Related papers: Quantum Channels, Wavelets, Dilations and Represen…
We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations.
Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known are the representations of quantum channels by Choi matrices…
We investigate the set of quantum channels acting on a single qubit. We provide an alternative, compact generalization of the Fujiwara-Algoet conditions for complete positivity to non-unital qubit channels, which we then use to characterize…
Let ${\cal O}_n$ denote the Cuntz algebra for $n\geq 2$. We introduce an embedding $f$ of ${\cal O}_m$ into ${\cal O}_n$ arising from a geometric progression of Cuntz generaters of ${\cal O}_n$. By identifying ${\cal O}_m$ with $f({\cal…
Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices…
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps…
This work shows an approach to reduce the dimensionality of matrix representations of quantum channels. It is achieved by finding a base of the cone of positive semidefinite matrices which represent quantum channels. Next, this is…
A linear map $L$ from ${\mathbb C}^{n \times n}$ into ${\mathbb C}^{n \times n}$ is called a quantum channel if it is completely positive and trace preserving. The set ${\cal L}_n$ of all such quantum channels is known to be a compact…
For a quantum channel (completely positive, trace-preserving map), we prove a generalization to the infinite dimensional case of a result by Baumgartner and Narnhofer. This result is, in a probabilistic language, a decomposition of a…
We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge…
Coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model: (i) classical realizations which are the starting point for geometric quantization; (ii) induced…
We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional $C^*$-algebras) carry an action of a compact…
A new technique for proving fixed point theorems for families of holomorphic transformations of operator balls is developed. One of these theorems is used to show that a bounded representation in a real or complex Hilbert space is…
The dynamics of quantum systems are generally described by a family of quantum channels (linear, completely positive and trace preserving maps). In this note, we mainly study the range of all possible values of…
We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of…
A quantum holonomy reflects the curvature of some underlying structure of quantum mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace…
We introduce a class of linear maps irreducibly covariant with respect to the finite group generated by the Weyl operators. This group provides a direct generalization of the quaternion group. In particular, we analyze the irreducibly…
We present and investigate a new class of quantum channels, what we call `universal collective rotation channels', that includes the well-known class of collective rotation channels as a special case. The fixed point set and noise commutant…
To each discrete product system E of finite-dimensional Hilbert spaces we associate a C*-algebra O_E. When E is the n-dimensional product system over N, O_E is the Cuntz algebra O_n, and the irrational rotation algebras appear as O_E for…
The coadjoint orbits for the series $B_l,\ C_l$ and $D_l$ are considered in the case when the base point is a multiple of a fundamental weight. A quantization of the big cell is suggested by means of introducing a $\ast$-algebra generated…