Gaussian Input States Research Articles

Dynamics of quantum entanglement in Gaussian open systems

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal environment. Using the Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a repeated collapse and revival of entanglement takes place. We determine the asymptotic Gaussian maximally entangled mixed states (GMEMS) and their corresponding asymptotic maximal logarithmic negativity.

Entanglement and mixedness in open systems with continuous variables*

Within the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the dynamics of entanglement for a system consisting of two uncoupled modes interacting with a thermal environment. Using the Peres–Simon necessary and sufficient criterion of separability for two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. We determine the asymptotic Gaussian maximum entangled mixed states (GMEMS) and their corresponding asymptotic maximum logarithmic negativity, which characterizes the degree of entanglement. Using the symplectic eigenvalues of the asymptotic covariance matrix, we obtain the expressions of von Neumann entropy and mutual information of asymptotic GMEMS.

Entanglement dynamics of two-mode Gaussian states in a thermal environment

Within the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous variable entanglement for a system consisting of two uncoupled modes interacting with a thermal environment. Using the Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type — separable or entangled. In other cases, entanglement generation, entanglement sudden death, or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.

Entanglement Generation and Evolution in Open Quantum Systems

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we study the continuous variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we show that for certain values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.

Entanglement in open quantum dynamics

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. Using the Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation or entanglement collapse (entanglement sudden death) takes place or even a periodic collapse and revival of entanglement takes place. We show that for certain classes of environments, the initial state evolves asymptotically to an entangled equilibrium bipartite state, whereas for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.

ASYMPTOTIC ENTANGLEMENT IN OPEN QUANTUM SYSTEMS

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we solve in the asymptotic long-time regime the master equation for two independent harmonic oscillators interacting with an environment. We give a description of the continuous-variable asymptotic entanglement in terms of the covariance matrix of the considered subsystem for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.

Second-order spectra for quadratic nonlinear systems by Volterra functional series: Analytical description and numerical simulation

Higher-order spectral analysis techniques are often used to identify nonlinearities in complex dynamical systems. More specifically, the auto- and cross-bispectrum have proven to be useful tools in testing for the presence of quadratic nonlinearities based on knowledge of a system's input and output. In this paper, analytical expressions for the auto- and cross-bispectrum are developed using a Volterra functional approach under the assumption of a zero-mean, stationary Gaussian input; proper simplifications are presented when the whiteness of the input signal is also imposed. These formulae show the contributions of the bispectrum in terms of the system frequency response function and elementary physical properties of the system. Simulations based on a stochastic numerical integration technique accompany the analytical solutions for a mechanical mass–spring–damper system possessing quadratic damping and stiffness coefficients and subjected to Gaussian white noise excitation. Subsequent estimates of the bispectrum based on the simulated signals show excellent agreement with theory. These results show how modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation. The presence and extent of nonlinear interactions between frequency components are identified. Advantages of using higher-order spectra techniques will be revealed and pertinent conclusions will be outlined.

Decoherence and asymptotic entanglement in open quantum dynamics

Within the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we determine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath. It is found that the system manifests a quantum decoherence which is more and more significant in time. We also calculate the decoherence time and show that it has the same scale as the time after which thermal fluctuations become comparable with quantum fluctuations. We solve the master equation for two independent harmonic oscillators interacting with an environment in the asymptotic long-time regime. We give a description of the continuous-variable asymptotic entanglement in terms of the covariance matrix of quantum states of the considered system for an arbitrary Gaussian input state. Using the Peres-Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that the two noninteracting systems immersed in a common environment become asymptotically entangled for certain environments, so that in the long-time regime they manifest nonlocal quantum correlations.

Selective cloning of Gaussian states by linear optics

We investigate the performance of a selective cloning machine based on linear optical elements and Gaussian measurements, which allows one to clone at will one of the two incoming input states. This machine is a complete generalization of a 1{yields}2 cloning scheme demonstrated by Andersen et al. [Phys. Rev. Lett. 94, 240503 (2005)]. The input-output fidelity is studied for a generic Gaussian input state, and the effect of nonunit quantum efficiency is also taken into account. We show that, if the states to be cloned are squeezed states with known squeezing parameter, then the fidelity can be enhanced using a third suitable squeezed state during the final stage of the cloning process. A binary communication protocol based on the selective cloning machine is also discussed.

Optimized interferometry with Gaussian states

We revisit Mach—Zehnder interferometry using a suitable phase-space analysis and present a rigorous optimization of the sensitivity in realistic condition, i.e., for Gaussian input states, and taking into account nonunit quantum efficiency in the detection stage. The working regime of the interferometer is optimized at fixed input energy versus the squeezing phases and amplitudes as well as the distribution of squeezing in the two input signals. For ideal detection we find the known result that the squeezing resource allows to beat the shot-noise limit. For nonunit detection efficiency, we show that for fixed input energy one can always optimize the squeezing fraction in order to enhance the sensitivity with respect to the case of no squeezing, even in cases when one cannot go beyond the shot-noise limit.

Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps

We investigate non-Gaussian states of light as ancillary inputs for generating nonlinear transformations required for quantum computing with continuous variables. We consider a recent proposal for preparing a cubic phase state, find the exact form of the prepared state and perform a detailed comparison to the ideal cubic phase state. We thereby identify the main challenges to preparing an ideal cubic phase state and describe the gates implemented with the non-ideal prepared state. We also find the general form of operations that can be implemented with ancilla Fock states, together with Gaussian input states, linear optics, squeezing transformations, and homodyne detection with feed forward, and discuss the feasibility of continuous variable quantum computing using ancilla Fock states.

Quantum fluctuations in holographic teleportation of optical images

We investigate in detail the quantum fluctuations in the quantum holographic teleportation protocol that we recently proposed [11]. This protocol implements a continuous variable teleportation scheme that enables the transfer of the quantum state of spatially multimode electromagnetic fields, preserving their quantum correlations in space-time, and can be used to perform teleportation of 2D optical images. We derive a characteristic functional, which provides any arbitrary spatio-temporal correlation function of the teleported field, and calculate the fidelity of the teleportation scheme for multimode Gaussian input states. We show that for multimode light fields one has to distinguish between a global and a reduced fidelity. While the global fidelity tends to vanish for teleportation of fields with many degrees of freedom, the reduced fidelity can be made close to unity by choosing properly the number of essential degrees of freedom and the spatial bandwidth of the EPR beams used in the teleportation scheme.

Non-geometric spectral moments for frequency varying filtered input processes

The spectral moments have been introduced in the literature for the case of structural systems subjected to stationary Gaussian input processes and were defined as the geometric moments of the one-sided power spectral density function of the response. It has been proved that the spectral moments play an important role for the safety prediction of structural systems. Extension of the definition of spectral moments for non-stationary Gaussian input processes has been provided by means of the geometric interpretation applied to the one-sided evolutionary power spectral density function of the response. In the case of white input process, such a geometric approach leads to divergences and needs adjustments in order to be exploited for the safety assessment in the non-stationary case. The exact formulation for the spectral moments has been proved to be based rather on a non-geometric time domain approach aiming at the evaluation of the co-variances of the pre-envelope response process. The geometric and the non-geometric approaches lead to the same spectral moments for the stationary case only. In this paper, a procedure for the evaluation of the non-geometric spectral moments is presented with particular attention to the case of non-stationary filtered Gaussian white noise which is both amplitude and frequency-modulated. The presented procedure is based on the formulation of the pre-envelope co-variance differential equations, whose solution is obtained by means of an approximated expression of the time-dependent transition matrix of the system constituted by the structure and the time varying frequency filter.

An improved statistical analysis of the least mean fourth (LMF) adaptive algorithm

The paper presents an improved statistical analysis of the least mean fourth (LMF) adaptive algorithm behavior for a stationary Gaussian input. The analysis improves previous results in that higher order moments of the weight error vector are not neglected and that it is not restricted to a specific noise distribution. The analysis is based on the independence theory and assumes reasonably slow learning and a large number of adaptive filter coefficients. A new analytical model is derived, which is able to predict the algorithm behavior accurately, both during transient and in steady-state, for small step sizes and long impulse responses. The new model is valid for any zero-mean symmetric noise density function and for any signal-to-noise ratio (SNR). Computer simulations illustrate the accuracy of the new model in predicting the algorithm behavior in several different situations.

Convergence analysis of the sign algorithm with badly behaved noise

The paper analyzes the convergence of the sign algorithm when the noise distribution has a dead zone that includes the origin. The analysis is done in the context of the identification of a plant with a stationary Gaussian input. Upper bounds of the time-averaged mean absolute excess estimation error and the time-averaged mean norm of the weight misalignment vector are derived. The former bound does not depend on the data correlation while the latter one does. The bounds hold for all values of the algorithm step size μ. Both bounds tend to zero as μ tends to zero. The bounds are significantly dependent on the width of the dead zone while they are weakly dependent on μ when μ is less than some threshold. The threshold is proportional to the width of the dead zone. The speed-accuracy trade-off of the algorithm is found to be poor in comparison with that in the case of a Gaussian noise. The wider is the dead zone, the worse is the trade-off. The theoretical results of the paper are supported by simulations.

Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement

A beam splitter is a simple, readily available device which can act to entangle the output optical fields. We show that a necessary condition for the fields at the output of the beam splitter to be entangled is that the pure input states exhibit nonclassical behavior. We generalize this proof for arbitrary (pure or impure) Gaussian input states. Specifically, nonclassicality of the input Gaussian fields is a necessary condition for entanglement of the field modes with the help of the beam splitter. We conjecture that this is a general property of the beam splitter: Nonclassicality of the inputs is a necessary condition for entangling fields in the beam splitter.

Achievable rates for the Gaussian quantum channel

We study the properties of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional Hilbert space. The stabilizer group of such a code is associated with a symplectically integral lattice in the phase space of 2N canonical variables. From the existence of symplectically integral lattices with suitable properties, we infer a lower bound on the quantum capacity of the Gaussian quantum channel that matches the one-shot coherent information optimized over Gaussian input states.

Anisotropy-based performance analysis of linear discrete time invariant control systems

The anisotropic norm of a linear discrete-time-invariant system measures system output sensitivity to stationary Gaussian input disturbances of bounded mean anisotropy. Mean anisotropy characterizes the degree of predictability (or colouredness) and spatial non-roundness of the noise. The anisotropic norm falls between the H

Transient performance degradation of the LMS, RLS, sign, signed regressor, and sign-sign algorithms with data correlation

The transient performance degradation, for correlated input data, is studied for various adaptive algorithms. The algorithms are least mean square (LMS), recursive least squares (RLS), sign (SA), signed regressor (SRA), and sign-sign (SSA). Analysis is performed for adaptive plant identification with stationary Gaussian inputs. A closed-form expression for the mean convergence time is derived for each algorithm. The degradation measure used is the ratio of convergence time for correlated data to the convergence time for white data. The LMS and SRA degradations are the same. The SA and SSA degradations are also the same. The smallest degradation occurs for RLS, the largest for LMS and SRA. The SRA, RLS, and LMS degradations are independent of plant-noise variance. The SA and SSA degradations increase with increased noise variance. The LMS and SRA degradations do not depend upon the weight initialization, RLS (SA and SSA) depends weakly (significantly) upon the weight initialization. The analytical results are supported by simulations.

A robust variable step-size LMS-type algorithm: analysis and simulations

A number of time-varying step-size algorithms have been proposed to enhance the performance of the conventional LMS algorithm. Experimentation with these algorithms indicates that their performance is highly sensitive to the noise disturbance. This paper presents a robust variable step-size LMS-type algorithm providing fast convergence at early stages of adaptation while ensuring small final misadjustment. The performance of the algorithm is not affected by existing uncorrelated noise disturbances. An approximate analysis of convergence and steady-state performance for zero-mean stationary Gaussian inputs and for nonstationary optimal weight vector is provided. Simulation results comparing the proposed algorithm to current variable step-size algorithms clearly indicate its superior performance for cases of stationary environments. For nonstationary environments, our algorithm performs as well as other variable step-size algorithms in providing performance equivalent to that of the regular LMS algorithm.