• Corpus ID: 118309233

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

  title={Quantum Channels, Wavelets, Dilations and Representations of \$O\_n\$},
  author={David W. Kribs},
  journal={arXiv: Operator Algebras},
  • D. Kribs
  • Published 23 September 2003
  • Mathematics
  • arXiv: Operator Algebras
We show that the representations of the Cuntz C$^\ast$-algebras $O_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, an application in quantum information theory is obtained; namely, a structure theorem for the fixed point set of a unital quantum channel. We also include some open problems motivated by this work. 

Quantum channels arising from abstract harmonic analysis

We present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over

Multiplicative properties of quantum channels

In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the

Quantum error correction and Young tableaux

A new class of quantum channels is presented that includes the class of collective rotation channels as a special case and classical representation theory of the symmetric group via Young tableaux is applied and a computationally amenable method for explicitly finding this structure is given.

Bures Contractive Channels on Operator Algebras

In a unital C*-algebra with a faithful trace functional $\tau$, the set $D_\tau(A)$ of positive $\rho\in A$ of trace \tau(\rho)=1 is an algebraic analogue of the space of density matrices (the set of

Universal collective rotation channels and quantum error correction

We present and investigate a new class of quantum channels, what we call “universal collective rotation channels,” that includes the class of collective rotation channels as a special case. The fixed

Noncommutative Poisson boundaries of unital quantum operations

In this paper, Poisson boundaries of unital quantum operations (also called Markov operators) are investigated. In the case of unital quantum channels, compact operators belonging to Poisson

Quantum operations fixing a convex cone of density operators on T(H)?>

The unital quantum operation acting on infinite dimensional quantum states fixing a convex cone of density operators is completely characterized. Based on this result, we classify the commutativity

Noiseless Subsystems for Collective Rotation Channels in Quantum Information Theory

Abstract.Collective rotation channels are a fundamental class of channels in quantum computing and quantum information theory. The commutant of the noise operators for such a channel is a C*-algebra

Closed subspaces which are attractors for representations of the Cuntz algebras

We analyze the structure of co-invariant subspaces for representations of the Cuntz algebras \({\mathcal{O}_N} \) for N = 2,3 ... , N < ∞ with special attention to the representations which are



Pure states on O_d

. We study representations of the Cuntz algebras O d and their associated decompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra

Wavelet filters and infinite-dimensional unitary groups

In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis

Wavelet representations and Fock space on positive matrices

On Some Additivity Problems in Quantum Information Theory

A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum


Abstract Orthogonal wavelets, or wavelet frames, for L2( R ) are associated with quadrature mirror filters (QMF), a set of complex numbers which relate the dyadic scaling of functions on R to the Z

Isometric dilations for infinite sequences of noncommuting operators

This paper develops a dilation theory for {T,}n=l an infinite sequence of noncommuting operators on a Hilbert space, when the matrix [T1, T2, ... ] is a contraction. A Wold decomposition for an

Additivity of the classical capacity of entanglement-breaking quantum channels

We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the

Compactly supported wavelets and representations of the Cuntz relations II

We show that compactly supported wavelets in L2 (R) of scale N may be effectively parameterized with a finite set of spin vectors in CN, and conversely that every set of spin vectors corresponds to a

General state changes in quantum theory