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:1407.0870 · Quantum Information and Computation

Characterization and properties of weakly optimal entanglement witnesses

Bang-Hai Wang, Hai-Ru Xu, Steve Campbell, Simone Severini B

Published Oct 1, 2015• 4 citations
arXivHTMLPDF

Abstract

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max} I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.

DOI
10.26421/qic15.13-14-2

Related papers

Separability in terms of a single entanglement witness

Piotr Badziąg, Paweł Horodecki, Ryszard Horodecki +1 more

The separability problem is formulated in terms of a characterization of a single entanglement witness. More specifically, we show that any (in general multipartite) state $\ensuremath{\varrho}$ is separable if and only if a specially constructed entanglement witness ${W}_{\ensuremath{\varrho}}$ is weakly optimal, i.e., its expectation value vanishes on at least one product vector. Interestingly, the witness can always be chosen to be decomposable. Our result changes the conceptual aspect of the separability problem and raises some questions about the properties of positive maps.

arXiv:quant-ph/0703097•Physical Review A•Jul 3, 2013•12 citations

Optimal decomposable witnesses without the spanning property

Remigiusz Augusiak, Gniewomir Sarbicki, Maciej Lewenstein

One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces ${\mathcal{H}}_{m,n}={\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ such that the product vectors obeying $\ensuremath{\langle}e,f|W|e,f\ensuremath{\rangle}=0$ do not span ${\mathcal{H}}_{m,n}$. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of $2\ensuremath{\bigotimes}n$ optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces ${\mathcal{H}}_{m,n}$ with $m,n\ensuremath{\ge}3$.

arXiv:1107.0505•Physical Review A•Nov 21, 2011•24 citations

Extremal entanglement witnesses

Leif Ove Hansen, Andreas Hauge, Jan Myrheim +1 more

We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus $\textrm{Tr}\,Ωρ\geq 0$ when $Ω$ is a witness and $ρ$ a pure product state, $ρ=ψψ^{\dagger}$ with $ψ=ϕ\otimesχ$. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value $f(ϕ,χ)=\mathrm{Tr}\,Ωρ$ as a function of $ϕ$ and $χ$ is a nonnegative biquadratic form. Every zero of $f(ϕ,χ)$ imposes real-linear constraints on $f$ and $Ω$. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of $f(ϕ,χ)$ is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions lead to an extremal witness which is usually quadratic with the minimum number of zeros. We discuss some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the structural physical approximation (SPA) of an optimal witness. We discuss how to treat the entanglement witnesses on a complex Hilbert space as witnesses on a real Hilbert space.

arXiv:1305.2385•May 10, 2013•3 citations

The structure of ultrafine entanglement witnesses

Mariami Gachechiladze, Nikolai Wyderka, Otfried Gühne

Abstract An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification of entanglement. Second, we present necessary and sufficient conditions that two product observables are able to detect entanglement. Finally, we explain some fallacies in the original construction of ultrafine entanglement witnesses (Shahandeh et al 2017 Phys. Rev. Lett . 118 110502).

arXiv:1805.06404•Journal of Physics A Mathematical and Theoretical•Jul 17, 2018•3 citations

Extremal entanglement witnesses

Leif Ove Hansen, Andreas Hauge, Jan Myrheim +1 more

We present a study of extremal entanglement witnesses on a bipartite composite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus [Formula: see text] when [Formula: see text] is a witness and [Formula: see text] is a pure product state, [Formula: see text] with [Formula: see text]. The set of witnesses of unit trace is a compact convex set, uniquely defined by its extremal points. The expectation value [Formula: see text] as a function of vectors [Formula: see text] and [Formula: see text] is a positive semidefinite biquadratic form. Every zero of [Formula: see text] imposes strong real-linear constraints on f and [Formula: see text]. The real and symmetric Hessian matrix at the zero must be positive semidefinite. Its eigenvectors with zero eigenvalue, if such exist, we call Hessian zeros. A zero of [Formula: see text] is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic if it has at least one quartic zero. A main result we prove is that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, then the constraints defined by its zeros and Hessian zeros determine all directions in which we may search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions, by numerical methods, leads to an extremal witness which is nearly always quadratic and has the minimum number of zeros. We discuss briefly some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the so-called structural physical approximation (SPA) of an optimal witness. Finally, we discuss how to treat the entanglement witnesses on a complex Hilbert space as a subset of the witnesses on a real Hilbert space.

International Journal of Quantum Information•Dec 1, 2015•11 citations

Spectral characterizations of entanglement witnesses

Zhiwei Song, Lin Chen

We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-partial-transpose state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.

arXiv:2508.11308•Journal of Mathematical Physics•Jul 1, 2026•0 citations
We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of Wwopt=σ−cσmaxIW^{wopt}=\sigma-c_{\sigma}^{max} IWwopt=σ−cσmax​I, where cσmaxc_{\sigma}^{max}cσmax​ is a non-negative number and III is the identity matrix. We show the relation between the weakly optimal witness WwoptW^{wopt}Wwopt and the eigenvalues of the separable states σ\sigmaσ. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.
The separability problem is formulated in terms of a characterization of a single entanglement witness. More specifically, we show that any (in general multipartite) state $\ensuremath{\varrho}$ is separable if and only if a specially constructed entanglement witness ${W}_{\ensuremath{\varrho}}$ is weakly optimal, i.e., its expectation value vanishes on at least one product vector. Interestingly, the witness can always be chosen to be decomposable. Our result changes the conceptual aspect of the separability problem and raises some questions about the properties of positive maps.
One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces ${\mathcal{H}}_{m,n}={\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ such that the product vectors obeying $\ensuremath{\langle}e,f|W|e,f\ensuremath{\rangle}=0$ do not span Hm,n{\mathcal{H}}_{m,n}Hm,n​. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of $2\ensuremath{\bigotimes}n$ optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces Hm,n{\mathcal{H}}_{m,n}Hm,n​ with $m,n\ensuremath{\ge}3$.
We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus Tr Ωρ≥0\textrm{Tr}\,Ωρ\geq 0TrΩρ≥0 when ΩΩΩ is a witness and ρρρ a pure product state, ρ=ψψ†ρ=ψψ^{\dagger}ρ=ψψ† with ψ=ϕ⊗χψ=ϕ\otimesχψ=ϕ⊗χ. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value f(ϕ,χ)=Tr Ωρf(ϕ,χ)=\mathrm{Tr}\,Ωρf(ϕ,χ)=TrΩρ as a function of ϕϕϕ and χχχ is a nonnegative biquadratic form. Every zero of f(ϕ,χ)f(ϕ,χ)f(ϕ,χ) imposes real-linear constraints on fff and ΩΩΩ. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of f(ϕ,χ)f(ϕ,χ)f(ϕ,χ) is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions lead to an extremal witness which is usually quadratic with the minimum number of zeros. We discuss some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the structural physical approximation (SPA) of an optimal witness. We discuss how to treat the entanglement witnesses on a complex Hilbert space as witnesses on a real Hilbert space.
Abstract An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification of entanglement. Second, we present necessary and sufficient conditions that two product observables are able to detect entanglement. Finally, we explain some fallacies in the original construction of ultrafine entanglement witnesses (Shahandeh et al 2017 Phys. Rev. Lett . 118 110502).
We present a study of extremal entanglement witnesses on a bipartite composite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus [Formula: see text] when [Formula: see text] is a witness and [Formula: see text] is a pure product state, [Formula: see text] with [Formula: see text]. The set of witnesses of unit trace is a compact convex set, uniquely defined by its extremal points. The expectation value [Formula: see text] as a function of vectors [Formula: see text] and [Formula: see text] is a positive semidefinite biquadratic form. Every zero of [Formula: see text] imposes strong real-linear constraints on f and [Formula: see text]. The real and symmetric Hessian matrix at the zero must be positive semidefinite. Its eigenvectors with zero eigenvalue, if such exist, we call Hessian zeros. A zero of [Formula: see text] is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic if it has at least one quartic zero. A main result we prove is that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, then the constraints defined by its zeros and Hessian zeros determine all directions in which we may search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions, by numerical methods, leads to an extremal witness which is nearly always quadratic and has the minimum number of zeros. We discuss briefly some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the so-called structural physical approximation (SPA) of an optimal witness. Finally, we discuss how to treat the entanglement witnesses on a complex Hilbert space as a subset of the witnesses on a real Hilbert space.
We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-partial-transpose state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.