W. A. Majewski
We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A method is developed for realizing entanglement and general second quantized fermionic and bosonic fields in the framework of the fermionic projector.
We consider questions related to quantizing complex valued functions defined on a locally compact topological group. In the case of bounded functions, we generalize R. Werner's approach to prove the characterization of the associated normal covariant quantization maps.
For a very ample line bundle L on a compact connected complex manifold X, with a real structure, we discuss entanglement properties of certain sequences of vectors in tensor products of spaces of holomorphic sections of powers of L.
In this thesis work, we have studied the role of positive and completely positive maps in detecting entanglement.
In the recent paper [Chruściński and Wudarski, arXiv:1105.4821], it was conjectured that the entanglement witnesses arising from some generalized Choi maps are optimal. We show that this conjecture is true. Furthermore, we show that they provide a one parameter family of indecomposable optimal entanglement witnesses.
We construct a new class of positive indecomposable maps in the algebra of 'd x d' complex matrices. Each map is uniquely characterized by a cyclic bistochastic matrix. This class generalizes a Choi map for d=3. It provides a new reach family of indecomposable entanglement witnesses which define important tool for investigating quantum entanglement.
We provide a generalization of the reduction and Robertson positive maps in matrix algebras. They give rise to a new class of optimal entanglement witnesses. Their structural physical approximation is analyzed. As a byproduct we provide a new examples of PPT (Positive Partial Transpose) entangled states.
We elaborate the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. The entanglement is characterized in terms of generalized Segre maps, supplementing thus an algebraic approach to the problem by a more geometric point of view.
We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
We introduce an explicit construction for realizing of the space of invariant deformation quantizations on an arbitrary symmetric bounded domain.
This paper studies the class of stochastic maps, or channels, whose action (when tensored with the identity) on an entangled state always yields a separable state. Such maps have a canonical form introduced by Holevo. Such maps are called entanglement breaking, and can always be written in a canonical form introduced by Holevo. Some special classes of these maps are considered and several equivalent characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be described by its extreme points. The only extreme points of the set of completely positive…
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…
We describe $ω$-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.
In \cite{CMW19}, the authors introduced $k$-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of $k$-entanglement breaking maps. We prove many equivalent conditions for a $k$-positive linear map to be $k$-entanglement breaking, thereby study the mapping cone structure of $k$-entanglement breaking maps. We discuss examples of $k$-entanglement breaking maps and some of their significance. As an application of our study, we characterize completely positive maps that reduce…
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…
Discrete Mixed Quantization for Cat Maps
本文提出一种结合半经典分析与量子猫映射的混合量子化方法，用于图向量丛，并应用于研究Kesten--McKay定律和量子遍历性。
We apply the topological quantization method to some gravitational fields which can be represented as generalized harmonic maps. This representation extends the well-known concept of harmonic maps and allows us to describe some solutions to the Einstein field equations as generalized strings, where the base space for the generalized harmonic map is a two-dimensional manifold. The results obtained here point to an incompatibility of the classical description of the gravitational field and its symmetries with a discretization of the parameters that enter in its description.
Squashed entanglement [Christandl and Winter, J. Math. Phys. 45(3):829-840 (2004)] is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information [Fawzi and Renner, Commun. Math. Phys. 340(2):575-611 (2015)] greatly extended and simplified in various work since, we show the converse, that a small value of squashed entanglement implies that the state is close to a highly extendible state. As a corollary, we establish an alternative proof of the fait…
In this note, we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of quantizations is parametrized by the formal G-systems introduced by the authors in a previous paper and allows us to obtain classical invariant Hamiltonians that quantize without anomalies with respect to the quantizations of the action prescribed by the formal G-systems.
We study equivariant linear maps between finite-dimensional matrix algebras, as introduced by Bhat. These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat and as a generalization of the equivariant maps studied b…
Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain the geometric meaning of this series as degrees of maps of some grand configuration spaces; the associativity proof is also interpreted in purely homological terms. An interpretation in terms of degrees of maps shows that any other 1-form on the circle also leads to a *-product and allows one to compare these products.
Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state $ρ$ on $H\otimes K$ is separable if and only …
This paper continues the study of stochastic maps, or channels, which break entanglement. We give a detailed description of entanglement-breaking qubit channels, and show that such maps are precisely the convex hull of those known as classical-quantum channels. We also review the complete positivity conditions in a canonical parameterization and show how they lead to entanglement-breaking conditions.
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。
Every PPT channel has finite entanglement breaking index
该研究证明所有PPT线性映射有有限破纠缠指数，还得出一类含2-超正映射的PPT映射族破纠缠指数不超3，为PPT-cubed猜想成立提供有力证据。
We construct a new class of positive indecomposable maps in the algebra of `d x d' complex matrices. These maps are characterized by the `weakest' positivity property and for this reason they are called atomic. This class provides a new reach family of atomic entanglement witnesses which define important tool for investigating quantum entanglement. It turns out that they are able to detect states with the `weakest' quantum entanglement.
We characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of 3 x 3 complex matrices. It turns out that boundary of this set displays elegant representation in terms of SO(2) rotations. We conjecture that maps parameterized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parameterized by improper rotations from the orthogonal group O(2).
Entanglement witnesses and separability criteria based on generalized equiangular tight frames
该研究利用广义等角测量算子构造正映射以生成涵盖文献多类特例的纠缠见证，同时提出基于相关矩阵的可分性判据并针对多类测量展开分析。
We characterize a set of positive maps in matrix algebra of 4x4 complex matrices. Equivalently, we provide a subset of entanglement witnesses parameterized by the rotation group SO(3). Interestingly, these maps/witnesses define two intersecting convex cones in the 3-dimensional parameter space. The existence of two cones is related to the topological structure of the underlying orthogonal group. We perform detailed analysis of the corresponding geometric structure.
Indecomposable entanglement witnesses from symmetric measurements
本文提出基于对称测量的正映射家族，用于定义包含其他常用方法的纠缠见证，特别关注可检测正偏转态的不可分解见证。
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of t…
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…
A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement
本文证明了有限维矩阵代数上双随机完全正线性上循环的乘法遍历定理，据此分类随机bcp映射组合的渐近纠缠破缺情况，发现偶尔PPT的bcp映射渐近纠缠破缺，且某类bcp上循环有限时间内几乎必然纠缠破缺。
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。
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the condition…
This paper has been removed and is superceded by a pair of papers on stochastic maps, or channels, which break entanglement (EBT). quant-ph/0302031 General Entanglement Breaking Channels by Michael Horodecki, Peter W. Shor and Mary Beth Ruskai includes the results originally in section 3, as well as a number of new results about the extreme points. (The remarks in previously in section 3 about "sign change condition" are incorrect and have been omitted from quant-ph/0302031.) The new paper contains a to the conjecture that all extreme points of EBT maps are CQ when d > 2. quant-ph/0302032 Qubi…
Pointwise Weyl law for graphs from quantized interval maps
该研究证明了一维区间映射量子化得到的酉矩阵族的逐点Weyl定律类似结论，强化了已有量子遍历定理，证明随机扰动量子化的本征向量在半经典极限下近似高斯，并研究了加倍映射的特例。
We analyze the behaviour of two quantum dynamical entropies in connection with the classical limit. Using strongly chaotic classical dynamical systems as models (Arnold Cat Maps and Sawtooth Maps), we also propose a discretization procedure that resembles quantization; even in this case, studies of quantum dynamical entropy production are carried out and the connection with the continuous limit is explored. In both case (quantization and discretization) the entropy production converge to the Kolmogorov-Sinai invariant on time-scales that are logarithmic in the quantization (discretization) par…
In 1998, A.Alekseev and E.Meinrenken construct an explicit $G$-differential space homomorphism $\mathcal{Q}$, called the quantization map, between the Weil algebra $\Weil{\g}= \sym{\co{\g}} \otimes \ext{\co{\g}}$ and $\NWeil{\g}=\U{\g} \otimes \Cl{\g}$ (which they called the noncommutative Weil algebra) for any quadratic Lie algebra $\g$. They showed that $\mathcal{Q}$ induces an algebra isomorphism between the basic cohomology rings $H^{\ast}_{bas}(\Weil{g})$ and $H^{\ast}_{bas}(\NWeil{\g})$. In this paper, I interpret the quantization map $\mathcal{Q}$ as the super Duflo map between the symm…
This paper is one of a series of papers on coherent spaces and their applications, defined in the recent book 'Coherent Quantum Mechanics' by the first author. The paper studies coherent quantization -- the way operators in the quantum space of a coherent space can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric…
A theory of nonunitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as fractional-linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation is mapped to another phase space representation thereby extending the standard concept of covariance. This extended covariance allows Dirac-Jordan transformation theory to naturally emerge from the Hilbert…
Deformation quantization for systems with second-class constraints in deformed fermionic phase space
本文研究了在变形费米相空间中使用Dirac括号进行量子化的方法，分析了振荡子系统并计算了能级和Wigner函数，同时探讨了变形费米相空间引起的纠缠熵。
M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sh…
In a previous paper we introduced examples of Hamiltonian mappings with phase space structures resembling circle packings. It was shown that a vast number of periodic orbits can be found using special properties. We now use this information to explore the semiclassical quantization of one of these maps.
Given a finite quiver, its double may be viewed as its non-commutative "cotangent" space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while Schedler constructed its quantization. We show that the non-commutative quantization and reduction commute with each other. Via the quantum and classical trace maps, such a commutativity induces the commutativity of the quantization and reduction on the space of quiver representations.
C*-extreme entanglement breaking maps on operator systems
本文研究算子系统上C*-凸的幺正纠缠破坏映射集合，通过膨胀与延拓方法刻画其C*-极端点的结构，证明其为Arveson极大映射且与线性极端点等价，并推广了相关结构定理。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。