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Chinese Journal of Quantum Electronics

Entanglement detection and measurement of quantum systems

Shuang Cong

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Various kinds of methods of entanglement detection and definitions of entanglement measurement in quantum systems were reviewed.For entanglement of two or more particles,some separability criterions were summarized.Their relationships with positive map and the relationships between themselves were also analyzed.Particularly,concentrations on entanglement witness are paid which is a special criterion involving its definition and construction methods.Further,its applications are also investigated from the viewpoint of experiment.Based on axiomatic hypothesis of entanglement measurement,the theoretical definitions of two-particle and multipartite entanglement,and their estimations in experiment were discussed.At last, various nonlinear separability criterions were analyzed.

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Various kinds of methods of entanglement detection and definitions of entanglement measurement in quantum systems were reviewed.For entanglement of two or more particles,some separability criterions were summarized.Their relationships with positive map and the relationships between themselves were also analyzed.Particularly,concentrations on entanglement witness are paid which is a special criterion involving its definition and construction methods.Further,its applications are also investigated from the viewpoint of experiment.Based on axiomatic hypothesis of entanglement measurement,the theoretical definitions of two-particle and multipartite entanglement,and their estimations in experiment were discussed.At last, various nonlinear separability criterions were analyzed.
Entanglement finds wide application in the development of quantum information theory. After reviewing states in classical mechanics, this chapter discusses the fundamental aspects of characterization of quantum entanglement like the definition, the detection, the classification and quantification of entanglement. The density operator is used to describe the state of a quantum system in a more general way than provided by the state vector of the Hilbert space. Multipartite entanglement theory is much complex compared to the bipartite case. One cannot generalize the results of bipartite entanglement to multipartite entanglement. The Peres criterion known as positive partial transpose criterion is an important criterion to find the separability of mixed states at least for low-dimensional systems. B.W. Terhal introduced the term entanglement witnesses for operators detecting quantum entanglement states. It is found that any entangled state can be detected by some entanglement witnesses.
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.
Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
[EN] In this Bachelor Thesis, I present some of the fundaments of entanglement detection. I introduce basic entanglement criteria namely the PPT and CCNR criteria. Entanglement witnesses are defined and used in illustrative examples. I also explain Bell inequalities and their relation with other entanglement witnesses. Lastly, in the last section of the theoretical background I introduce some collective observables and show how we can use them to detect entanglement in our experiments. Once the basics have been explained, I present a new criterion that gives us a lower bound in the amount of entanglement particles of a system. This caracterizes the quantum system from a new perspective and can be of great interest in modern multipartite entanglement experiments. Some results will be used to validate our approach, demostrating that it complements previous methods.
In this review, we introduce some methods for detecting or measuring entanglement. Several nonlinear entanglement witnesses are presented. We derive a series of Bell inequalities whose maximally violations for any multipartite qubit states can be calculated by using our formulas. Both the nonlinear entanglement witnesses and the Bell inequalities can be operated experimentally. Thus they supply an effective way for detecting entanglement. We also introduce some experimental methods to measure the entanglement of formation, and the lower bound of the convex-roof extension of negativity.