arXiv:quant-ph/0403219 · Physical Review A

Entropic uncertainty relations and entanglement

Otfried Gühne, Maciej Lewenstein

Published Aug 24, 2004• 195 citations

Abstract

We discuss the relationship between entropic uncertainty relations and entanglement. We present two methods for deriving separability criteria in terms of entropic uncertainty relations. In particular, we show how any entropic uncertainty relation on one part of the system results in a separability condition on the composite system. We investigate the resulting criteria using the Tsallis entropy for two and three qubits.

DOI: 10.1103/physreva.70.022316

Related papers

Separability conditions from entropic uncertainty relations

Vittorio Giovannetti

We derive a collection of separability conditions for bipartite systems of dimension $d\ifmmode\times\else\texttimes\fi{}d$ which is based on the entropic version of the uncertainty relations. A detailed analysis of the two-qubit case is given by comparing the new separability conditions with existing criteria.

arXiv:quant-ph/0307171Physical Review AJul 8, 2004106 citations

Conditional entropic uncertainty relations for Tsallis entropies

Dariusz Kurzyk, Łukasz Pawela, Zbigniew Puchała

The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem 1 . Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.

arXiv:1707.09278Quantum Information ProcessingJun 22, 201813 citations

Separability Criteria from Uncertainty Relations

Otfried Gühne, Maciej Lewenstein

Abstract. We explain several separability criteria which rely on uncertainty relations. For the derivation of these criteria uncertainty relations in terms of variances or entropies can be used. We investigate the strength of the separability conditions for the case of two qubits and show how they can improve entanglement witnesses.

Jan 1, 200410 citations

Separability Criteria from Uncertainty Relations

Otfried Gühne

We explain several separability criteria which rely on uncertainty relations. For the derivation of these criteria uncertainty relations in terms of variances or entropies can be used. We investigate the strength of the separability conditions for the case of two qubits and show how they can improve entanglement witnesses.

arXiv:quant-ph/0409140AIP conference proceedingsJan 1, 200412 citations

Detecting quantum entanglement : entanglement witnesses and uncertainty relations

Otfried Gühne

This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.

Jan 1, 20043 citations

Entanglement criteria via concave-function uncertainty relations

Yichen Huang

A general theorem as a necessary condition for the separability of quantum states in both finite and infinite dimensional systems, based on concave-function uncertainty relations, is derived. Two special cases of the general theorem are stronger than two known entanglement criteria based on the Shannon entropic uncertainty relation and the Landau-Pollak uncertainty relation, respectively; other special cases are able to detect entanglement where some famous entanglement criteria fail.

Physical Review AJul 29, 2010107 citations