Hastings [12] recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. In this paper we present an expanded version of Hastings' proof. In addition to a careful elucidation of the details of the proof, we also present bounds for the minimal dimensions needed to obtain a counterexample.where the supremum runs over ensembles of input states, and where S(ρ) denotes the von Neumann entropy of the state ρ: S(ρ) = −Trρ log ρ (1.2)

Forests are the planet’s main CO2 filtering agent as well as important economical, environmental and social assets. Climate change is exerting an increased stress, resulting in a need for improved research methodologies to study their health, composition or evolution. Traditionally, information about forests has been collected using expensive and work-intensive field inventories, but in recent years unoccupied autonomous vehicles (UAVs) have become very popular as they represent a simple and inexpensive way to gather high resolution data of large forested areas. In addition to this trend, deep learning (DL) has also been gaining much attention in the field of forestry as a way to include the knowledge of forestry experts into automatic software pipelines tackling problems such as tree detection or tree health/species classification. Among the many sensors that UAVs can carry, RGB cameras are fast, cost-effective and allow for straightforward data interpretation. This has resulted in a large increase in the amount of UAV-acquired RGB data available for forest studies. In this review, we focus on studies that use DL and RGB images gathered by UAVs to solve practical forestry research problems. We summarize the existing studies, provide a detailed analysis of their strengths paired with a critical assessment on common methodological problems and include other information, such as available public data and code resources that we believe can be useful for researchers that want to start working in this area. We structure our discussion using three main families of forestry problems: (1) individual Tree Detection, (2) tree Species Classification, and (3) forest Anomaly Detection (forest fires and insect Infestation).

This paper contains several new results concerning covariant quantum channels in d ≥ 2 dimensions. The first part, Sec. 3, based on [4], is devoted to unitarily covariant channels, namely depolarizing and transpose-depolarizing channels. The second part, Sec. 4, based on [10], studies Weyl-covariant channels. These results are preceded by Sec. 2 in which we discuss various representations of general completely positive maps and channels. In the first part of the paper we compute complementary channels for depolarizing and transpose-depolarizing channels. This method easily yields minimal Kraus representations from non-minimal ones. We also study properties of the output purity of the tensor product of a channel and its complementary. In the second part, the formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weyl-covariant maps and channels. We then extend a result in [16] concerning a bound for the maximal output 2-norm of a Weyl-covariant channel. A class of maps which attain the bound is introduced, for which the multiplicativity of the maximal output 2-norm is proven. The complementary channels are described which have the same multiplicativity properties as the Weyl-covariant channels.

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter.

We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results.Comment: 5 page