Non-decomposable quantum dynamical semigroups and bound entangled states

Stabilizing Entangled States with Quasi-Local Quantum Dynamical Semigroups

We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under fixed locality constraints. Beside recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.

Optimization of entanglement witnesses

An entanglement witness (EW) is an operator that allows to detect entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e. to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the non-decomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES, i.e. states that violate the range separability criterio…

Exact calculation of robustness of entanglement via convex semi-definite programming

In general the calculation of robustness of entanglement for the mixed entangled quantum states is rather difficult to handle analytically. Using the the convex semi-definite programming method, the robustness of entanglement of some mixed entangled quantum states such as: $2\otimes 2$ Bell decomposable (BD) states, a generic two qubit state in Wootters basis, iso-concurrence decomposable states, $2\otimes 3$ Bell decomposable states, $d\otimes d$ Werner and isotropic states, a one parameter $3\otimes 3$ state and finally multi partite isotropic state, is calculated exactly, where thus obtaine…

Optimal Lewenstein-Sanpera Decomposition for some Biparatite Systems

It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of product set) on the separable-entangled boundary, optimal Lewenstein-Sanpera (L-S) decomposition can be obtained via optimization for a generic entangled density matrix. Based on this, We obtain optimal L-S decomposition for some bipartite systems such as $2\otimes 2$ and $2\otimes 3$ Bell decomposable states, generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one parameter and three parameters local operations and classical …

生成非可分解映射的可微半定规划

Generating Non-Decomposable Maps with Differentiable Semidefinite Programming

本文提出基于可微半定规划的优化框架，用于生成非可分解正映射，通过灵活的Choi矩阵结构约束，系统搜索不同输入输出维度的映射，并揭示了新的数值例子和参数化映射家族。

两qudit系统的一对镜像最优不可分解纠缠见证确实存在

A mirrored pair of optimal non-decomposable entanglement witnesses for two qudits does exist

针对双qudit态可分性判定中的纠缠见证结构问题，利用镜像纠缠见证概念，推翻已有猜想，证明存在成对不可分解最优纠缠见证且可推广至高维，揭示了纠缠见证的新结构。

Non-Positive Semigroup Dynamics in Continuous Variable Models

Non-positive, Markovian semigroups are sometimes used to describe the time evolution of subsystems immersed in an external environment. A widely adopted prescription to avoid the appearance of negative probabilities is to eliminate from the admissible initial conditions those density matrices that would not remain positive by the action of the semigroup dynamics. Using a continuous variable model, we show that this procedure leads to physical inconsistencies when two subsystems are considered and their initial state is entangled.

矩阵代数上的可分解动力学

Decomposable dynamics on matrix algebras

研究了可分解可除的量子演化的性质，提出高对称性qubit和qudit动态映射的必要和充分条件，通过重构演化生成器将可分解可除性编码为时间依赖系数的正性。

Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups $IGL(n, \mathbb{R})$ and $GL(n, \mathbb{R})$ respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities ar…

Universal Quantum Entanglement Concentration Gate

We construct a Universal Quantum Entanglement Concentration Gate (QEC-Gate). Special times operations of QEC-Gate can transform a pure 2-level bipartite entangled state to nearly maximum entanglement. The transformation can attain any required fidelity with optimal probability by adjusting concentration step. We also generate QEC-Gate to the Schmidt decomposable multi-partite system.

Role of Partial Transpose and Generalized Choi maps in Quantum Dynamical Semigroups involving Separable and Entangled States

Power symmetric matrices defned and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studie…

Ipso-information-transfer

A tripartite system with entangled non-orthogonal states is used to transfer retrievable or usable information without requiring an external channel, ipso-information-transfer (IIT). The non-Schmidt decomposable entanglement couples two independent interactions through which the information is transferred between a pair of non-interacting components of the system. We outline a dynamical model of IIT using localised particles. Implications for quantum nonlocality are raised.

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.

Witnessing genuine multipartite entanglement with positive maps

We derive a general framework that connects every positive map with a corresponding witness for partial separability in multipartite quantum systems. We show that many previous approaches were intimately connected to the witnesses derived from partial transposition and that such criteria can easily be outperformed in higher dimensions by non-decomposable maps. As an exemplary case we present a witness that is capable of detecting genuine multipartite entanglement in bound entangled states.

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。

General fixed points of quasi-local frustration-free quantum semigroups: from invariance to stabilization

We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, whic…

Entanglement in two-mode continuous variable open quantum systems

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable entanglement for a system consisting of two noninteracting modes embedded in a thermal environment. By using the Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the …

Balance between quantum Markov semigroups

The concept of balance between two state preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly e…

Dynamics of quantum entanglement in Gaussian open systems

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal environment. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type: separable or entangled. In other cas…

Constructing positive maps from block matrices

Positive maps are useful for detecting entanglement in quantum information theory. Any entangled state can be detected by some positive map. In this paper, the relation between positive block matrices and completely positive trace-preserving maps is characterized. Consequently, a new method for constructing decomposable maps from positive block matrices is derived. In addition, a method for constructing positive but not completely positive maps from Hermitian block matrices is also obtained.

Exact stabilization of entangled states in finite time by dissipative quantum circuits

Open quantum systems evolving according to discrete-time dynamics are capable, unlike continuous-time counterparts, to converge to a stable equilibrium in finite time with zero error. We consider dissipative quantum circuits consisting of sequences of quantum channels subject to specified quasi-locality constraints, and determine conditions under which stabilization of a pure multipartite entangled state of interest may be exactly achieved in finite time. Special emphasis is devoted to characterizing scenarios where finite-time stabilization may be achieved robustly with respect to the order o…

k-正映射：新的特征化和生成方法

$k$-Positive Maps: New Characterizations and a Generation Method

本文提出了一种基于李群的方法生成k-正非可分解映射，并通过优化条件特征化k-正性，同时验证了PPT平方猜想。

Rényi-2 quantum correlations of two-mode Gaussian systems in a thermal environment

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of continuous variable Gaussian Rényi-2 (GR2) quantum discord for a system consisting of two non-interacting non-resonant bosonic modes embedded in a thermal environment. In the case of both an entangled initial squeezed vacuum state and squeezed thermal state, GR2 quantum discord has non-negative values for all temperatures of the environment. We describe also the time evolution of GR2 classical correlations and GR2 quantum mutual information of the quantum system. A…

Asymptotic entanglement in open quantum systems

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we solve in the asymptotic long-time regime the master equation for two independent harmonic oscillators interacting with an environment. We give a description of the continuous-variable asymptotic entanglement in terms of the covariance matrix of the considered subsystem for an arbitrary Gaussian input state. Using Peres--Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that for certain classes of environments the initial state evolves asympto…

A universal framework for entanglement detection under group symmetry

One of the most fundamental questions in quantum information theory is PPT-entanglement of quantum states, which is an NP-hard problem in general. In this paper, however, we prove that all PPT $(\overlineπ_A\otimes π_B)$-invariant quantum states are separable if and only if all extremal unital positive $(π_B,π_A)$-covariant maps are decomposable where $π_A,π_B$ are unitary representations of a compact group and $π_A$ is irreducible. Moreover, an extremal unital positive $(π_B,π_A)$-covariant map $\mathcal{L}$ is decomposable if and only if $\mathcal{L}$ is completely positive or completely cop…

On the structure of positive maps II: low dimensional matrix algebras

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 …

The geometry of density states, positive maps and tomograms

The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomogr…

源自共正矩阵的正映射与可延拓性层级

Positive maps and extendibility hierarchies from copositive matrices

本文引入PCOP等新型凸锥搭建共正矩阵与正映射的桥梁，构造图参数化正映射族，解答DPS04公开问题，给出跨任意PPT玻色可延拓层级的纠缠Dicke态构造。

量子退极化过程的纠缠代价

Entanglement cost of quantum depolarization

本研究确定了量子比特各向同性态和两量子比特贝尔对角态的精确纠缠代价，并利用半群方法和张量积定理推导出qudit态及退极化信道的一般下界，显著改进了现有结果。