Schmidt states and positivity of linear maps

Facial structures for various notions of positivity and applications to the theory of entanglement

In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.

qutrit上的稀疏正映射：精确的非可分解阈值与PPT纠缠转变

Sparse positive maps on qutrits with exact nondecomposability thresholds and PPT-entanglement transitions

研究qutrit上稀疏正映射族，通过Choi矩阵的块结构将正性简化为厄米双二次型，精确确定三个代表性参数族的正性边界、可分解与非可分解映射的转变，并构造两类PPT纠缠态，给出可分离性与束缚纠缠的精确阈值。

Positivity in the presence of initial system-environment correlation

The constraints imposed by the initial system-environment correlation can lead to nonpositive Dynamical maps. We find the conditions for positivity and complete positivity of such dynamical maps by using the concept of an assignment map. Any initial system-environment correlations make the assignment map nonpositive, while the positivity of the dynamical map depends on the interplay between the assignment map and the system-environment coupling. We show how this interplay can reveal or hide the nonpositivity of the assignment map. We discuss the role of this interplay in Markovian models.

Positivity and complete positivity of differentiable quantum processes

We study quantum processes, as one parameter families of differentiable completely positive and trace preserving (CPTP) maps. Using different representations of the generator, and the Sylvester criterion for positive semi-definite matrices, we obtain conditions for the divisibility of the process into completely positive (CP-divisibility) and positive (P-divisibility) infinitesimal maps. Both concepts are directly related to the definition of quantum non-Markovianity. For the single qubit case we show that CP- and P-divisibility only depend on the dissipation matrix in the master equation form…

k-正 maps 的家族与来自广义等角测量的 Schmidt 数见证

Families of $k$-positive maps and Schmidt number witnesses from generalized equiangular measurements

本文提出利用广义等角测量定义k-正线性映射及其对应的Schmidt数见证，有效检测任意维数中Schmidt数为k+1的双参数纠缠态，相较对称测量算符构造的现有方法更具效率。

Linear Assignment Maps for Correlated System-Environment States

An assignment map is a mathematical operator that describes initial system-environment states for open quantum systems. We reexamine the notion of assignments, introduced by Pechukas, and show the conditions assignments can account for correlations between the system and the environment, concluding that assignment maps can be made linear at the expense of positivity or consistency is more reasonable. We study the role of other conditions, such as consistency and positivity of the map, and show the effects of relaxing these. Finally, we establish a connection between the violation of positivity…

Tensor power of dynamical maps and P- vs. CP-divisibility

The are several non-equivalent notions of Markovian quantum evolution. In this paper we show that the one based on the so-called CP-divisibility of the corresponding dynamical map enjoys the following stability property: the dynamical map $Λ_t$ is CP-divisible iff the second tensor power $Λ_t\otimesΛ_t$ is CP-divisible as well. Moreover, the P-divisibility of the map $Λ_t\otimesΛ_t$ is equivalent to the CP-divisibility of the map $Λ_t$. Interestingly, the latter property is no longer true if we replace the P-divisibility of $Λ_t\otimesΛ_t$ by simple positivity and the CP-divisibility of $Λ_t$ …

用于高维量子态的 Schmidt 数见证族

Families of Schmidt-number witnesses for high dimensional quantum states

该研究利用 SIC-POVM 与相互无偏基构造 k-正映射及相应的 Schmidt 数见证，并据此量化两体态与低 Schmidt 数态集合的距离。

Kubo-Ando 均值与量子正性锥的刚性

Kubo-Ando Means and Rigidity of Quantum Positivity Cones

本文研究非线性算子均值下量子正性锥的稳定性，发现加权算术均值是唯一在所有维度下保持可分锥的 Kubo-Ando 均值，并证明非算术均值会破坏 PPT 条件并增加 Schmidt 数。

Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose

Motivated by the desire to better understand the class of quantum operations on bipartite systems that preserve positivity of partial transpose (PPT operations) and its relation to the class LOCC (local operations and classical communication), we present some results on deterministic bipartite pure state transformations by PPT operations. Restricting our attention to the case where we start with a rank K maximally entangled state, we give a necessary condition for transforming it into a given pure state, which we show is also sufficient when K is two and the final state has Schmidt rank three.…

$k$-正定性与在辛群对称下的高维纠缠

$k$-Positivity and high-dimensional bound entanglement under symplectic group symmetries

研究了在辛群对称下高维纠缠的$k$-正定性和Schmidt数特性，并首次构造了最优$k$-正定不可分解线性映射。

Impact of positivity and complete positivity on accessibility of Markovian dynamics

We consider a two-dimensional quantum control system evolving under an entropy-increasing irreversible dynamics in the semigroup form. Considering a phenomenological approach to the dynamics, we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive. In particular, we characterize the family of maps having different accessibility and show the impact of that property on observable quantities by means of a simple physical model.

A general framework for complete positivity

Complete positivity of quantum dynamics is often viewed as a litmus test for physicality, yet it is well known that correlated initial states need not give rise to completely positive evolutions. This observation spurred numerous investigations over the past two decades attempting to identify necessary and sufficient conditions for complete positivity. Here we describe a complete and consistent mathematical framework for the discussion and analysis of complete positivity for correlated initial states of open quantum systems. This formalism is built upon a few simple axioms and is sufficiently …

Distillability via protocols respecting the positivity of partial transpose

We show that all quantum states that do not have a positive partial transpose are distillable via channels, which preserve the positivity of the partial transpose. The question whether NPT bound entanglement exist is therefore closely related to the connection between the set of separable superoperators and PPT-preserving maps.

Positive Maps From Irreducibly Covariant Operators

In this paper, we discuss positive maps induced by (irreducibly) covariant linear operators for finite groups. The application of group theory methods allows deriving some new results of a different kind. In particular, a family of necessary conditions for positivity, for such objects is derived, stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map. In the low-dimensional cases, for the permutation group $S(3)$ and the quaternion group $Q$, the necessary and sufficient conditions are given, together with the discussion on their decompo…

对角扰动下的Tomiyama型映射

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。

Geometrical aspects of entanglement

We study geometrical aspects of entanglement, with the Hilbert--Schmidt norm defining the metric on the set of density matrices. We focus first on the simplest case of two two-level systems and show that a ``relativistic'' formulation leads to a complete analysis of the question of separability. Our approach is based on Schmidt decomposition of density matrices for a composite system and non-unitary transformations to a standard form. The positivity of the density matrices is crucial for the method to work. A similar approach works to some extent in higher dimensions, but is a less powerful to…

利用广义正映射的矩进行高维纠缠检测与量子信道表征

Higher-dimensional entanglement detection and quantum channel characterization using moments of generalized positive maps

提出利用广义正映射的矩检测高维纠缠、判定 Schmidt 数并识别 PPT/NPT 纠缠态的免层析判据，且可用于表征非破坏型量子信道与信道区分。

On a family of a linear maps from $M_{n}(\mathbb{C})$ to $M_{n^{2}}(\mathbb{C})$

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $Φ_{α, β,n} : M_{n}(\mathbb{C}) …

分数$k$-正性：$k$-正尺度的连续细化

Fractional $k$-positivity: a continuous refinement of the $k$-positive scale

本文提出分数$k$-正性概念，通过连续细化$k$-正尺度，揭示了新的量子映射结构和计算方法。

Criteria of positivity for linear maps constructed from permutation pairs

In this paper, we show that a $D$-type map $Φ_D:M_n\rightarrow M_n$ with $D=(n-2)I_n+P_{π_1}+P_{π_2}$ induced by a pair $\{π_1,π_2\}$ of permutations of $(1,2,..., n)$ is positive if $\{π_1,π_2\}$ has property (C). The property (C) is characterized for $\{π_1,π_2\}$, and an easy criterion is given for the case that $π_1=π^p$ and $π_2=π^q$, where $π$ is the permutation defined by $π(i)=i+1$ mod $n$ and $1\leq p<q\leq n$.

A simple proof of the Jamiolkowski criterion for complete positivity of linear maps

We give a simple direct proof of the Jamiolkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accesible for physicists than others found in the literature and provides a systematic method to give any set of Kraus matrices of its Kraus decomposition.

超越正部分转置平方猜想：三量子比特情形

Beyond the Positive Partial Transpose Squared Conjecture: The Qutrit Case

研究超出正部分转置（PPT）设置的映射复合问题，针对三量子比特完全正定映射，证明特定条件下映射复合是纠缠破缺的，确定1-不可蒸馏CP映射锥的性质，还证明特定态在任意选择性测量下不能产生终端纠缠。

$k$-正映射的刻画

Characterization of $k$-positive maps

本文利用 Kraus 算子矩阵的 Ky Fan 范数刻画正映射的 $k$-正性，并构造参数化正映射族，得到更优的 $k$-正性参数界及可分解性区域的精确界。

Positive tensor products of qubit maps and n-tensor-stable positive qubit maps

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 …

Entangled states with positive partial transpose arising from indecomposable positive linear maps

We construct entangled states with positive partial transposes using indecomposable positive linear maps between matrix algebras. We also exhibit concrete examples of entangled states with positive partial transposes arising in this way, and show that they generate extreme rays in the cone of all positive semi-definite matrices with positive partial transposes. They also have Schmidt numbers two.

Implementing positive maps with multiple copies of an input state

Valid transformations between quantum states are necessarily described by completely positive maps, instead of just positive maps. Positive but not completely positive maps such as the transposition map cannot be implemented due to the existence of entanglement in composite quantum systems, but there are classes of states for which the positivity is guaranteed, e.g., states not correlated to other systems. In this paper, we introduce the concept of N-copy extension of maps to quantitatively analyze the difference between positive maps and completely positive maps. We consider implementations o…

严格正性与$D$-优化排序

Strict Positivity and $D$-Majorization

该研究受量子热力学启发，探究严格正性的判定与性质，并将其用于推广$D$-优化排序，给出二维情形下该排序的迹范数刻画及序性质。

Merging of positive maps: a construction of various classes of positive maps on matrix algebras

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 …

Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy

Consider the set $\mathcal{S}=\lbraceρ_{SE}\rbrace$ of possible initial states of the system-environment, steered from a tripartite reference state $ω_{RSE}$. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)] showed that the reduced dynamics of the system, for each $ρ_{S}\in \mathrm{Tr}_{E}\mathcal{S}$, is always completely positive if and only if $ω_{RSE}$ is a Markov state. There, during the proof, it has been assumed that the dimensions of the system and the environment can vary through the evolution. Here, we show that this assumption is necessary: we give an example for which, tho…

$M_n$ 中一类最优正映射

A class of optimal positive maps in $M_n$

本文证明了矩阵代数 $M_n$ 中某类正映射为最优映射，并推广了开创性的 Choi 正映射。

Multiplicativity properties of entrywise positive maps

Multiplicativity of certain maximal p -> q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity (CP) is either augmented by, or replaced by, the requirement that the entries of a matrix representative of the map are non-negative (EP). In particular, for integer t, multiplicativity holds for the maximal 2 -> 2t norm of a product of two maps, whenever one of the pair is EP; for the maximal 1 -> t norm for pairs of CP maps when one of them is also EP; and for the maximal 1 -> 2t norm for the product of an EP and a 2-posi…

Positivity of linear maps under tensor powers

We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with $n$ copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every $n\in\mathbb{N}$ there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all $n$. For higher dimensions we reduce the existence question of such non-trivial "tensor-stable positive maps" to a one-parameter family of maps and show that an affirmative answe…

Measure of not-completely-positive qubit maps: the general case

We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full positivity domain. The definition is motivated by specific examples. We prove that for valid NCP maps, the eigenvalue spectrum of the corresponding dynamical matrix is not bounded. Based on this, we argue that in general the volume measure of qubit maps, including NCP maps, is not well defined.

When Do Composed Maps Become Entanglement Breaking?

For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in d…

Scaling maps of $s$-ordered quasiprobabilities are either nonpositive or completely positive

Continuous-variable systems in quantum theory can be fully described through any one of the ${\rm s}$-ordered family of quasiprobabilities $Λ_{\rm s}(α)$, ${\rm s} \in [-1,1]$. We ask for what values of $({\rm s}, a)$ is the scaling map $Λ_{\rm s}(α) \rightarrow a^{-2} Λ_{\rm s}(a^{-1}α)$ a positive map? Our analysis based on a duality we establish settles this issue (i) the scaling map generically fails to be positive, showing that there is no useful entanglement witness of the scaling type beyond the transpose map, and (ii) in the two particular cases $({\rm s}=1, |a| \leq 1)$ and $({\rm s}=…

量子通道的Hilbert-Schmidt范数与单位球上的矩阵积分

The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere

本文研究量子通道的Hilbert-Schmidt范数范围及其互补通道的关系，并探讨了无限维系统上保持纯态的完全正映射以及相关矩阵积分的等价性与扩展。

Maximum-likelihood reconstruction of CP maps

We present a method for the determination of the completely positive (CP) map describing a physical device based on random preparation of the input states, random measurements at the output, and maximum-likelihood principle. In the numerical implementation the constraint of completely positivity can be imposed by exploiting the isomorphism between linear transformations from Hilbert spaces $\cal H$ to $\cal K$ and linear operators in ${\cal K}\otimes {\cal H}$. The effectiveness of the method is shown on the basis of some examples of reconstruction of CP maps related to quantum communication c…

Affine Maps of the Polarization Vector for Quantum Systems of Arbitrary Dimension

The operator-sum decomposition (OS) of a mapping from one density matrix to another has many applications in quantum information science. To this mapping there corresponds an affine map which provides a geometric description of the density matrix in terms of the polarization vector representation. This has been thoroughly explored for qubits since the components of the polarization vector are measurable quantities (corresponding to expectation values of Hermitian operators) and also because it enables the description of map domains geometrically. Here we extend the OS-affine map correspondence…

New multiplicativity results for qubit maps

Let $Φ$ be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex $2 \times 2$ matrices, and let $Ω$ be any finite-dimensional completely positive map. For $p=2$ and $p \geq 4$, we prove that the maximal $p$-norm of the product map $Φ\ot Ω$ is the product of the maximal $p$-norms of $Φ$ and $Ω$. Restricting $Φ$ to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values $p \geq 4$.

Structure and Parametrization of Stochastic Maps of Density Matrices

The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of hermiticity of density matrices renders an associated supermatrix hermitian and hence diagonalizable; but the positivity of the density matrix does not make this associated supermatrix positive. If it is positive, the map is called completely positive and they have a simple parametrization. This…

Complete positivity on the subsystems level

We consider complete positivity of dynamics regarding subsystems of an open composite quantum system, which is subject of a completely positive dynamics. By "completely positive dynamics", we assume the dynamical maps called the completely positive and trace preserving maps, with the constraint that domain of the map is the whole Banach space of the system's density matrices. We provide a technically simple and conceptually clear proof for the subsystems' completely positive dynamics. Actually, we prove that every subsystem of a composite open system can be subject of a completely positive dyn…

On partially entanglement breaking channels

Using well known duality between quantum maps and states of composite systems we introduce the notion of Schmidt number of a quantum channel. It enables one to define classes of quantum channels which partially break quantum entanglement. These classes generalize the well known class of entanglement breaking channels.

Bloch Equations and Completely Positive Maps

The phenomenological dissipation of the Bloch equations is reexamined in the context of completely positive maps. Such maps occur if the dissipation arises from a reduction of a unitary evolution of a system coupled to a reservoir. In such a case the reduced dynamics for the system alone will always yield completely positive maps of the density operator. We show that, for Markovian Bloch maps, the requirement of complete positivity imposes some Bloch inequalities on the phenomenological damping constants. For non-Markovian Bloch maps some kind of Bloch inequalities involving eigenvalues of the…

连接经典与量子世界：映射、态与演化

Bridging Classical and Quantum Worlds: Maps, States, and Evolutions

本文通过分析希尔伯特空间算子空间线性映射中正性与完全正性的中间概念，探讨经典与量子理论的相互作用，提出从经典对应物推导量子态与演化及反之的构造方法。

Spectral properties of entanglement witnesses

Entanglement witnesses are observables which when measured, detect entanglement in a measured composed system. It is shown what kind of relations between eigenvectors of an observable should be fulfilled, to allow an observable to be an entanglement witness. Some restrictions on the signature of entaglement witnesses, based on an algebraic-geometrical theorem will be given. The set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive, but not completely positive. A translation of the results to the language of positive maps is also give…

单位 Kadison-Schwarz 映射

Unital Kadison-Schwarz Maps

本文研究 $M_2(\mathbb{C})$ 上的单位 Kadison-Schwarz 映射，推导其成立条件并给出非平凡例子，从而扩展了可分类正映射的范围，为纠缠检测提供理论工具。

从密度矩阵的谱界限纠缠态

Bounding the entanglement of a state from its spectrum

研究通过线性映射和其逆运算，基于密度矩阵的谱特征，推导出纠缠度的约束条件，聚焦于负性与Schmidt数，提出全秩态纠缠含量的量化方法。

Complete positivity of the map from a basis to its dual basis

The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of the n by n matrices to its dual basis is a complete order isomorphism and complete co-order isomorphism. In the case of the standard matrix units this map is a complete order isomorphism and this is a restatement of the correspondence between completely positive maps and the Choi matrix. However, we exhibit natural orthonormal bases for the matrices such that this map is an order…

Topology of the cone of positive maps on qubit systems

An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra $M_{2}(\mathbb{C})$ has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of $M_{2}(\mathbb{C})$ have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of $M_{n}(\…

Criteria and new classes of k-positive maps

We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered.

Vanishing quantum discord is not necessary for completely-positive maps

The description of the dynamics of a system that may be correlated with its environment is only meaningful within the context of a specific framework. Different frameworks rely upon different assumptions about the initial system-environment state. We reexamine the connections between complete-positivity and quantum discord within two different sets of assumptions about the relevant family of initial states. We present an example of a system-environment state with non-vanishing quantum discord that leads to a completely-positive map. This invalidates an earlier claim on the necessity of vanishi…

$M_3$上幺正正线性映射的Kadison-Schwarz性质的充分条件

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。

Diagonal unitary and orthogonal symmetries in quantum theory

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular…