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:0910.4567 · Physical Review A

Detecting entanglement with non-Hermitian operators

Mark S. Hillery, Ho Trung Dung, Julien Niset

Published Nov 30, 2009• 22 citations
arXivHTMLPDF

Abstract

We derive several entanglement conditions employing non-Hermitian operators. We start with two conditions that were derived previously for field mode operators and use them to derive conditions that can be used to show the existence of field-atom entanglement and entanglement between groups of atoms. The original conditions can be strengthened by making them invariant under certain sets of local unitary transformations such as Gaussian operations. We then apply these conditions to several examples such as the Dicke model. We conclude with a short discussion of how local uncertainty relations with non-Hermitian operators can be used to derive entanglement conditions.

DOI
10.1103/physreva.80.052335

Related papers

Detecting entanglement with non-hermitian operators

Mark S. Hillery

There is no single test that will determine whether a general quantum state is separable or entangled. There are, however, a number of sufficient conditions, including the partial-transpose condition and entanglement witnesses. Here we present a class of sufficient conditions based on expectation values of non-hermitian operators. Initially we examine bipartite systems. The application of these conditions to multimode states of the electromagnetic field and to spin systems is discussed. We show that our original conditions can be strengthened if we make them invariant under certain classes of local unitary transformations. We then go on to study multipartite systems and present conditions that can be used to show that a state is not totally separable. Some simple applications of these conditions are also presented.

International Conference on Quantum Information•Jan 1, 2011•0 citations

Uncertainty Relations of Non-Hermitian Operators: Theory and Experimental Scheme

Xinzhi Zhao, Chengjie Zhang

The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.

Frontiers in Physics•Apr 19, 2022•9 citations

Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: Detection of non-Gaussian entangled states

Hyunchul Nha, Jaewan Kim

We derive a class of inequalities, from the uncertainty relations of the su(1,1) and the su(2) algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su(2) operators ${J}_{x}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b+a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2$, ${J}_{y}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b\ensuremath{-}a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2i$, and the total photon number $⟨{N}_{a}+{N}_{b}⟩$. They include as special cases the inequality derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su(2) minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.

arXiv:quant-ph/0512180•Physical Review A•Jul 21, 2006•70 citations

Entanglement quantification through local observable correlations

Christian Kothe, Gunnar G. Bjork

We present a significantly improved scheme of entanglement detection inspired by local uncertainty relations for a system consisting of two qubits. Developing the underlying idea of local uncertainty relations, namely correlations, we demonstrate that it is possible to define a measure which is invariant under local unitary transformations and which is based only on local measurements. It is quite simple to implement experimentally and it allows entanglement quantification in a certain range for mixed states and exactly for pure states, without first obtaining full knowledge (e.g., through tomography) of the state.

arXiv:quant-ph/0608041•Physical Review A•Jan 30, 2007•22 citations

Experimental Investigation of Uncertainty Relations for Non-Hermitian Operators

Xinzhi Zhao, Xinglei Yu, Wenting Zhou +4 more

Uncertainty relations for Hermitian operators have been confirmed through many experiments. However, previous experiments have only tested the special case of non-Hermitian operators, i.e., uncertainty relations for unitary operators. In this study, we explore uncertainty relations for general non-Hermitian operators, which include Hermitian and unitary operators as special cases. We perform experiments with both real and complex non-Hermitian operators for qubit states, and confirm the validity of the uncertainty relations within the experimental error. Our results provide experimental evidence of uncertainty relations for non-Hermitian operators. Furthermore, our methods for realizing and measuring non-Hermitian operators are valuable in characterizing open-system dynamics and enhancing parameter estimation.

arXiv:2508.12214•Physical Review Letters•Feb 15, 2024•10 citations

Conditions for entanglement in multipartite systems

Mark S. Hillery, Ho Trung Dung, Hongjun Zheng

We introduce two entanglement conditions that take the form of inequalities involving expectation values of operators. These conditions are sufficient conditions for entanglement; that is, if they are satisfied the state is entangled, but if they are not, one can say nothing about the entanglement of the state. These conditions are quite flexible, because the operators in them are not specified, and they are particularly useful in detecting multipartite entanglement. We explore the range of utility of these conditions by considering a number of examples of entangled states, and seeing under what conditions entanglement in them can be detected by the inequalities presented here.

arXiv:1005.4453•Physical Review A•Jun 17, 2010•47 citations
We derive several entanglement conditions employing non-Hermitian operators. We start with two conditions that were derived previously for field mode operators and use them to derive conditions that can be used to show the existence of field-atom entanglement and entanglement between groups of atoms. The original conditions can be strengthened by making them invariant under certain sets of local unitary transformations such as Gaussian operations. We then apply these conditions to several examples such as the Dicke model. We conclude with a short discussion of how local uncertainty relations with non-Hermitian operators can be used to derive entanglement conditions.
There is no single test that will determine whether a general quantum state is separable or entangled. There are, however, a number of sufficient conditions, including the partial-transpose condition and entanglement witnesses. Here we present a class of sufficient conditions based on expectation values of non-hermitian operators. Initially we examine bipartite systems. The application of these conditions to multimode states of the electromagnetic field and to spin systems is discussed. We show that our original conditions can be strengthened if we make them invariant under certain classes of local unitary transformations. We then go on to study multipartite systems and present conditions that can be used to show that a state is not totally separable. Some simple applications of these conditions are also presented.
The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.
We derive a class of inequalities, from the uncertainty relations of the su(1,1) and the su(2) algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su(2) operators ${J}_{x}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b+a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2$, ${J}_{y}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b\ensuremath{-}a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2i$, and the total photon number ⟨Na+Nb⟩⟨{N}_{a}+{N}_{b}⟩⟨Na​+Nb​⟩. They include as special cases the inequality derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su(2) minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.
We present a significantly improved scheme of entanglement detection inspired by local uncertainty relations for a system consisting of two qubits. Developing the underlying idea of local uncertainty relations, namely correlations, we demonstrate that it is possible to define a measure which is invariant under local unitary transformations and which is based only on local measurements. It is quite simple to implement experimentally and it allows entanglement quantification in a certain range for mixed states and exactly for pure states, without first obtaining full knowledge (e.g., through tomography) of the state.
Uncertainty relations for Hermitian operators have been confirmed through many experiments. However, previous experiments have only tested the special case of non-Hermitian operators, i.e., uncertainty relations for unitary operators. In this study, we explore uncertainty relations for general non-Hermitian operators, which include Hermitian and unitary operators as special cases. We perform experiments with both real and complex non-Hermitian operators for qubit states, and confirm the validity of the uncertainty relations within the experimental error. Our results provide experimental evidence of uncertainty relations for non-Hermitian operators. Furthermore, our methods for realizing and measuring non-Hermitian operators are valuable in characterizing open-system dynamics and enhancing parameter estimation.
We introduce two entanglement conditions that take the form of inequalities involving expectation values of operators. These conditions are sufficient conditions for entanglement; that is, if they are satisfied the state is entangled, but if they are not, one can say nothing about the entanglement of the state. These conditions are quite flexible, because the operators in them are not specified, and they are particularly useful in detecting multipartite entanglement. We explore the range of utility of these conditions by considering a number of examples of entangled states, and seeing under what conditions entanglement in them can be detected by the inequalities presented here.