Extending Resource Monotones using Kan Extensions

In this paper we generalize the framework proposed by Gour and Tomamichel regarding extensions of monotones for resource theories. A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monontone extensions using Kan extensions, we introduce partitioned categories (pCat) as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into $([0,\infty], \leq)$, and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.Comment: Accepted at Applied Category Theory 2022, 19 page

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

Examination of the Constant Trace Value Condition of a SIC-POVM and Various Majorization Relations from Games of Chance

The constant-trace-value condition in the definition of a symmetric informationally complete positive operator-valued measure (SIC-POVM) is often overlooked, since it may give the impression that this condition is redundant. We show that this condition is necessary, otherwise the resulting mathematical object is no longer a SIC-POVM. This observation has led us to define a broader class of measurements which we call the semi-SIC POVMs. In dimension two we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open. Majorization relations have played an important role in quantum Shannon theory. We construct families of games of chance whose pay-off gives rise to three different types of majorization relations: the standard vector majorization, conditional majorization, and channel majorization. We show that our definition of conditional majorization and channel entropy is consistent with previous literature and find a channel entropy that reduces Shannon entropy and asymptotically continuous