Searcharxiv⌕ Search

arXiv subjects

Seung-Hyeok Kye

Publications and source records attributed to Seung-Hyeok Kye.

At least 19 recordsLinked to original sources
arXiv2026

Local geometry for Schmidt number witnesses

Suppose that $F_E$ is the face of the convex set of all $m\otimes n$ bi-partite states which consists of states with ranges contained in a subspace $E$. For generic subspaces $E$ with a specific dimension, we use the result in [Phys. Rev. A 112 (2025), 032426] to see that there exists a number $κ$, depending only on the dimension of $E$, such that there exist Schmidt number $\ell$ witnesses outside of $F_{E^\perp}$ if and only if $\ell\leκ$. In this generic case, we show in this paper that there exist Schmidt number $\ell$ witnesses for $\ell>κ$ around the projection states located at the center of $F_{E^\perp}$.

quant-ph↗
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.

quant-ph↗
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.

quant-ph↗
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.

quant-ph↗
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.

quant-ph↗
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.

math.OA↗
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.

quant-ph↗
arXiv2023

Exposedness of elementary positive maps between matrix algebras

The positive linear maps $\ad_s$ which send matrices $x$ to $s^*xs$ play important roles in quantum information theory as well as matrix theory. It was proved by Marciniak [Linear Multilinear Alg. 61 (2013), 970--975] that the map $\ad_s$ generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

quant-ph↗
arXiv2022

Compositions and tensor products of linear maps between matrix algebras

In this semi-expository paper, we first explain key notions from current quantum information theory and criteria for them in a coherent way. These include separability/entanglement, Schmidt numbers of bi-partite states and block-positivity, together with various kinds of positive maps between matrix algebras like entanglement breaking maps, $k$-superpositive maps, completely positive maps, $k$-positive maps. We will begin with concrete examples of elementary positive maps given by $x\mapsto s^*xs$, and use Choi matrices and duality to explain all the notions mentioned above. We also show that the Choi matrix can be defined free from coordinates. The above notions of positive maps give rise to mapping cones, whose dual cones are characterized in terms of compositions or tensor products of linear maps. Through the discussion, we exhibit an identity which connects tensor products and compositions of linear maps between matrix algebras through the Choi matrices. Using this identity, we show that the description of the dual cone with tensor products is possible only when the involving cones are mapping cones, and recover various known criteria with ampliation for the notions mentioned above. As another applications of the identity, we construct various mapping cones arising from ampliation and factorization, and provide several equivalent statements to PPT (positive partial transpose) square conjecture in terms of tensor products.

quant-ph↗
arXiv2022

Choi matrices revisited

A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras, whose definition depends on the matrix units. Paulsen and Shultz [J. Math. Phys. {\bf 54} (2013), 072201] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of Choi matrix to retain the correspondence between complete positivity of maps and positivity of Choi matrices, and gave a sufficient condition on basis under which this is true. In this note, we provide necessary and sufficient conditions, to see that the Paulsen--Shultz condition is also necessary.

math.OA↗
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.

quant-ph↗
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.

quant-ph↗
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.

quant-ph↗
arXiv2020

Entangled edge states of corank one with positive partial transposes

We construct a parameterized family of $n\otimes n$ PPT (positive partial transpose) states of corank one for each $n\ge 3$. With a suitable choice of parameters, we show that they are $n\otimes n$ PPT entangled edge states of corank one for $3\le n\le 1000$. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by $2n-3$, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first explicit examples of such states for $n\ge 4$.

quant-ph↗
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.

quant-ph↗
arXiv2020

Convex cones in mapping spaces between matrix algebras

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The duals of such convex cones can be characterized in terms of ampliation maps, which can also be used to characterize many notions from quantum information theory---such as separability, entanglement-breaking maps, Schmidt numbers, as well as decomposable maps and $k$-positive maps in functional analysis. In fact, such characterizations hold if and only if the involved cone is a one-sided mapping cone. Through this analysis, we obtain mapping properties for compositions of cones from which we also obtain several equivalent statements of the PPT (positive partial transpose) square conjecture.

math.OA↗
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.

quant-ph↗