Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1110.2532 · International Journal of Quantum Information

GAUSSIAN GEOMETRIC DISCORD

Gerardo Adesso, Davide Girolami

Published Oct 1, 2011• 56 citations
arXivHTMLPDF

Abstract

We extend the geometric measure of quantum discord, introduced and computed for two-qubit states, to quantify non-classical correlations in composite Gaussian states of continuous variable systems. We lay the formalism for the evaluation of a Gaussian geometric discord in two-mode Gaussian states, and present explicit formulas for the class of two-mode squeezed thermal states. In such a case, under physical constraints of bounded mean energy, geometric discord is shown to admit upper and lower bounds for a fixed value of the conventional (entropic) quantum discord. We finally discuss alternative geometric approaches to quantify Gaussian quadrature correlations.

DOI
10.1142/s0219749911008192

Related papers

Gaussian geometric discord in terms of Hellinger distance

Serban Suciu, Аурелиан Исар

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we address the quantification of general non-classical correlations in Gaussian states of continuous variable systems from a geometric perspective. We give a description of the Gaussian geometric discord by using the Hellinger distance as a measure for quantum correlations between two non-interacting non-resonant bosonic modes embedded in a thermal environment. We evaluate the Gaussian geometric discord by taking two-mode squeezed thermal states as initial states of the system and show that it has finite values between 0 and 1 and that it decays asymptotically to zero in time under the effect of the thermal bath.

AIP conference proceedings•Jan 1, 2015•6 citations

Quantum versus Classical Correlations in Gaussian States

Gerardo Adesso, Animesh Datta

Quantum discord, a measure of genuinely quantum correlations, is generalized to continuous variable systems. For all two-mode Gaussian states, we calculate analytically the quantum discord and a related measure of classical correlations, solving an optimization over all Gaussian measurements. Almost all two-mode Gaussian states are shown to have quantum correlations, while for separable states, the discord is smaller than unity. For a given amount of entanglement, it admits tight upper and lower bounds. Via a duality between entanglement and classical correlations, we derive a closed formula for the Gaussian entanglement of formation of a family of three-mode Gaussian states.

arXiv:1003.4979•Physical Review Letters•Jul 15, 2010•490 citations

Hellinger distance as a measure of Gaussian discord

Paulina Marian, Tudor A. Marian

Especially investigated in recent years, the Gaussian discord can tentatively be quantified by a distance between a given two-mode Gaussian state (GS) and the set of all the zero-discord two-mode GSs. However, as this set consists only of product states, such a distance captures all the correlations (quantum and classical) between modes. Therefore it is merely an upper bound for the geometric discord, no matter which is the employed distance. In this work we choose for this purpose the Hellinger metric that is known to have many beneficial properties recommending it as a good measure of quantum behaviour. In general, this metric is determined by affinity, a relative of the Uhlmann fidelity with which it shares many important features. As a first step of our work, the affinity of a pair of n -mode GSs is written. Then, in the two-mode case, we succeeded in determining exactly the closest Gaussian product state and computed the Gaussian discord accordingly. The obtained general formula is remarkably simple and becomes still friendlier in the significant case of symmetric two-mode GSs. We then analyse in detail two special classes of two-mode GSs of theoretical and experimental interest as well: the squeezed thermal states and the mode-mixed thermal ones. The former are separable under a well-known threshold of squeezing, while the latter are always separable. It is worth stressing that for symmetric states belonging to either of these classes, we find consistency between their geometric Hellinger discord and the originally defined discord in the Gaussian approach. At the same time, the Gaussian Hellinger discord of such a state turns out to be a reliable measure of the total amount of its cross correlations.

arXiv:1408.4477•Journal of Physics A Mathematical and Theoretical•Feb 19, 2015•34 citations

Gaussian geometric discord of two-mode systems in a thermal environment

Serban Suciu, Аурелиан Исар

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the Gaussian geometric discord for a system consisting of two non-interacting non-resonant bosonic modes embedded in a thermal environment. We take as initial state of the system a two-mode squeezed thermal state and describe the time evolution of the Gaussian geometric discord under the influence of the thermal bath. By tracing the distance between the state of the considered subsystem and the closest classical-quantum Gaussian state we evaluate the Gaussian geometric discord for all times and temperatures. The geometric discord has finite values between 0 and 1 and decreases asymptotically to zero at large times and temperatures with oscillations on the time axis.

AIP conference proceedings•Jan 1, 2014•2 citations

Quantum discord for Gaussian states with non-Gaussian measurements

Paolo Giorda, Michele Allegra, Matteo G. A. Paris

In recent years the paradigm based on entanglement as the unique measure of quantum correlations has been challenged by the rise of new correlation concepts, such as quantum discord, able to reveal quantum correlations that are present in separable states. It is in general difficult to compute quantum discord, because it involves a minimization over all possible local measurements in a bipartition. In the realm of continuous-variable (CV) systems, a Gaussian version of quantum discord has been put forward upon restricting to Gaussian measurements. It is natural to ask whether non-Gaussian measurements can lead to a stronger minimization than Gaussian ones. Here we focus on two relevant classes of two-mode Gaussian states: squeezed thermal states and mixed thermal states, and allow for a range of experimentally feasible non-Gaussian measurements, comparing the results with the case of Gaussian measurements. We provide evidence that Gaussian measurements are optimal for Gaussian states.

arXiv:1206.1807•Physical Review A•Nov 27, 2012•31 citations

Gaussian Quantum Discord

Paolo Giorda, Matteo G. A. Paris

We extend the quantum discord to continuous variable systems and evaluate Gaussian quantum discord C(ϱ) for bipartite Gaussian states. In particular, for squeezed-thermal states, we explicitly maximize the extractable information over Gaussian measurements: C(ϱ) is minimized by a generalized measurement rather than a projective one. Almost all squeezed-thermal states have nonzero Gaussian discord: They may be either separable or entangled if the discord is below the threshold C(ϱ) = 1, whereas they are all entangled above the threshold. We elucidate the general role of state parameters in determining the discord and discuss its evolution in noisy channels.

arXiv:1003.3207•Physical Review Letters•Jul 7, 2010•488 citations
We extend the geometric measure of quantum discord, introduced and computed for two-qubit states, to quantify non-classical correlations in composite Gaussian states of continuous variable systems. We lay the formalism for the evaluation of a Gaussian geometric discord in two-mode Gaussian states, and present explicit formulas for the class of two-mode squeezed thermal states. In such a case, under physical constraints of bounded mean energy, geometric discord is shown to admit upper and lower bounds for a fixed value of the conventional (entropic) quantum discord. We finally discuss alternative geometric approaches to quantify Gaussian quadrature correlations.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we address the quantification of general non-classical correlations in Gaussian states of continuous variable systems from a geometric perspective. We give a description of the Gaussian geometric discord by using the Hellinger distance as a measure for quantum correlations between two non-interacting non-resonant bosonic modes embedded in a thermal environment. We evaluate the Gaussian geometric discord by taking two-mode squeezed thermal states as initial states of the system and show that it has finite values between 0 and 1 and that it decays asymptotically to zero in time under the effect of the thermal bath.
Quantum discord, a measure of genuinely quantum correlations, is generalized to continuous variable systems. For all two-mode Gaussian states, we calculate analytically the quantum discord and a related measure of classical correlations, solving an optimization over all Gaussian measurements. Almost all two-mode Gaussian states are shown to have quantum correlations, while for separable states, the discord is smaller than unity. For a given amount of entanglement, it admits tight upper and lower bounds. Via a duality between entanglement and classical correlations, we derive a closed formula for the Gaussian entanglement of formation of a family of three-mode Gaussian states.
Especially investigated in recent years, the Gaussian discord can tentatively be quantified by a distance between a given two-mode Gaussian state (GS) and the set of all the zero-discord two-mode GSs. However, as this set consists only of product states, such a distance captures all the correlations (quantum and classical) between modes. Therefore it is merely an upper bound for the geometric discord, no matter which is the employed distance. In this work we choose for this purpose the Hellinger metric that is known to have many beneficial properties recommending it as a good measure of quantum behaviour. In general, this metric is determined by affinity, a relative of the Uhlmann fidelity with which it shares many important features. As a first step of our work, the affinity of a pair of n -mode GSs is written. Then, in the two-mode case, we succeeded in determining exactly the closest Gaussian product state and computed the Gaussian discord accordingly. The obtained general formula is remarkably simple and becomes still friendlier in the significant case of symmetric two-mode GSs. We then analyse in detail two special classes of two-mode GSs of theoretical and experimental interest as well: the squeezed thermal states and the mode-mixed thermal ones. The former are separable under a well-known threshold of squeezing, while the latter are always separable. It is worth stressing that for symmetric states belonging to either of these classes, we find consistency between their geometric Hellinger discord and the originally defined discord in the Gaussian approach. At the same time, the Gaussian Hellinger discord of such a state turns out to be a reliable measure of the total amount of its cross correlations.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the Gaussian geometric discord for a system consisting of two non-interacting non-resonant bosonic modes embedded in a thermal environment. We take as initial state of the system a two-mode squeezed thermal state and describe the time evolution of the Gaussian geometric discord under the influence of the thermal bath. By tracing the distance between the state of the considered subsystem and the closest classical-quantum Gaussian state we evaluate the Gaussian geometric discord for all times and temperatures. The geometric discord has finite values between 0 and 1 and decreases asymptotically to zero at large times and temperatures with oscillations on the time axis.
In recent years the paradigm based on entanglement as the unique measure of quantum correlations has been challenged by the rise of new correlation concepts, such as quantum discord, able to reveal quantum correlations that are present in separable states. It is in general difficult to compute quantum discord, because it involves a minimization over all possible local measurements in a bipartition. In the realm of continuous-variable (CV) systems, a Gaussian version of quantum discord has been put forward upon restricting to Gaussian measurements. It is natural to ask whether non-Gaussian measurements can lead to a stronger minimization than Gaussian ones. Here we focus on two relevant classes of two-mode Gaussian states: squeezed thermal states and mixed thermal states, and allow for a range of experimentally feasible non-Gaussian measurements, comparing the results with the case of Gaussian measurements. We provide evidence that Gaussian measurements are optimal for Gaussian states.
We extend the quantum discord to continuous variable systems and evaluate Gaussian quantum discord C(ϱ) for bipartite Gaussian states. In particular, for squeezed-thermal states, we explicitly maximize the extractable information over Gaussian measurements: C(ϱ) is minimized by a generalized measurement rather than a projective one. Almost all squeezed-thermal states have nonzero Gaussian discord: They may be either separable or entangled if the discord is below the threshold C(ϱ) = 1, whereas they are all entangled above the threshold. We elucidate the general role of state parameters in determining the discord and discuss its evolution in noisy channels.