these random orientations is given by non-positive spin quasi-distribution functions. It is shown that the application of these spin quasi-distribution functions to the Einstein-Podolsky-Rosen correlations provide an insight into the quantum theory of measurement.

Stochastic decoherence of qubits.

We study the stochastic decoherence of qubits using the Bloch equations and the Bloch sphere description of a two-level atom. We show that it is possible to describe a general decoherence process of a qubit by a stochastic map that is dependent on 12 independent parameters. Such a stochastic map is constructed with the help of the damping basis associated with a Master equation that describes the decoherence process of a qubit.

Quantum interference in the Kirkwood–Rihaczek representation

Abstract We discuss the Kirkwood–Rihaczek phase space distribution and analyze a whole class of quasi-distributions connected with this function. All these functions have the correct marginals. We construct a coherent state representation of such functions, discuss which operator ordering corresponds to the Kirkwood–Rihaczek distribution and their generalizations, and show how such states are connected to squeezed states. Quantum interference in the Kirkwood–Rihaczek representation is discussed.