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Local quora for entanglement detection

Giacomo Mauro D’Ariano, Chiara Macchiavello, Matteo G. A. Paris

Published Nov 22, 2002• 0 citations
OpenAlex

Abstract

We present an explicit construction of entanglement witnesses for depolarized states in arbitrary finite dimension. For infinite dimension we generalize the construction to twin-beams perturbed by Gaussian noises in the phase and in the amplitude of the field. We show that entanglement detection for all these families of states requires only three local measurements. The explicit form of the corresponding set of local observables (quorom) needed for entanglement witness is derived.

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We present an explicit construction of entanglement witnesses for depolarized states in an arbitrary finite dimension. For an infinite dimension, we generalize the construction to twin beams perturbed by Gaussian noises in the phase and in the amplitude of the field. We show that entanglement detection for all these families of states require only three local measurements. The explicit form of the corresponding set of local observables needed for an entanglement witness is derived.

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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.

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Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.

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Construction of a Non-Linear Entanglement Witness Operator in Arbitrary Dimension Using a Given Linear Witness Operator

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Entanglement detection is one of the important problems in quantum information theory. To deal with this problem, many entanglement detection criteria have been proposed. Among the proposed criteria, the detection of entanglement through witness operator (also known as linear entanglement witness (LEW) operator) may be considered as the most practical. Although the witness operator approach to detect entanglement is experimentally friendly, the construction of these operators is not a very simple task. Even if we are able to construct a LEW operator, our problem is not solved as it may either detect a few entangled states or not a single entangled state from a given family of entangled states. Thus, we need a constructive approach in order to tackle this type of problem. In this work, we provide a few constructions of the non-linear entanglement witnesses (NLEW) for $d_1\otimes d_2$ dimensional system from any linear entanglement witness (LEW) operator. The advantage of these constructions is that, if a LEW is unable to detect any particular entangled state described by the density operator $ρ^{ent}$ then our construction of NLEW may detect the same entangled state $ρ^{ent}$. Further, we have constructed NLEW operator that may detect not only a class of bipartite negative partial transpose entangled state (NPTES), but also positive partial transpose entangled state (PPTES). Moreover, we have shown that the constructed NLEW operators may be decomposed in terms of the tensor product of local observables and hence may be realizable in an experiment.

quant-pharXiv:2605.03605•May 6, 2026•0 citations
We present an explicit construction of entanglement witnesses for depolarized states in arbitrary finite dimension. For infinite dimension we generalize the construction to twin-beams perturbed by Gaussian noises in the phase and in the amplitude of the field. We show that entanglement detection for all these families of states requires only three local measurements. The explicit form of the corresponding set of local observables (quorom) needed for entanglement witness is derived.
We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of a 3x3 bound entangled state [P. Horodecki, Phys. Lett. A 232, 333 (1997)] is explicitly constructed and a family of dxd bound entangled states is introduced, whose entanglement can be detected by permuting local orthogonal observables. The proposed criterion of separability can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables.
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.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
Entanglement detection and estimation are fundamental problems in quantum information science. Compared with discrete-variable states, for which lots of efficient entanglement detection criteria and lower bounds of entanglement measures have been proposed, the continuous-variable entanglement is much less understood. Here we shall present a family of entanglement witnesses based on continuous-variable local orthogonal observables (CVLOOs) to detect and estimate entanglement of Gaussian and non-Gaussian states, especially for bound entangled states. By choosing an optimal set of CVLOOs, our entanglement witness is equivalent to the realignment criterion and can be used to detect bound entanglement of a class of 2+2 mode Gaussian states. Via our entanglement witness, lower bounds of two typical entanglement measures for arbitrary two-mode continuous-variable states are provided.
Entanglement detection is one of the important problems in quantum information theory. To deal with this problem, many entanglement detection criteria have been proposed. Among the proposed criteria, the detection of entanglement through witness operator (also known as linear entanglement witness (LEW) operator) may be considered as the most practical. Although the witness operator approach to detect entanglement is experimentally friendly, the construction of these operators is not a very simple task. Even if we are able to construct a LEW operator, our problem is not solved as it may either detect a few entangled states or not a single entangled state from a given family of entangled states. Thus, we need a constructive approach in order to tackle this type of problem. In this work, we provide a few constructions of the non-linear entanglement witnesses (NLEW) for d1⊗d2d_1\otimes d_2d1​⊗d2​ dimensional system from any linear entanglement witness (LEW) operator. The advantage of these constructions is that, if a LEW is unable to detect any particular entangled state described by the density operator ρentρ^{ent}ρent then our construction of NLEW may detect the same entangled state ρentρ^{ent}ρent. Further, we have constructed NLEW operator that may detect not only a class of bipartite negative partial transpose entangled state (NPTES), but also positive partial transpose entangled state (PPTES). Moreover, we have shown that the constructed NLEW operators may be decomposed in terms of the tensor product of local observables and hence may be realizable in an experiment.