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Kil-Chan Ha

Publications and source records attributed to Kil-Chan Ha.

At least 19 recordsLinked to original sources
arXiv2021

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↗
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.

quant-ph↗
arXiv2014

Construction of exposed indecomposable positive linear maps between matrix algebras

We construct a large class of indecomposable positive linear maps from the $2\times 2$ matrix algebra into the $4\times 4$ matrix algebra, which generate exposed extreme rays of the convex cone of all positive maps. We show that extreme points of the dual faces for separable states arising from these maps are parametrized by the Riemann sphere, and the convex hulls of the extreme points arising from a circle parallel to the equator have the exactly same properties with the convex hull of the trigonometric moment curve studied from combinatorial topology. Any interior points of the dual faces are $2\otimes 4$ boundary separable states with full ranks. We exhibit concrete examples of such states.

math.OA↗
arXiv2014

Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose

We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decomposition, and show that this is true in most cases if the number of product vectors is sufficiently small. In the three qubit case, generic five dimensional spaces give rise to faces of the convex set consisting of all separable states, which are affinely isomorphic to the five dimensional simplex with six vertices. As a byproduct, we construct three qubit entangled PPT edge states of rank four with explicit formulae. This covers those entanglement which cannot be constructed from unextendible product basis.

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arXiv2013

Global geometric difference between separable and Positive partial transpose states

In the convex set of all $3\ot 3$ states with positive partial transposes, we show that one can take two extreme points whose convex combinations belong to the interior of the convex set. Their convex combinations may be even in the interior of the convex set of all separable states. In general, we need at least $mn$ extreme points to get an interior point by their convex combination, for the case of the convex set of all $m\ot n$ separable states. This shows a sharp distinction between PPT states and separable states. We also consider the same questions for positive maps and decomposable maps.

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arXiv2013

Separable states with unique decompositions

We search for faces of the convex set consisting of all separable states, which are affinely isomorphic to simplices, to get separable states with unique decompositions. In the two-qutrit case, we found that six product vectors spanning a five dimensional space give rise to a face isomorphic to the 5-dimensional simplex with six vertices, under suitable linear independence assumption. If the partial conjugates of six product vectors also span a 5-dimensional space, then this face is inscribed in the face for PPT states whose boundary shares the fifteen 3-simplices on the boundary of the 5-simplex. The remaining boundary points consist of PPT entangled edge states of rank four. We also show that every edge state of rank four arises in this way. If the partial conjugates of the above six product vectors span a 6-dimensional space then we have a face isomorphic to 5-simplex, whose interior consists of separable states with unique decompositions, but with non-symmetric ranks. We also construct a face isomorphic to the 9-simplex. As applications, we give answers to questions in the literature \cite{chen_dj_semialg,chen_dj_ext_PPT}, and construct $3\ot 3$ PPT states of type (9,5). For the qubit-qudit cases with $d\ge 3$, we also show that $(d+1)$-dimensional subspaces give rise to faces isomorphic to the $d$-simplices, in most cases.

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arXiv2013

Geometry for separable states and construction of entangled states with positive partial transposes

We construct faces of the convex set of all $2\otimes 4$ bipartite separable states, which are affinely isomorphic to the simplex $Δ_{9}$ with ten extreme points. Every interior point of these faces is a separable state which has a unique decomposition into 10 product states, even though ranks of the state and its partial transpose are 5 and 7, respectively. We also note that the number 10 is greater than $2\times 4$, to disprove a conjecture on the lengths of qubit-qudit separable states. This face is inscribed in the corresponding face of the convex set of all PPT states so that sub-simplices $Δ_k$ of $Δ_{9}$ share the boundary if and only if $k\le 5$. This enables us to find a large class of $2\otimes 4$ PPT entangled edge states with rank five.

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arXiv2013

Notes on extremality of the Choi map

It is widely believed that the Choi map generates an extremal ray in the cone $\mathcal P(M_3)$ of all positive linear maps between $C^*$-algebra $M_3$ of all $n\times n$ matrices over the complex field. But the only proven fact is that the Choi map generates the extremal ray in the cone of all positive linear map preserving all real symmetric $3\times 3$ matrices. In this note, we show that the Choi map is indeed extremal in the cone $\mathcal P(M_3)$. We also clarify some misclaims about the correspondence between positive semi-definite biquadratic real forms and postive linear maps, and discuss possible positive linear maps which coincide with the Choi map on symmetric matrices.

math.OA↗
arXiv2012

Separability of qubit-qudit quantum states with strong positive partial transposes

We show that all $2\otimes 4$ states with strong positive partial transposes (SPPT) are separable. We also construct a family of $2\otimes 5$ entangled SPPT states, so the conjecture on the separability of SPPT states are completely settled. In addition, we clarify the relation between the set of all $2\otimes d$ separable states and the set of all $2\otimes d$ SPPT states for the case of $d=3,4$.

quant-ph↗
arXiv2012

The structural physical approximations and optimal entanglement witnesses

We introduce the notions of positive and copositive types for entanglement witnesses, depending on the distance to the positive part and copositive part. An entanglement witness $W$ is of positive type if and only if its partial transpose $W^Γ$ is of copositive type. We show that if the structural physical approximation of $W$ is separable then $W$ should be of copositive type, and the SPA of $W^Γ$ is never separable unless $W$ is of both positive and copositive type. This shows that the SPA conjecture is meaningful only for those of copositive type. We provide examples to show that the SPA conjecture fails even for the case of copositive types.

quant-ph↗
arXiv2012

Optimal indecomposable witnesses without extremality as well as spanning property

One of the interesting problems on optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable witness which neither has the spanning property nor is associated with extremal positive linear map. Here, we answer this question negatively by examining the extremality of the positive linear maps constructed by Qi and Hou [J. Phys. A {\bf 44}, 215305 (2100)].

quant-ph↗
arXiv2012

Entanglement witnesses arising from Choi type positive linear maps

We construct optimal PPTES witnesses to detect $3\otimes 3$ PPT entangled edge states of type $(6,8)$ constructed recently \cite{kye_osaka}. To do this, we consider positive linear maps which are variants of the Choi type map involving complex numbers, and examine several notions related to optimality for those entanglement witnesses. Through the discussion, we suggest a method to check the optimality of entanglement witnesses without the spanning property.

quant-ph↗
arXiv2012

Optimal PPTES witnesses for states in $\mathbb C^n\otimes \mathbb C^n$

Recently, X. Qi and J. Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated to indecomposable positive linear maps, but it is not checked whether partial transposes of these witnesses are also optimal. We show that partial transposes of these entanglement witnesses have spanning property, and these witnesses are indeed optimal PPTES witnesses (non-decomposable optimal entanglement witnesses).

quant-ph↗
arXiv2012

Optimality for indecomposable entanglement witnesses

We examine various notions related with the optimality for entanglement witnesses arising from Choi type positive linear maps. We found examples of optimal entanglement witnesses which are non-decomposable, but which are not `non-decomposable optimal entanglement witnesses' in the sense of [M. Lewenstein, B. Kraus, J. Cirac, and P. Horodecki, Phys. Rev. A 62, 052310 (2000)]. We suggest to use the term `PPTES witness' and `optimal PPTES witness' in the places of `non-decomposable entanglement witness' and `non-decomposable optimal entanglement witnesses' in order to avoid possible confusion. We also found examples of non-extremal optimal entanglement witnesses which are indecomposable.

quant-ph↗
arXiv2012

Geometry of the faces for separable states arising from generalized Choi maps

We exhibit examples of separable states which are on the boundary of the convex cone generated by all separable states but in the interior of the convex cone generated by all PPT states. We also analyze the geometric structures of the smallest face generated by those examples. As a byproduct, we obtain a large class of entangled states with positive partial transposes.

quant-ph↗
arXiv2012

Entanglement witnesses arising from exposed positive linear maps

We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entanglement can be detected by one of these witnesses, and this witness detects a unique set of entanglement among those. Therefore, they provide a minimal set of witnesses to detect all entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entanglement with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps between three dimensional matrix algebra.

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