In the framework of the project IST-1999-13071 (QUICOV) our group in Olomouc has solved several recent problems in quantum information processing with continuous variables (CVs). We mainly work on two recent important problems:

1. protection and concetration of entanglement in CV quantum information protocols,

2. direct measurement and Max-Lik estimation of CV states.

Central recent problem of CV quantum communication is deep analysis and feasible elimination of influence of noise and decoherence affecting quantum channel and quantum memory. It is necessary for any realistic and efficient implementation of CV quantum communication protocols. We theoretically analysed two different sources of decoherence affecting the entangled two-mode squeezed vacuum state when it is transmitted through damping channel. First, the non-Gaussian entangled state given by the CV analogue of the Werner state (a mixture of the entangled state and local thermal radiations) was analysed [L. Mista, Jr., R. Filip, and J. Fiurasek, Continuous-variable Werner state: separability, nonlocality, squeezing and teleportation, Phys. Rev. A 65, 062315 (2002)]. The entanglement and violation of Bell’s inequalities for the state were studied in dependence on quality of the damping channel. Efficiency of the standard CV teleportation of coherent states with the CV Werner state was also discussed. Second, the two-mode squeezed vacuum state whose single mode propagates throught Gaussian lossy channel was investigated. As shown the entangled state can violate Bell’s inequalities at a substantially larger distance in comparison with previous claims. This fact is of a particular importance in quantum cryptography with CV entangled states [R. Filip and L. Mista, Jr., Violation of Bell's inequalities for a two-mode squeezed vacuum state in lossy transmission lines, Phys. Rev. A 66, 044309 (2002)].

An open question is how to distill or concentrate CV entanglement using only experimentally feasible methods. Active Gaussian operations rely on second-order nonlinear interactions and homodyne detection which are feasible with recent technology. Using CV Gaussian operations many CV quantum information protocols previously suggested for discrete variables were realized experimentally. At the initial stage of the programme our group worked on the central problem of distillation of Gaussian entagled states with Gaussian operations. The machinery of Gaussian completely positive maps was used to show that Gaussian operations together with classical communication do not suffice for distilling more entaglement from a single copy of a two-mode entangled Gaussian state [J. Fiurasek, Gaussian transformations and distillation of entangled Gaussian states, Phys. Rev. Lett. 89, 137904 (2002)].

Inefficiency of many-copy Gaussian distillation procedure stimulates another question: which collective task can be resolved rely only on Gaussian operations in a better way using more copies of input Gaussian state. Our further activity was focused on manipulation with quantum noise in CV entangled states based on the local application of phase-sensitive optical amplifiers. The QND amplifiers were shown to be essential ingredient of a multiple-copy protocol capable of squeezing concentration in two-mode Gaussian entangled states with unknown mean value of a certain quadrature. In this way the usefulness of multiple copies of entangled Gaussian states was demonstrated in tasks when these states are known only partially. This result is in a marked difference with the results obtained under the assumption of complete knowledge of the processed entangled state when Gaussian multiple-copy protocols can be substituted only by LOCC operations on a single copy [R. Filip and L. Mista, Jr., Squeezing concentration for Gaussian states with unknown parameter, J. Opt. B: Quantum Semiclass. Opt. 5, 387 (2002)].

Another method how to reduce noise is to use a fact that a distribution of information from one quantum system to enviroment can be reversible using only local measurement on the enviroment, classical communication and local operation on the system. Thus under certain conditions we can partially or even almost precisely erase the effect of the effect of environmental decoherence. In CV domain where we cannot use entanglement destillation by feasible Gaussian methods it is a feasible method how protect a quantum state against loss of information. A simple demonstration of quantum erasing using CV systems and only Gaussian operations was proposed [R. Filip, Continuous-variable quantum erasing, Phys. Rev. A 67, 042111 (2003)]. This theoretical proposal is now followed by an experimental implementation of the CV erasing of vacuum state with help squeezed bright beam in laboratory of UNI Erlanglen.

From the our analysis followed that to implement distillation of Gaussian beams we need non-Gaussian methods and therefore two kinds of third-order nonlinear interactions were considered with respect to manipulations and measurement of CV Gaussian states: the interaction of an atom with a cavity field and the cross-Kerr interaction with strong coherent probe beam. With respect to the distillation of CV entanglement two interactions were used in proposals of two probabilistic entanglement concentration schemes working with a single copy of a two-mode entangled state. Further, it was shown that the entanglement concentration allows us to improve the fidelity of CV teleportation of coherent states [J. Fiurasek, L. Mista, Jr., and R. Filip, Entanglement concentration of continuous-variable quantum states, Phys. Rev. A 67, 022304 (2003)].

Another method of non-Gaussian entanglement concentration could be based on Gaussian coupling between the system and ancilla, followed by appropriate non-Gaussian measurement on ancilla. To perform non-Gaussian measurements we would like to implement a detector resolving photon number in a CV state. We proposed and experimentally implemented a kind of this detector using fiber loop simulating a multi-port detection device [J. Rehacek, Z. Hradil, O. Haderka, J. Perina, Jr., M. Hamar: Multiple-photon resolving fiber-loop detector , Phys. Rev. A 67, 061801 (2003)]. The Max-Lik reconstruction techniques enable us to obtain the experimental result even for few cykles of the pulse in the fiber loop detector. Recently, the experimental investigation of non-Gaussian detectors is focused on exploitation of CCD camera as a multi-chanel detector.

To quantify an efficiency of CV information protocols in presence of noise we need sophisticated method how to measure and estimate quantum states and channels.

In the domain of quantum continuous variables the algorithm for efficient estimation of single mode Gaussian fields has been devised. Homodyne detection serves here for the observation of phase sensitive data corresponding to detection of appropriate quadrature component. Our other activities touched several problems related to reconstruction of continuous variables which is necesseary to perform a fair quantification of information protocols. The effort has been focused on question how classical MaxLik estimation of images can be implemented for quantum tasks, since the mathematical formulations both classical and quantum procedure are similar [M. Jezek, Z. Hradil, Image processing as state reconstruction in optics, preprint arXiv.org:physics/0309126]. As an example, using Max-Lik reconstruction technique on entangled states we obtain in principle an optimal measurement which discriminates entangled states from the disentangled ones [J. Rehacek, Z. Hradil: Quantification of entanglement by means of convergent iterations, Phys. Rev. Lett. 90, 127904 (2003)].

Further we demonstrate that using non-Gaussian detection method we have a wider class of direct feasible measurements of CV quantum states. Therefore our attention was fixed on the exploitation of the third-order nonlinearity in direct measurement of fundamental quantum information quantities. A direct overlap measurement using non-Gaussian method and interferometry were devised for Gaussian as well as non-Gaussian states [R. Filip, Overlap and entanglement witness measurements, Phys. Rev. A 65, 062320 (2002)]. The identical setup can be used to implement measurement of purity of Gaussian or even non-Gaussian states if we have two identical copies. The proposed method is based on non-linear coupling between CV system and qubit system and it can be straightforwardly implemented in experiments with high-Q cavities storing microwave radiation. An optical implementation of this scheme requires an efficient coupling of two light pulses on quantum level in Kerr-like media which could be enhanced by recently intensively investigated electromagnatically induced transparency.