Strong subadditivity for log-determinant of covariance matrices and its applications

Adesso, Gerardo; Simon, R.

doi:10.1088/1751-8113/49/34/34lt02

Cited by 25 publications

(40 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We further present for the first time an experimental observation of a 'reverse' steerability, where the party being steered comprises more than one mode. With this ability, we precisely validate four types of monogamy relations recently proposed for Gaussian steering (see Table I) in the presence of loss [33][34][35][36][37]. Our study helps quantify how steering can be distributed among different parties in cluster states and link the amount of steering to the security of channels in a communication network.…”

supporting

confidence: 59%

“…In this Letter, we experimentally investigate properties of bipartite steering within a CV four-mode square Gaussian cluster state (see Fig. 1), and quantitatively test its monogamy relations [33][34][35][36][37]. By reconstructing the covariance matrix of the cluster state, we measure the quantifier of EPR steering under Gaussian measurements introduced in [15], for various bipartite splits.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Demonstration of Monogamy Relations for Einstein-Podolsky-Rosen Steering in Gaussian Cluster States

et al. 2017

Self Cite

View full text Add to dashboard Cite

A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Understanding how quantum resources can be quantified and distributed over many parties has profound applications in quantum communication. As one of the most intriguing features of quantum mechanics, Einstein-Podolsky-Rosen (EPR) steering is a useful resource for secure quantum networks. By reconstructing the covariance matrix of a continuous variable four-mode square Gaussian cluster state subject to asymmetric loss, we quantify the amount of bipartite steering with a variable number of modes per party, and verify recently introduced monogamy relations for Gaussian steerability, which establish quantitative constraints on the security of information shared among different parties. We observe a very rich structure for the steering distribution, and demonstrate one-way EPR steering of the cluster state under Gaussian measurements, as well as one-to-multimode steering. Our experiment paves the way for exploiting EPR steering in Gaussian cluster states as a valuable resource for multiparty quantum information tasks.

show abstract

supporting

confidence: 59%

mentioning

confidence: 99%

Demonstration of Monogamy Relations for Einstein-Podolsky-Rosen Steering in Gaussian Cluster States

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…This manuscript, following a recent series of contributions [20], [26], [18], casts further light on the connections between matrix analysis (in particular determinantal inequalities) and information theory in both classical and quantum settings. In future work, within the context of continuous variable quantum information with Gaussian states [19], it could be interesting to establish whether the equivalence between the Gaussian Rényi-2 squashed entanglement defined here and the Gaussian Rényi-2 entanglement of formation defined in [20] further extends to a third measure of entanglement, namely the recently introduced Gaussian intrinsic entanglement [68].…”

Section: Discussionmentioning

confidence: 96%

“…Therefore, in the relevant case of tripartite quantum Gaussian states, the general inequality (5) for log-det entropy takes the form of a SSA inequality for the Rényi-2 entropy [20], [21], [26], holding in addition to the standard one for Rényi-1 entropy aka von Neumann entropy, which is valid for arbitrary (Gaussian or not) tripartite quantum states. The usefulness of inequalities like (5) in quantum optics and quantum information was acknowledged in a series of recent papers.…”

Section: Introductionmentioning

confidence: 99%

From Log-Determinant Inequalities to Gaussian Entanglement via Recoverability Theory

Lami

Hirche

Adesso

et al. 2017

IEEE Trans. Inform. Theory

Self Cite

View full text Add to dashboard Cite

Abstract-Many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialised to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields ln det VAC + ln det VBC − ln det VABC − ln det VC ≥ 0 for all 3 × 3-block matrices VABC , where subscripts identify principal submatrices. We shall refer to the above inequality as SSA of log-det entropy. In this paper we develop further insights on the properties of the above inequality and its applications to classical and quantum information theory.In the first part of the paper, we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs. Subsequently, we discuss the role of the classical transpose channel (also known as Petz recovery map) in this problem and find its action explicitly. We then prove some extensions of the saturation theorem, by finding faithful lower bounds on a log-det conditional mutual information.In the second part, we focus on quantum Gaussian states, whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation. For Gaussian states, the log-det entropy is equivalent to the Rényi entropy of order 2. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian Rényi-2 entanglement of formation, a well-behaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a log-det entropy equivalent of the squashed entanglement measure, which is remarkably shown to coincide with the Gaussian Rényi-2 entanglement of formation. This allows us to establish useful properties of such measure(s), like monogamy, faithfulness, and additivity on Gaussian states.

show abstract

“…This is, however, not the only special feature for Gaussian steering. Namely, within the Gaussian regime one can also prove monogamy relations for steering with more than two parties (Adesso and Simon, 2016;Ji et al, 2015a;Lami et al, 2016;Reid, 2013). One should note that the monogamy can break when one is allowed to perform non-Gaussian measurements (Ji et al, 2016).…”

Section: A Criterion For Steering Of Gaussian Statesmentioning

confidence: 99%

Quantum steering

et al. 2020

View full text Add to dashboard Cite

Quantum correlations between two parties are essential for the argument of Einstein, Podolsky, and Rosen in favour of the incompleteness of quantum mechanics. Schrödinger noted that an essential point is the fact that one party can influence the wave function of the other party by performing suitable measurements. He called this phenomenon quantum steering and studied its properties, but only in the last years this kind of quantum correlation attracted significant interest in quantum information theory. In this paper we review the theory of quantum steering. We first present the basic concepts of steering and local hidden state models and their relation to entanglement and Bell nonlocality. Then, we describe the various criteria to characterize steerability and structural results on the phenomenon. A detailed discussion is given on the connections between steering and incompatibility of quantum measurements. Finally, we review applications of steering in quantum information processing and further related topics. 34 J. Quantum teleportation 34 K. Resource theory of steering 35 L. Post-quantum steering 36 M. Historical aspects of steering 36 1. Discussions between Schrödinger and Einstein 36 2. The two papers by Schrödinger 37 3. Impact of these papers 38 VI. Conclusion 38 References 39 arXiv:1903.06663v2 [quant-ph]

show abstract

Strong subadditivity for log-determinant of covariance matrices and its applications

Cited by 25 publications

References 59 publications

Demonstration of Monogamy Relations for Einstein-Podolsky-Rosen Steering in Gaussian Cluster States

Demonstration of Monogamy Relations for Einstein-Podolsky-Rosen Steering in Gaussian Cluster States

From Log-Determinant Inequalities to Gaussian Entanglement via Recoverability Theory

Quantum steering

Contact Info

Product

Resources

About