Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ...., Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital or / and trace preserving on the map ? are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum information science are mentioned.
Holomorphic Interpolation of Multivariate Completely Monotone Functions
Indian Institute of Technology, Bombay（印度理工学院孟买分校）；Indian Institute of Technology Palakkad（印度理工学院帕拉卡德分校）；University of California at Santa Barbara（加利福尼亚大学圣塔芭芭拉分校）
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.
In this work we show that, given a linear map from a general operator space into the dual of a C$^*$-algebra, its completely bounded norm is upper bounded by a universal constant times its $(1,cb)$-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication.
We prove a local convex version of Arveson's extension theorem and of Wittstock's extension theorem. Also we prove a Stinespring type theorem for unbounded local completely contractive maps.
Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is completely regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of tri-linear maps. We investigate the relation between Arens regularity of bounded bilinear maps and close-to-regularity bounded tri-linear maps. We give a simple criterion for the completely regularity of tri-linear maps. We provide a necessary and suf…
A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes A$. This gives a unified treatment for the different cases of the result obtained by researchers using different techniques.
We introduce a preorder relation in the collection of all operator valued completely positive maps on a full Hilbert C*-module and characterize this relation in terms of the Stinespring construction associated to each completely positive map.
Let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. In this paper, we show that every bijection completely preserving quadratic operators from $\mathcal{A}$ onto $\mathcal{B}$ is either an isomorphism or (in the complex case) a conjugate isomorphism.
In this article, we introduce local completely positive $k$-linear maps between locally $C^{\ast}$-algebras and obtain Stinespring type representation by adopting the notion of "invariance" defined by J. Heo for $k$-linear maps between $C^{\ast}$-algebras. Also, we supply the minimality condition to make certain that minimal representation is unique up to unitary equivalence. As a consequence, we prove Radon-Nikodým theorem for unbounded operator-valued local completely positive invariant $k$-linear maps. The obtained Radon-Nikodým derivative is a positive contraction on some Hilbert space wit…
In this paper we generalize the results shown by Das and Peterson. Let $M$ be a ${\rm II}_1$-factor acting on $L^2(M)$. We consider certain unital normal completely positive maps on $B(L^2(M))$ which are identity on $M$. We investigate their fixed point spaces and obtain a rigidity result. As an application, we show some results of subfactors.
We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If $\mathbb{G}$ is a locally compact quantum group, we characterise the completely bounded $L^{\infty}(\mathbb{G})'$-bimodule maps that send $C_0(\hat{\mathbb{G}})$ into $L^{\infty}(\hat{\mathbb{G}})$ in terms of the properties of the corresponding elements of the normal Haagerup tensor product $L^{\infty}(\mathbb{G}) \otimes_{σ{\rm h}} L^{\infty}(\mathbb{G})$. As a consequence, we obtain an intrinsic characterisation of the normal completely bounded $L^{\infty}(\mathbb{G})'…
A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $\mathbb R$-linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete $\mathbb R$-isomorphic embeddability (in particular, weaker than complete $\mathbb C$-isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space $X$ …
A linear map $Φ:\mathbb{M}_n \to \mathbb{M}_k$ is called completely copositive if the resulting matrix $[Φ(A_{j,i})]_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}]_{i,j=1}^m$. In this paper, we present some applications of the completely copositive maps $Φ(X)=(\mathrm{tr} X)I+X$ and $Ψ(X)= (\mathrm{tr} X)I-X$. Some new extensions about traces inequalities of positive semidefinite $3\times 3$ block matrices are included.
It is a well-known result due to E. Størmer that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by S. Fillipov and K. Magadov. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show …
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…
Nevanlinna-Pick interpolation developed from a topic in classical complex analysis to a useful tool for solving various problems in control theory and electrical engineering. Over the years many extensions of the original problem were considered, including extensions to different function spaces, nonstationary problems, several variable settings and interpolation with matrix and operator points. Here we discuss a variation on Nevanlinna-Pick interpolation for positive real odd functions evaluated in real matrix points. This problem was studied by Cohen and Lewkowicz using convex invertible con…
Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually after taking enough tensor powers. Moreover, we establish explicit bounds to quantify when this happens. To prove these results we use a symmetrization technique from the theory of entanglement distillation, and analyze when certain symmetric …
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 develop a completely bounded counterpart to the non-commutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps on an operator space admitting a dilation of the same norm which is multiplicative on the generated $C^*$-algebra. We view such maps as analogues of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps which are extremal …
We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the $\ast$-representations of the $\mathrm{C}^{\ast}$-envelope that are approximately unitarily equivalent to one that possesses the unique extension property. Although these $\ast$-representations do not necessarily have the unique extension property themselves, we show that their unital completely positive extensions display significant restrictions. When the underlyi…
The tetrablock is the set $$ \mathcal{E}=\{x \in \mathbb{C}^3: \quad 1-x_1z-x_2w+x_3z w \neq 0 \quad whenever \quad |z|\leq 1, |w|\leq 1\}. $$ The closure of $\mathcal{E}$ is denoted by $\overline{\mathcal{E}}$. A tetra-inner function is an analytic map $x$ from the unit disc $ \mathbb{D} $ to $\overline{\mathcal{E}}$ such that, for almost all points $λ$ of the unit circle $ \mathbb{T}$, \[ \lim_{r\uparrow 1} x(r λ) \mbox{ exists and lies in } b \overline{\mathcal{E}}, \] where $b \overline{\mathcal{E}}$ denotes the distinguished boundary of $\overline{\mathcal{E}}$. There is a natural notion …
We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the $n^2$ by $n^2$ matrix $\sum{\overline{X_i}}\otimes{X_i}.$ We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely postive map, including the special case …
