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Quantum Information Processing

Quantum discord is not extremalized by Gaussian states

Seungmin Lee, Changsuk Noh, Jiyong Park

Published Mar 8, 2025• 2 citations
Publisher

Abstract

Abstract Quantum discord is an important measure of quantum correlations that goes beyond the paradigm of quantum entanglement. However, calculating quantum discord involves optimization over measurements, which is computationally challenging and often infeasible. This raises the intriguing question of Gaussian extremality—whether the quantum discord of a reference Gaussian state can provide a meaningful bound to the quantum discord of the original state. In this paper, we investigate this question by comparing the Gaussian discord of a reference Gaussian state with the quantum discord.

DOI
10.1007/s11128-025-04693-x

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Abstract Quantum discord is an important measure of quantum correlations that goes beyond the paradigm of quantum entanglement. However, calculating quantum discord involves optimization over measurements, which is computationally challenging and often infeasible. This raises the intriguing question of Gaussian extremality—whether the quantum discord of a reference Gaussian state can provide a meaningful bound to the quantum discord of the original state. In this paper, we investigate this question by comparing the Gaussian discord of a reference Gaussian state with the quantum discord.
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.
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.
Abstract Quantum discord is a measure of non-classical correlations, which are excess correlations inherent in quantum states that cannot be accessed by classical measurements. For multipartite states, the classically accessible correlations can be defined by the mutual information of the multipartite measurement outcomes. In general the quantum discord of an arbitrary quantum state involves an optimisation of over the classical measurements which is hard to compute. In this paper, we examine the quantum discord in the experimentally relevant case when the quantum states are Gaussian and the measurements are restricted to Gaussian measurements. We perform the optimisation over the measurements to find the Gaussian discord of the bipartite EPR state and tripartite GHZ state in the presence of different types of noise: uncorrelated noise, multiplicative noise and correlated noise. We find that by adding uncorrelated noise and multiplicative noise, the quantum discord always decreases. However, correlated noise can either increase or decrease the quantum discord. We also find that for low noise, the measurements that maximise the classically accessible correlations are single quadrature measurements. As the noise increases, a dual quadrature measurement becomes optimal.
In this Letter we exploit the recently solved conjecture on the bosonic minimum output entropy to show the optimality of Gaussian discord, so that the computation of quantum discord for bipartite Gaussian states can be restricted to local Gaussian measurements. We prove such optimality for a large family of Gaussian states, including all two-mode squeezed thermal states, which are the most typical Gaussian states realized in experiments. Our family also includes other types of Gaussian states and spans their entire set in a suitable limit where they become Choi matrices of Gaussian channels. As a result, we completely characterize the quantum correlations possessed by some of the most important bosonic states in quantum optics and quantum information.
In this Letter we exploit the recently-solved conjecture on the bosonic minimum output entropy to show the optimality of Gaussian discord, so that the computation of quantum discord for bipartite Gaussian states can be restricted to local Gaussian measurements. We prove such optimality for a large family of Gaussian states, including all two-mode squeezed thermal states, which are the most typical Gaussian states realized in experiments. Our family also includes other types of Gaussian states and spans their entire set in a suitable limit where they become Choi-matrices of Gaussian channels. As a result, we completely characterize the quantum correlations possessed by some of the most important bosonic states in quantum optics and quantum information.
We introduce an operational discord-type measure for quantifying nonclassical correlations in bipartite Gaussian states based on using Gaussian measurements. We refer to this measure as operational Gaussian discord (OGD). It is defined as the difference between the entropies of two conditional probability distributions associated to one subsystem, which are obtained by performing optimal local and joint Gaussian measurements. We demonstrate the operational significance of this measure in terms of a Gaussian quantum protocol for extracting maximal information about an encoded classical signal. As examples, we calculate OGD for several Gaussian states in the standard form.