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Register a new userThe outer spectral radius and dynamics of completely positive maps
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Authors
J. E. Pascoe
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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 of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescus theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.
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We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)otimes_{rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{sharp} = {(s,t) : stin E}$ and show that if $E^{sharp}$ is a set of spectral synthesis for $A(G)otimes_{rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{sharp}$ is a set of spectral synthesis for $A(G)otimes_{rm h} A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G)$-bimodule maps with the dual of $A(G)otimes_{rm h} A(G)$, we show that, in the case $G$ is weakly amenable, such a map leaves the multiplication algebra of $L^{infty}(G)$ invariant if and only if its support is contained in the antidiagonal of $G$.
For a tuple $A= (A_0, A_1, ldots , A_n)$ of elements in a unital Banach algebra $mathcal{B}$, its textit{projective (joint) spectrum} $p(A)$ is the collection of $zinmathbb{P}^{n}$ such that $A(z)=z_0A_0+z_1 A_1 + ldots z_n A_n$ is not invertible. If the tuple $A$ is associated with the generators of a finitely generated group, then $p(A)$ is simply called the projective spectrum of the group. This paper investigates a connection between self-similar group representations and an induced polynomial map on the projective space that preserves the projective spectrum of the group. The focus is on two groups: the infinite dihedral group $D_infty$ and the Grigorchuk group ${mathcal G}$ of intermediate growth. The main theorem shows that for $D_infty$ the Julia set of the induced rational map $F$ is equal to the union of the projective spectrum with the extended indeterminacy set. Moreover, the limit function of the iteration sequence ${F^{circ n}}$ on the Fatou set is determined explicitly. The result has an application to the group ${mathcal G}$ and gives rise to a conjecture about its associated Julia set.
We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.
D. Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between $C^*$-algebras by D. Kretschmann, D. Schlingemann and R. F. Werner. We present a Hilbert $C^*$-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.
Let $sigma(A)$, $rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$rho(AB)le r(A)r(B) quadtext{ for all bounded linear operators } B$$ if and only if there is a unique $mu in sigma (A)$ satisfying $|mu| = rho(A)$ and $A = frac{mu(I + L)}{2}$ for a contraction $L$ with $1insigma(L)$. One can get the same conclusion on $A$ if $rho(AB) le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x) ge 1$ for all unit vector $x$.
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