Dariusz Chruściński, Gniewomir Sarbicki
Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a sufficient condition for a positive map to be exposed. This is an analog of a spanning property which guaranties that a positive map is optimal. We analyze a class of decomposable maps for which this condition is also necessary.
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It is well known that so called Breuer-Hall positive maps used in entanglement theory are optimal. We show that these maps possess much more subtle property --- they are exposed. As a byproduct it proves that a Robertson map in the algebra of 4 x 4 complex matrices is not only extreme, which was already shown by Robertson, but also exposed.
Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a class of indecomposable positive maps in the algebra of 2n x 2n complex matrices with n>1. It is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states.
It is shown that the family of so called Breuer-Hall maps in M_4(C) possesses strong spanning property and hence they are exposed in the convex cone of positive maps in M_4(C)
Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.
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 indecomposabl…
A natural and intrinsic characterization of the structure of the set $\mathfrak{C}$ of positive unital maps is given, i.e. it is shown that $\mathfrak{C}$ is isometrically isomorphic to the subset $\gD$ of bp-positive density matrices endowed with the geometry given by the norm $α$ dual to the Grothendieck projective norm $π$, the structure of $\gD$ is determined by the set of its exposed points, and finally a characterization of exposed points of $\gD$ in terms of convex analysis is presented. This seems to be an answer to an old open problem, characterization of the structure of the set of p…
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.
Let $\cK$ and $\cH$ be finite dimensional Hilbert spaces and let $\fP$ denote the cone of all positive linear maps acting from $\fB(\cK)$ into $\fB(\cH)$. We show that each map of the form $ϕ(X)=AXA^*$ or $ϕ(X)=AX^TA^*$ is an exposed point of $\fP$. We also show that if a map $ϕ$ is an exposed point of $\fP$ then either $ϕ$ is rank 1 non-increasing or $\rankϕ(P)>1$ for any one-dimensional projection $P\in\fB(\cK)$.
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 a…
We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on $M_2(C)$ and $M_3(C)$. Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiolkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting we show that in $M_2(C)$ case a large class of nice …
We present a large class of indecomposable exposed positive linear maps between three dimensional matrix algebras. We also construct two qutrit separable states with lengths ten in the interior of their dual faces. With these examples, we show that the length of a separable state may decrease strictly when we mix it with another separable state.
We fully characterize bipartite entanglement-annihilating (EA) channels that destroy entanglement of any state shared by subsystems and, thus, should be avoided in any entanglement-enabled experiment. Our approach relies on extending the problem to EA positive maps, the cone of which remains invariant under concatenation with partially positive maps. Due to this invariancy, positive EA maps adopt a well characterization and their intersection with completely positive trace-preserving maps results in the set of EA channels. In addition to a general description, we also provide sufficient operat…
For two positive maps $ϕ_i:B(\mathcal{K}_i)\to B(\mathcal{H}_i)$, $i=1,2$, we construct a new linear map $ϕ:B(\mathcal{H})\to B(\mathcal{K})$, where $\mathcal{K}=\mathcal{K}_1\oplus\mathcal{K}_2\oplus\mathbb{C}$, $\mathcal{H}=\mathcal{H}_1\oplus\mathcal{H}_2\oplus\mathbb{C}$, by means of some additional ingredients such as operators and functionals. We call it a merging of maps $ϕ_1$ and $ϕ_2$. We discuss properties of this construction. In particular, we provide conditions for positivity of $ϕ$, as well as for $2$-positivity, complete positivity and nondecomposability. In particular, we show …
We analyze positivity of a tensor product of two linear qubit maps, $Φ_1 \otimes Φ_2$. Positivity of maps $Φ_1$ and $Φ_2$ is a necessary but not a sufficient condition for positivity of $Φ_1 \otimes Φ_2$. We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both $Φ_1$ and $Φ_2$ are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for $n$-tensor-stable positive qubit maps, i.e. such qubit maps $Φ$ that $Φ^{\otimes n}$ is positive. Particular cases of 2- and 3-tensor-stable positive qubit …
Sparse anisotropic positive maps for qutrit entanglement: exact indecomposability and PPT geometry
研究三量子比特纠缠，引入可精确求解的双参数稀疏双随机正映射族，分析其相图，构造特定PPT边缘态并推导检测区域，给出最优细化，能在单个框架中研究正性等多种性质。
We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of $n-1$ particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition.
Sufficiency and Petz recovery for positive maps
本文研究了量子态家族通过正、迹保持映射的互换，并通过最小充分Jordan代数的数学结构阐明其本质。证明Neyman-Pearson检验生成最小充分Jordan代数，并展示相对熵或α-z量子Rényi散度的数据处理不等式等式条件下的恢复映射存在性。
A system interacting with its environment will give rise to a quantum evolution. After tracing over the environment the net evolution of the system can be described by a linear Hermitian map. It has recently been shown that a necessary and sufficient condition for this evolution to be completely positive is for the initial state to have vanishing quantum discord. In this paper, we provide a sufficient condition for the map to be positive with respect to the initial system-environment correlation. This could lead to ways in which to identify positive but not completely positive maps. Illustrati…
Physical transformations are described by linear maps that are completely positive and trace preserving (CPTP). However, maps that are positive (P) but not completely positive (CP) are instrumental to derive separability/entanglement criteria. Moreover, the properties of such maps can be linked to entanglement properties of the states they detect. Here, we extend the results presented in [Phys. Rev A 93, 042335 (2016)], where sufficient separability criteria for bipartite systems were derived. In particular, we analyze the entanglement depth of an $N$-qubit system by proposing linear maps that…
Two long standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (1) a map between the initial and final system states, and (2) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information processing and open quantum systems theory. Here we settle both these questions by showing that the answer to the first is "all", with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from th…
We study families of positive and completely positive maps acting on a bipartite system $\mathbb{C}^M\otimes \mathbb{C}^N$ (with $M\leq N$). The maps have a property that when applied to any state (of a given entanglement class) they result in a separable state, or more generally a state of another certain entanglement class (e.g., Schmidt number $\leq k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary $M,N$, with a special emphasis on $M=2$. Our results can be viewed as generalizations of th…
Quantum supermaps are a higher-order generalization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive, trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link t…
Sufficient positive maps between von Neumann algebras: Rényi divergences
该研究针对冯·诺依曼代数间的正规幺正正映射，证明了α-z Rényi散度的恢复定理，将此前所需的2-正性放宽至单纯正性，还解决了Haagerup与Stormer提出的条件期望分解问题。
Sufficient conditions for the Kadison--Schwarz property of unital positive maps on $M_3$
本文研究了$M_3$上幺正正线性映射的Kadison-Schwarz性质，通过Bloch-Gell-Mann表示推导出显式分析充分条件，利用反称结构常数抵消机制，给出基于Bloch参数的结构准则。
Revival and instabilities of entanglement in monitoring maps with indefinite causal order
本文研究含环境噪声与控制端后选择的 quantum switch，证明不定因果序可逆转确定因果序下的纠缠突然死亡，并在高温及低温弱监测条件下实现纠缠恢复或重现。
A necessary and sufficient condition for 1-distillability is formulated in terms of decomposable positive maps. As an application we provide insight into why all states violating the reduction criterion map are distillable and demonstrate how to construct such maps in a systematic way. We establish a connection between a number of existing results, which leads to an elementary proof for the characterisation of distillability in terms of 2-positive maps.
Tomiyama-type maps with a diagonal perturbation
研究了一类由经典Tomiyama映射的对角扰动构造出的双参数线性映射，通过Choi矩阵方法和块正性技术，推导出任意维度下正性、完全正性和k正性的显式必要充分条件，提供参数空间中正性区域的几何刻画。
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