The fidelity of recovery is multiplicative

© 1963-2012 IEEE. Fawzi and Renner recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e., the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here, we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandão et al. of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions

PRISM: Sparse Recovery of the Primordial Power Spectrum

The primordial power spectrum describes the initial perturbations in the Universe which eventually grew into the large-scale structure we observe today, and thereby provides an indirect probe of inflation or other structure-formation mechanisms. Here, we introduce a new method to estimate this spectrum from the empirical power spectrum of cosmic microwave background (CMB) maps. A sparsity-based linear inversion method, coined \textbf{PRISM}, is presented. This technique leverages a sparsity prior on features in the primordial power spectrum in a wavelet basis to regularise the inverse problem. This non-parametric approach does not assume a strong prior on the shape of the primordial power spectrum, yet is able to correctly reconstruct its global shape as well as localised features. These advantages make this method robust for detecting deviations from the currently favoured scale-invariant spectrum. We investigate the strength of this method on a set of WMAP 9-year simulated data for three types of primordial power spectra: a nearly scale-invariant spectrum, a spectrum with a small running of the spectral index, and a spectrum with a localised feature. This technique proves to easily detect deviations from a pure scale-invariant power spectrum and is suitable for distinguishing between simple models of the inflation. We process the WMAP 9-year data and find no significant departure from a nearly scale-invariant power spectrum with the spectral index $n_s = 0.972$. A high resolution primordial power spectrum can be reconstructed with this technique, where any strong local deviations or small global deviations from a pure scale-invariant spectrum can easily be detected

Spectral Properties of Tensor Products of Channels

We investigate spectral properties of the tensor products of two quantum channels defined on matrix algebras. This leads to the important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels $\mathcal{E}_1$ and $\mathcal{E}_2$ the multiplicative domain of $\mathcal{E}_1\otimes\mathcal{E}_2$ splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper $^*$-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the 'multiplicative index' of a unital channel which was recently introduced. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which can not be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum error correction.Comment: Proofs of Section 3 are simplified using a result of Ola Bratteli. Some references have been update

Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients

We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods

A fidelity measure for quantum channels

We propose a fidelity measure for quantum channels in a straightforward analogy to the corresponding mixed-state fidelity of Jozsa. We describe properties of this fidelity measure and discuss some applications of it to quantum information science.Comment: 14 pages; elsart.st