We show that quantum designs characterize the general structure of the optimal approximation of the transpose map on quantum states. Based on this characterization, we propose an implementation of the approximate transpose map by a measurement-and-preparation scheme. The results show that state manipulation in quantum two-designs suffices for transpose-based quantum-information applications. In particular, we present how these results can be applied to the framework of detecting multipartite entangled states, for instance, when local measurements or interferometry-based experimental approaches are applied.We show that quantum designs characterize the general structure of the optimal approximation of the transpose map on quantum states. Based on this characterization, we propose an implementation of the approximate transpose map by a measurement-and-preparation scheme. The results show that state manipulation in quantum two-designs suffices for transpose-based quantum-information applications. In particular, we present how these results can be applied to the framework of detecting multipartite entangled states, for instance, when local measurements or interferometry-based experimental approaches are applied.

arXiv:1303.3096Physical Review AJun 12, 201324 citations

Detecting Quantum Entanglement: Positive Maps and Entanglement Witnesses

Dariusz Chruciski

InTech eBooksJul 24, 20131 citation

Steering Maps and Their Application to Dimension-Bounded Steering

Tobias Moroder, Oleg Gittsovich, Marcus Huber +2 more

The existence of quantum correlations that allow one party to steer the quantum state of another party is a counterintuitive quantum effect that was described at the beginning of the past century. Steering occurs if entanglement can be proven even though the description of the measurements on one party is not known, while the other side is characterized. We introduce the concept of steering maps, which allow us to unlock sophisticated techniques that were developed in regular entanglement detection and to use them for certifying steerability. As an application, we show that this allows us to go beyond even the canonical steering scenario; it enables a generalized dimension-bounded steering where one only assumes the Hilbert space dimension on the characterized side, with no description of the measurements. Surprisingly, this does not weaken the detection strength of very symmetric scenarios that have recently been carried out in experiments.The existence of quantum correlations that allow one party to steer the quantum state of another party is a counterintuitive quantum effect that was described at the beginning of the past century. Steering occurs if entanglement can be proven even though the description of the measurements on one party is not known, while the other side is characterized. We introduce the concept of steering maps, which allow us to unlock sophisticated techniques that were developed in regular entanglement detection and to use them for certifying steerability. As an application, we show that this allows us to go beyond even the canonical steering scenario; it enables a generalized dimension-bounded steering where one only assumes the Hilbert space dimension on the characterized side, with no description of the measurements. Surprisingly, this does not weaken the detection strength of very symmetric scenarios that have recently been carried out in experiments.

arXiv:1412.2623Physical Review LettersMar 3, 201642 citations

Bound Entanglement Detection in 4 ⊗ 4 Systems via Generalized Choi Maps

Mazhar Ali

We construct a family of positive but not completely positive linear maps acting on $M_4(\mathcal{C})$ , obtained as a natural extension of Kye’s indecomposable maps defined for $M_3(\mathcal{C})$ . We rigorously prove positivity of these maps on 'X' states and employ them to identify a one-parameter family of bipartite quantum states living in the vicinity of maximally mixed state to be entangled even though states are positive under partial transposition (PPT). This provides an explicit example of a family of quantum states with both bound entangled states and free entangled states in $4 \otimes 4$ systems, a regime that remains less characterized compared to lower-dimensional cases. The proposed maps detect entanglement and reveal new structural features of the PPT entangled region in higher dimensions. Our results extend the applicability of positive-map–based entanglement detection and contribute to the systematic understanding of bound entanglement beyond the $3\otimes 3$ and $2 \otimes 4$ systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for $2 \otimes 4$ systems.We construct a family of positive but not completely positive linear maps acting on $M_4(\mathcal{C})$, obtained as a natural extension of Kye’s indecomposable maps defined for $M_3(\mathcal{C})$. We rigorously prove positivity of these maps on 'X' states and employ them to identify a one-parameter family of bipartite quantum states living in the vicinity of maximally mixed state to be entangled even though states are positive under partial transposition (PPT). This provides an explicit example of a family of quantum states with both bound entangled states and free entangled states in $4 \otimes 4$ systems, a regime that remains less characterized compared to lower-dimensional cases. The proposed maps detect entanglement and reveal new structural features of the PPT entangled region in higher dimensions. Our results extend the applicability of positive-map–based entanglement detection and contribute to the systematic understanding of bound entanglement beyond the $3\otimes 3$ and $2 \otimes 4$ systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for $2 \otimes 4$ systems.

Feb 6, 20260 citations

Quantum Mappings and Characterization of Entangled Quantum States

Sergey N. Filippov

Journal of Mathematical SciencesAug 1, 201916 citations

We construct a family of positive but not completely positive linear maps acting on

M_4(\mathcal{C})

, obtained as a natural extension of Kye’s indecomposable maps defined for

M_3(\mathcal{C})

. We rigorously prove positivity of these maps on 'X' states and employ them to identify a one-parameter family of bipartite quantum states living in the vicinity of maximally mixed state to be entangled even though states are positive under partial transposition (PPT). This provides an explicit example of a family of quantum states with both bound entangled states and free entangled states in

4 \otimes 4

systems, a regime that remains less characterized compared to lower-dimensional cases. The proposed maps detect entanglement and reveal new structural features of the PPT entangled region in higher dimensions. Our results extend the applicability of positive-map–based entanglement detection and contribute to the systematic understanding of bound entanglement beyond the

3\otimes 3

and

2 \otimes 4

systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for

2 \otimes 4

systems.