QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Abstract

We show that the representations of the Cuntz C * -algebras On which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finitedimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work. There has been considerable recent interest in the analysis of completely positive maps on finite-dimensional space. There are a number of reasons for this including connections with wavelet analysis [3,4,15], dilation theory [11,16], representation theory of the Cuntz C * -algebras O n [3,4,10,11], and quantum information theory [1,17,21,22]. The results obtained in the current paper have implications for each of these areas. In presenting this work, another goal we have is to push further the connections between the various areas mentioned above.In § 1 we establish a result for completely positive maps. While we focus on the finitedimensional setting, this is not necessary in the proof. A structure theorem for the fixedpoint set of a unital quantum channel is contained in § 2. In particular, we prove that the fixed-point set is a C * -algebra which is equal to the commutant of the algebra generated by any choice of row contraction which determines the channel. We discuss the twodimensional channels [17,21], and use the theorem to classify them by their fixed-point sets. The representation theory for O n is considered in § 3. We focus on a subclass of representations arising in dilation theory and wavelet analysis [3, 4, 10, 11, 15]. Each of these representations determines a completely positive map on finite-dimensional space. We ask if these representations can be classified just by examining the map. An affirmative answer is provided by the result on completely positive maps from the first section. Finally, in § 4 we pose some open questions which are motivated by this work.

Asymptotic evolution of random unitary operations

“…In the case of random unitary operations the following powerful theorem can be proved which allows us to specify the space of attractors of the RUO Φ. In this context it should be also mentioned that for the more general case of arbitrary unital quantum operations interesting general results have been derived by Kribs [13,14] recently. …”

“…Consider now the density operator ρ( ) = Φ (ρ(0)) describing the physical system after iterations and denote its decomposition coefficients (14) into the same basis by β ( ) . It is clear that the coefficients β ( ) corresponding to eigenvectors of eigenvalues λ ∈ σ |1| evolve simply as …”

Theory of initialization-free decoherence-free subspaces and subsystems

“…Of course, for T to be a proper quantum state it must be Hermitian and have unit trace, whence c ≥ 0 and L is Hermitian. Subject to these constraints we see that the aforementioned theorem [53] gives a sufficient, but not necessary characterization of the allowed DF states. Indeed, the form (27) arises as a special case of our considerations, where we allow for T to be a state with support in H DFS ⊥ , but not of the most general form allowed by Eq.…”

“…Recently it has been shown that the fixed point set of unital CP maps is the commutant of the algebra generated by Kraus operators [53]. In other words, if E is the set of all polynomials in {E α }, or E = Alg{E α }, then…”

“…(52) leads to Eq. (53). In the case of perfect initialization, since it does not involve H out , Eq.…”