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arXiv:1909.02335 · Quantum Science and Technology

Upper bounds for relative entropy of entanglement based on active learning

Shi‐Yao Hou, Chenfeng Cao, D L Zhou, Bei Zeng

Published Sep 1, 2020• 6 citations
arXivHTMLPDF

Abstract

Abstract Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement E R is an outstanding quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of E R were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of E R based on active learning, a subfield in machine learning, to generate an approximation of the set of separable states. We apply our method to calculate E R for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a reliable tool for entanglement measure calculation and deepens our understanding for the structure of separable states.

DOI
10.1088/2058-9565/abb412

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Abstract Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement E R is an outstanding quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of E R were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of E R based on active learning, a subfield in machine learning, to generate an approximation of the set of separable states. We apply our method to calculate E R for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a reliable tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement ERE_{R}ER​ is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of ERE_RER​ were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of ERE_RER​ based on the convex hull approximation of the set of separable states. We apply our method to calculate ERE_RER​ for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
Quantifying entanglement in quantum systems is an important yet challenging task due to its NP-hard nature. In this work, we propose an efficient algorithm for evaluating distance-based entanglement measures. Our approach builds on Gilbert's algorithm for convex optimization, providing a reliable upper bound on the entanglement of a given arbitrary state. We demonstrate the effectiveness of our algorithm by applying it to various examples, such as calculating the squared Bures metric of entanglement as well as the relative entropy of entanglement for GHZ states, W states, Horodecki states, and chessboard states. These results demonstrate that our algorithm is a versatile and accurate tool that can quickly provide reliable upper bounds for entanglement measures.
Abstract The relative entropy of entanglement ERE_RER​ E R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, ERE_RER​ E R is everywhere lower semi-continuous. We use this to derive a dual variational expression for ERE_RER​ E R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative λ\lambda λ λ -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for ERE_RER​ E R and closely related quantities in the presence of an energy constraint.
Abstract Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.
Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.
Abstract Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in Fawzi, Saunderson and Parrilo (2017 arXiv: 1705.00812). As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.