The structural physical approximation (SPA) to a positive map is considered to be one of the most important methods to detect entanglement in the real physical world. We first show that an arbitrary entanglement witness (EW) $W$ can be constructed from a separable density matrix $\ensuremath{\sigma}$ in the form of $W=\ensuremath{\sigma}\ensuremath{-}{c}_{\ensuremath{\sigma}}I$, where ${c}_{\ensuremath{\sigma}}$ is a non-negative number and $I$ is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [Phys. Rev. A 78, 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.The structural physical approximation (SPA) to a positive map is considered to be one of the most important methods to detect entanglement in the real physical world. We first show that an arbitrary entanglement witness (EW) $W$ can be constructed from a separable density matrix $\ensuremath{\sigma}$ in the form of $W=\ensuremath{\sigma}\ensuremath{-}{c}_{\ensuremath{\sigma}}I$, where ${c}_{\ensuremath{\sigma}}$ is a non-negative number and $I$ is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [Phys. Rev. A 78, 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.

arXiv:1106.5233Physical Review AJun 21, 201314 citations

Integral representations of separable states

Bronisław Jakubczyk, Gabriel Pietrzkowski

arXiv:0802.0291Reports on Mathematical PhysicsFeb 1, 20092 citations

On structural physical approximations and entanglement breaking maps

Remigiusz Augusiak, J Bae, Ł Czekaj +1 more

Very recently, a conjecture saying that the so-called structural physical approximations (SPAs) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed (Korbicz et al 2008 Phys. Rev. A 78 062105). The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses supporting the conjecture. Then, we ask whether SPAs constructed from other than the depolarizing channel maps also lead to EB maps and show that in general this is not the case. On the other hand, we prove an interesting fact that for any positive map Λ, there exists an EB channel Φ such that the SPA of Λ constructed with the aid of Φ is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of constructing SPA and prove that in the case of the transposition map it gives the EB channel.Very recently, a conjecture saying that the so-called structural physical approximations (SPAs) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed (Korbicz et al 2008 Phys. Rev. A 78 062105). The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses supporting the conjecture. Then, we ask whether SPAs constructed from other than the depolarizing channel maps also lead to EB maps and show that in general this is not the case. On the other hand, we prove an interesting fact that for any positive map Λ, there exists an EB channel Φ such that the SPA of Λ constructed with the aid of Φ is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of constructing SPA and prove that in the case of the transposition map it gives the EB channel.

arXiv:1008.5056Journal of Physics A Mathematical and TheoreticalApr 11, 201133 citations

Characterization of optimal entanglement witnesses

Xiaofei Qi, Jinchuan Hou

In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).

arXiv:1109.2300Physical Review AFeb 27, 201227 citations

The Structure of Quantum States

Gary E. Bowman

Oxford University Press eBooksNov 1, 20070 citations

W

can be constructed from a separable density matrix $\ensuremath{\sigma}$ in the form of $W=\ensuremath{\sigma}\ensuremath{-}{c}_{\ensuremath{\sigma}}I$, where ${c}_{\ensuremath{\sigma}}$ is a non-negative number and

I

is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [Phys. Rev. A 78, 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.