Additive monotones for resource theories of parallel-combinable processes with discarding

A partitioned process theory, as defined by Coecke, Fritz, and Spekkens, is a symmetric monoidal category together with an all-object-including symmetric monoidal subcategory. We think of the morphisms of this category as processes, and the morphisms of the subcategory as those processes that are freely executable. Via a construction we refer to as parallel-combinable processes with discarding, we obtain from this data a partially ordered monoid on the set of processes, with f > g if one can use the free processes to construct g from f. The structure of this partial order can then be probed using additive monotones: order-preserving monoid homomorphisms with values in the real numbers under addition. We first characterise these additive monotones in terms of the corresponding partitioned process theory. Given enough monotones, we might hope to be able to reconstruct the order on the monoid. If so, we say that we have a complete family of monotones. In general, however, when we require our monotones to be additive monotones, such families do not exist or are hard to compute. We show the existence of complete families of additive monotones for various partitioned process theories based on the category of finite sets, in order to shed light on the way such families can be constructed.Comment: In Proceedings QPL 2015, arXiv:1511.0118

There are no monotone homomorphisms out of the convolution semigroup

We prove that there is no nonzero way of assigning real numbers to probability measures on R in a way which is monotone under first-order stochastic dominance and additive under convolution

The asymptotic spectrum of LOCC transformations

We study exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in success probability, known as the converese error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterisation of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [Hayashi et al, 2003], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case we prove that this bound holds with equality.Comment: v1: 21 pages v2: 21 pages, Minor corrections v3: 17 pages, Minor corrections, new reference added, parts of Section 5 and the Appendix removed, the omitted material can be found in an extended form in arXiv:1808.0515

There are no monotone homomorphisms out of the convolution semigroup

We prove that there is no nonzero way of assigning real numbers to probability measures on R in a way which is monotone under first-order stochastic dominance and additive under convolution

An Algebraic Theory for Data Linkage

There are countless sources of data available to governments, companies, and citizens, which can be combined for good or evil. We analyse the concepts of combining data from common sources and linking data from different sources. We model the data and its information content to be found in a single source by an ordered partial monoid, and the transfer of information between sources by different types of morphisms. To capture the linkage between a family of sources, we use a form of Grothendieck construction to create an ordered partial monoid that brings together the global data of the family in a single structure. We apply our approach to database theory and axiomatic structures in approximate reasoning. Thus, ordered partial monoids provide a foundation for the algebraic study for information gathering in its most primitive form

The asymptotic spectrum of graphs and the Shannon capacity

We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional clique cover number, the complement of the fractional orthogonal rank and the fractional Haemers bounds