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Kyung Hoon Han

Publications and source records attributed to Kyung Hoon Han.

At least 19 recordsLinked to original sources
arXiv2026

Bi-qutrit entangled edge states of positive partial transposes with largest ranks

Whenever $E$ is an eight dimensional subspace of the bi-qutrit quantum system whose orthogonal complement is spanned by a vector of Schmidt rank three, we show that there exist PPT entangled edge states with the range space $E$ whose partial transposes are of rank six, which is the largest possible rank. In this way, we exhibit a huge family of bi-qutrit PPT entangled edge states of type $(8,6)$. They make faces of the convex set of all PPT states, and we find bi-qutrit PPT entangled edge states of other types on the boundaries of such faces.

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arXiv2025

Global locations of Schmidt number witnesses

We investigate global locations of Schmidt number witnesses which are outside of the convex set of all bipartite states. Their locations are classified by interiors of faces of the convex set of all states, by considering the line segments from them to the maximally mixed state. In this way, a nonpositive Hermitian matrix of trace 1 is located outside of one and only one face. Faces of the convex set of all states are classified by subspaces, which are range spaces of states belonging to specific faces. For a given subspace, we show that there exist Schmidt number $k+1$ witnesses outside of the face arising from this subspace if and only if every vector in the orthogonal complement of the subspace has Schmidt rank greater than $k$. Once we have Schmidt number $k+1$ witnesses outside of a face, we also have Schmidt number $2,3,\dots, k$ witnesses outside of the face.

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arXiv2025

Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses

We consider the compact convex set of all bi-partite states of Schmidt number less than or equal to $k$, together with that of $k$-blockpositive matrices of trace one, which play the roles of Schmidt number witnesses. In this note, we look for hyperplanes which support those convex sets and are perpendicular to a one parameter family through the maximally mixed state. We show that this is equivalent to determining the intervals for the dual objects on the one parameter family. We illustrate our results for the one parameter families including Werner states and isotropic states. Through the discussion, we give a simple decomposition of the separable Werner state into the sum of product states.

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arXiv2024

Choi matrices revisited. III

We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send $k$-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to $k$. They also send $k$-positive maps onto $k$-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between $k$-positivity and Schmidt numbers $\le k$. They also retain the duality between $k$-superpositivity and $k$-block-positivity.

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arXiv2024

Infinite dimensional analogues of Choi matrices

For a class of linear maps on a von Neumann factor, we associate two objects, bounded operators and trace class operators, both of which play the roles of Choi matrices. Each of them is positive if and only if the original map on the factor is completely positive. They are also useful to characterize positivity of maps as well as complete positivity. It turns out that such correspondences are possible for every normal completely bounded map if and only if the factor is of type I. As an application, we provide criteria for Schmidt numbers of normal positive functionals in terms of Choi matrices of $k$-positive maps, in infinite dimensional cases. We also define the notion of $k$-superpositive maps, which turns out to be equivalent to the property of $k$-partially entanglement breaking.

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arXiv2023

Choi matrices revisited. II

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between $k$-superpositivity and Schmidt number $\le k$ as well as $k$-positivity and $k$-block-positivity. We also compare de Pillis' definition [Pacific J. Math. 23 (1967), 129--137] and Choi's definition [Linear Alg. Appl. 10 (1975), 285--290], which arise from different bilinear forms.

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arXiv2022

There exist infinitely many kinds of partial separability/entanglement

In tri-partite systems, there are three basic biseparability, $A$-$BC$, $B$-$CA$ and $C$-$AB$ biseparability according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members, to see that there are infinitely many kinds of three qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three qubit Greenberger-Horne-Zeilinger diagonal states distinguishing those convex sets in the chain.

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arXiv2021

Polytope structures for Greenberger-Horne-Zeilinger diagonal states

We explore the polytope structures for genuine entanglement, biseparability, full biseparability and Bell inequality of multi-qubit GHZ diagonal states. We first show that biseparable GHZ diagonal states make hypersimplices inside the simplices consisting of all GHZ diagonal states. Next, we consider full biseparability which is equivalent to positive partial transpose for GHZ diagonal states, and show that they make the convex hulls of simplices and cubes. We also visualize which part of the simplex violates multipartite Bell inequality. Finally, we compute precise volumes for genuine entanglement, biseparability, full biseparability and states violating Bell inequality among all GHZ diagonal states.

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arXiv2021

Criteria for partial entanglement of three qubit states arising from distributive rules

It is known that the partial entanglement/separability violates distributive rules with respect to the operations of taking convex hull and intersection. In this note, we give criteria for three qubit partially entangled states arising from distributive rules, together with the corresponding witnesses. The criteria will be given in terms of diagonal and anti-diagonal entries. They actually characterize those partial entanglement completely when all the entries are zero except for diagonal and anti-diagonal entries. Important states like Greenberger-Horne-Zeilinger diagonal states fall down in this class.

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arXiv2020

Partial separability/entanglement violates distributive rules

We found three qubit Greenberger-Horne-Zeilinger diagonal states which tells us that the partial separability of three qubit states violates the distributive rules with respect to the two operations of convex sum and intersection. The gaps between the convex sets involving the distributive rules are of nonzero volume.

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arXiv2019

On the convex cones arising from classifications of partial entanglement in the three qubit system

In order to classify partial entanglement of multi-partite states, it is natural to consider the convex hulls, intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems. In this paper, we consider convex cones consisting of X-shaped three qubit states arising in this way. The class of X-shaped states includes important classes like Greenberger-Horne-Zeilinger diagonal states. We find all the extreme rays of those convex cones to exhibit corresponding partially separable states. We also give characterizations for those cones which give rise to necessary criteria in terms of diagonal and anti-diagonal entries for general three qubit states.

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arXiv2018

Separability of multi-qubit states in terms of diagonal and anti-diagonal entries

We give separability criteria for general multi-qubit states in terms of diagonal and anti-diagonal entries. We define two numbers which are obtained from diagonal and anti-diagonal entries, respectively, and compare them to get criteria. They give rise to characterizations of separability when all the entries are zero except for diagonal and anti-diagonal, like Greenberger-Horne-Zeilinger diagonal states. The criteria is strong enough to get nonzero volume of entanglement with positive partial transposes.

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arXiv2017

The role of phases in detecting three-qubit entanglement

We propose separability criteria for three-qubit states in terms of diagonal and anti-diagonal entries to detect entanglement with positive partial transposes. We report that the phases, that is, the angular parts of anti-diagonal entries, play a crucial role in determining whether a given three-qubit state is separable or entangled, and they must obey even an identity for separability in some cases. These criteria are strong enough to detect PPT (positive partial transpose) entanglement with nonzero volume. In several cases when all the entries are zero except for diagonal and anti-diagonal entries, we characterize separability using phases. These include the cases when anti-diagonal entries of such states share a common magnitude, and when ranks are less than or equal to six. We also compute the lengths of rank six cases, and find three-qubit separable states with lengths $8$ whose maximum ranks of partial transposes are $7$.

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arXiv2017

Separability criterion for three-qubit states with a four dimensional norm

We give a separability criterion for three qubit states in terms of diagonal and anti-diagonal entries. This gives us a complete characterization of separability when all the entries are zero except for diagonal and anti-diagonals. The criterion is expressed in terms of a norm arising from anti-diagonal entries. We compute this norm in several cases, so that we get criteria with which we can decide the separability by routine computations.

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arXiv2017

Separability of three qubit Greenberger-Horne-Zeilinger diagonal states

We characterize the separability of three qubit GHZ diagonal states in terms of entries. This enables us to check separability of GHZ diagonal states without decomposition into the sum of pure product states. In the course of discussion, we show that the necessary criterion of Gühne for (full) separability of three qubit GHZ diagonal states is sufficient with a simpler formula. The main tool is to use entanglement witnesses which are tri-partite Choi matrices of positive bi-linear maps.

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arXiv2016

A Kirchberg type tensor theorem for operator systems

We construct operator systems $\mathfrak C_I$ that are universal in the sense that all operator systems can be realized as their quotients. They satisfy the operator system lifting property. Without relying on the theorem by Kirchberg, we prove the Kirchberg type tensor theorem $$\mathfrak C_I \otimes_{\min} B(H) = \mathfrak C_I \otimes_{\max} B(H).$$ Combining this with a result of Kavruk, we give a new operator system theoretic proof of Kirchberg's theorem and show that Kirchberg's conjecture is equivalent to its operator system analogue $$\mathfrak C_I \otimes_{\min} \mathfrak C_I =\mathfrak C_I \otimes_{\rm c} \mathfrak C_I.$$ It is natural to ask whether the universal operator systems $\mathfrak C_I$ are projective objects in the category of operator systems. We show that an operator system from which all unital completely positive maps into operator system quotients can be lifted is necessarily one-dimensional. Moreover, a finite dimensional operator system satisfying a perturbed lifting property can be represented as the direct sum of matrix algebras. We give an operator system theoretic approach to the Effros-Haagerup lifting theorem.

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arXiv2016

Construction of multi-qubit optimal genuine entanglement witnesses

We interpret multi-partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them. We apply this result to multi-qubit {\sf X}-shaped Hermitian matrices, and characterize the conditions for them to be genuine entanglement witnesses, in terms of entries. Furthermore, we find all optimal ones among them. They turn out to have the spanning properties, and so they detect non-zero volume set of multi-qubit genuine entanglement. We also characterize decomposability for {\sf X}-shaped entanglement witnesses.

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