Quantum Channels, Wavelets, Dilations and Representations of $O_n$

Matrix N-dilations of quantum channels

We study unital quantum channels which are obtained via partial trace of a $*$-automorphism of a finite unital matrix $*$-algebra. We prove that any such channel, $q$, on a unital matrix $*$-algebra, $\mathcal{A}$, admits a finite matrix $N-$dilation, $α_N$, for any natural number N. Namely, $α_N$ is a $*$-automorphism of a larger bi-partite matrix algebra $\mathcal{A} \otimes \mathcal{B}$ so that partial trace of $M$-fold self-compositions of $α_N$ yield the $M$-fold self-compositions of the original quantum channel, for any $1\leq M \leq N$. This demonstrates that repeated applications of th…

Strong convergence of quantum channels: continuity of the Stinespring dilation and discontinuity of the unitary dilation

We show that a sequence $\{Φ_n\}$ of quantum channels strongly converges to a quantum channel $Φ_0$ if and only if there exist a common environment for all the channels and a corresponding sequence $\{V_n\}$ of Stinespring isometries strongly converging to a Stinespring isometry $V_0$ of the channel $Φ_0$. We also give a quantitative description of the above characterization of the strong convergence in terms of the appropriate metrics on the sets of quantum channels and Stinespring isometries. As a result, the uniform selective continuity of the complementary operation with respect to the str…

Unitary Dilations of Discrete-Time Quantum-Dynamical Semigroups

We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single ch…

A duality approach to representations of Baumslag-Solitar groups

We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the classical Baumslag-Solitar relations $u t = t^2 u$ where $t$ is a unitary operator in a Hilbert space $\mathcal H$ and $u$ is an isometry in $\mathcal H$. There are isometric dilations of this system into a bigger Hilbert space, relevant for wavelets. For a variety of carefully selected dilations, the ``bigger'' Hilbert space may be $L^2(\br)$, and the dilated o…

CPⁿ和更广阔的领域：2-范畴的膨胀理论

CP$^{\infty}$ and beyond: 2-categorical dilation theory

本文通过对CPⁿ-构造进行水平范畴化，从冯·诺依曼代数相关的2-范畴[W*]中恢复了所有冯·诺依曼代数和信道的范畴，并扩展了Choi关于极值信道的刻画至任意冯·诺依曼代数。